Toniolo and Linder, Equation (7)

Percentage Accurate: 34.2% → 85.4%

Time: 21.7s

Alternatives: 14

Speedup: 225.0×

Specification

Math

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

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)))))

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)));
}

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

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)));
}

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)))

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

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

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]

Initial Program: 34.2% accurate, 1.0× speedup

Math

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

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)))))

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)));
}

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

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)));
}

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)))

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

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

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]

Alternative 1: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ t_2 := \frac{t}{\sqrt{0.5 \cdot \left(t_1 + \left(t_1 + \left(2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{t \cdot -2}{\frac{x}{t}}\right)\right)\right)}}\\ t_3 := \sqrt{\frac{1 + x}{x + -1}}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{-t_3}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-246}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{t_3 \cdot \sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ l (/ x l)))
        (t_2
         (/
          t
          (sqrt
           (*
            0.5
            (+
             t_1
             (+
              t_1
              (- (* 2.0 (fma t t (* t (/ t x)))) (/ (* t -2.0) (/ x t)))))))))
        (t_3 (sqrt (/ (+ 1.0 x) (+ x -1.0)))))
   (if (<= t -3.5e+74)
     (/ 1.0 (- t_3))
     (if (<= t -3.1e-143)
       t_2
       (if (<= t -8.5e-246)
         (+ -1.0 (/ 1.0 x))
         (if (<= t 3.8e+50) t_2 (/ (sqrt 2.0) (* t_3 (sqrt 2.0)))))))))

C

double code(double x, double l, double t) {
	double t_1 = l / (x / l);
	double t_2 = t / sqrt((0.5 * (t_1 + (t_1 + ((2.0 * fma(t, t, (t * (t / x)))) - ((t * -2.0) / (x / t)))))));
	double t_3 = sqrt(((1.0 + x) / (x + -1.0)));
	double tmp;
	if (t <= -3.5e+74) {
		tmp = 1.0 / -t_3;
	} else if (t <= -3.1e-143) {
		tmp = t_2;
	} else if (t <= -8.5e-246) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 3.8e+50) {
		tmp = t_2;
	} else {
		tmp = sqrt(2.0) / (t_3 * sqrt(2.0));
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(l / Float64(x / l))
	t_2 = Float64(t / sqrt(Float64(0.5 * Float64(t_1 + Float64(t_1 + Float64(Float64(2.0 * fma(t, t, Float64(t * Float64(t / x)))) - Float64(Float64(t * -2.0) / Float64(x / t))))))))
	t_3 = sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t <= -3.5e+74)
		tmp = Float64(1.0 / Float64(-t_3));
	elseif (t <= -3.1e-143)
		tmp = t_2;
	elseif (t <= -8.5e-246)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 3.8e+50)
		tmp = t_2;
	else
		tmp = Float64(sqrt(2.0) / Float64(t_3 * sqrt(2.0)));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Sqrt[N[(0.5 * N[(t$95$1 + N[(t$95$1 + N[(N[(2.0 * N[(t * t + N[(t * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t * -2.0), $MachinePrecision] / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.5e+74], N[(1.0 / (-t$95$3)), $MachinePrecision], If[LessEqual[t, -3.1e-143], t$95$2, If[LessEqual[t, -8.5e-246], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e+50], t$95$2, N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.50000000000000014e74

   1. Initial program 24.7%

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

      1. associate-/l* 24.7%

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

      2. fma-neg 24.7%

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

      3. remove-double-neg 24.7%

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

      4. fma-neg 24.7%

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

      5. sub-neg 24.7%

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

      6. metadata-eval 24.7%

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

      7. remove-double-neg 24.7%

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

      8. fma-def 24.7%

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

   3. Simplified 24.7%

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

   4. Taylor expanded in t around -inf 95.1%

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

      1. mul-1-neg 95.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 95.1%

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

      3. +-commutative 95.1%

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

      4. sub-neg 95.1%

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

      5. metadata-eval 95.1%

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

      6. +-commutative 95.1%

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

   6. Simplified 95.1%

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

      1. div-inv 95.1%

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

      2. +-commutative 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. associate-*r/ 95.1%

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

      2. *-rgt-identity 95.1%

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

      3. associate-/r* 95.1%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 95.1%

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

   10. Simplified 95.1%

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

   if -3.50000000000000014e74 < t < -3.10000000000000007e-143 or -8.4999999999999998e-246 < t < 3.79999999999999987e50

   1. Initial program 43.7%

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

      1. associate-*l/ 43.7%

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

   3. Simplified 43.7%

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

   5. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 38.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 40.5%

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

      3. metadata-eval 40.5%

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

      4. sub-neg 40.5%

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

      5. associate-/l* 29.1%

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

      6. sub-neg 29.1%

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

      7. metadata-eval 29.1%

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

      8. +-commutative 29.1%

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

   7. Simplified 29.1%

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

   8. Taylor expanded in x around inf 73.1%

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

      1. distribute-lft-in 73.1%

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

      2. associate--l+ 73.1%

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

      3. unpow2 73.1%

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

      4. +-commutative 73.1%

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

      5. unpow2 73.1%

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

      6. unpow2 73.1%

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

      7. associate-*r/ 73.1%

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

   10. Simplified 73.1%

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

   11. Taylor expanded in t around 0 73.1%

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

       1. +-commutative 73.1%

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

       2. mul-1-neg 73.1%

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

       3. unsub-neg 73.1%

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

       4. *-commutative 73.1%

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

       5. unpow2 73.1%

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

       6. unpow2 73.1%

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

   13. Simplified 73.1%

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

       1. expm1-log1p-u 71.9%

          \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \left(\frac{t \cdot t}{x} \cdot -2 - \frac{\ell \cdot \ell}{x}\right)\right)}{2}}\right)\right)}} \]

       2. expm1-udef 34.4%

          \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \left(\frac{t \cdot t}{x} \cdot -2 - \frac{\ell \cdot \ell}{x}\right)\right)}{2}}\right)} - 1}} \]

   15. Applied egg-rr 42.6%

       \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right) \cdot 0.5}\right)} - 1}} \]
   16. Step-by-step derivation

       1. expm1-def 80.1%

          \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right) \cdot 0.5}\right)\right)}} \]

       2. expm1-log1p 82.1%

          \[\leadsto \frac{t}{\color{blue}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right) \cdot 0.5}}} \]

       3. *-commutative 82.1%

          \[\leadsto \frac{t}{\sqrt{\color{blue}{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right)}}} \]

       4. +-commutative 82.1%

          \[\leadsto \frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right)\right)}\right)}} \]

       5. associate-/r/ 82.1%

          \[\leadsto \frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right)\right)\right)}} \]

       6. associate-*l/ 82.1%

          \[\leadsto \frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \color{blue}{\frac{t \cdot -2}{\frac{x}{t}}}\right)\right)\right)}} \]

   17. Simplified 82.1%

       \[\leadsto \frac{t}{\color{blue}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{t \cdot -2}{\frac{x}{t}}\right)\right)\right)}}} \]

   if -3.10000000000000007e-143 < t < -8.4999999999999998e-246

   1. Initial program 7.6%

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

      1. associate-/l* 7.5%

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

      2. fma-neg 7.5%

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

      3. remove-double-neg 7.5%

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

      4. fma-neg 7.5%

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

      5. sub-neg 7.5%

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

      6. metadata-eval 7.5%

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

      7. remove-double-neg 7.5%

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

      8. fma-def 7.5%

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

   3. Simplified 7.5%

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

   4. Taylor expanded in t around -inf 69.7%

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

      1. mul-1-neg 69.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 69.7%

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

      3. +-commutative 69.7%

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

      4. sub-neg 69.7%

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

      5. metadata-eval 69.7%

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

      6. +-commutative 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in x around inf 69.9%

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

   if 3.79999999999999987e50 < t

   1. Initial program 28.3%

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

      1. associate-/l* 28.2%

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

      2. fma-neg 28.2%

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

      3. remove-double-neg 28.2%

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

      4. fma-neg 28.2%

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

      5. sub-neg 28.2%

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

      6. metadata-eval 28.2%

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

      7. remove-double-neg 28.2%

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

      8. fma-def 28.2%

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

   3. Simplified 28.2%

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

   4. Taylor expanded in l around 0 94.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 94.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 94.6%

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

      3. metadata-eval 94.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 94.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 94.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{t \cdot -2}{\frac{x}{t}}\right)\right)\right)}}\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-246}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{t \cdot -2}{\frac{x}{t}}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{1 + x}{x + -1}} \cdot \sqrt{2}}\\ \end{array} \]

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ t_2 := \frac{t}{\sqrt{0.5 \cdot \left(t_1 + \left(t_1 + \left(2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{t \cdot -2}{\frac{x}{t}}\right)\right)\right)}}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-246}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ l (/ x l)))
        (t_2
         (/
          t
          (sqrt
           (*
            0.5
            (+
             t_1
             (+
              t_1
              (- (* 2.0 (fma t t (* t (/ t x)))) (/ (* t -2.0) (/ x t))))))))))
   (if (<= t -3e+74)
     (/ 1.0 (- (sqrt (/ (+ 1.0 x) (+ x -1.0)))))
     (if (<= t -3.1e-143)
       t_2
       (if (<= t -8e-246)
         (+ -1.0 (/ 1.0 x))
         (if (<= t 4e+145)
           t_2
           (* t (/ (sqrt (/ (+ x -1.0) (+ 1.0 x))) t))))))))

C

double code(double x, double l, double t) {
	double t_1 = l / (x / l);
	double t_2 = t / sqrt((0.5 * (t_1 + (t_1 + ((2.0 * fma(t, t, (t * (t / x)))) - ((t * -2.0) / (x / t)))))));
	double tmp;
	if (t <= -3e+74) {
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.1e-143) {
		tmp = t_2;
	} else if (t <= -8e-246) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 4e+145) {
		tmp = t_2;
	} else {
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(l / Float64(x / l))
	t_2 = Float64(t / sqrt(Float64(0.5 * Float64(t_1 + Float64(t_1 + Float64(Float64(2.0 * fma(t, t, Float64(t * Float64(t / x)))) - Float64(Float64(t * -2.0) / Float64(x / t))))))))
	tmp = 0.0
	if (t <= -3e+74)
		tmp = Float64(1.0 / Float64(-sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))));
	elseif (t <= -3.1e-143)
		tmp = t_2;
	elseif (t <= -8e-246)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 4e+145)
		tmp = t_2;
	else
		tmp = Float64(t * Float64(sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x))) / t));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Sqrt[N[(0.5 * N[(t$95$1 + N[(t$95$1 + N[(N[(2.0 * N[(t * t + N[(t * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t * -2.0), $MachinePrecision] / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3e+74], N[(1.0 / (-N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -3.1e-143], t$95$2, If[LessEqual[t, -8e-246], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+145], t$95$2, N[(t * N[(N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3e74

   1. Initial program 24.7%

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

      1. associate-/l* 24.7%

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

      2. fma-neg 24.7%

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

      3. remove-double-neg 24.7%

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

      4. fma-neg 24.7%

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

      5. sub-neg 24.7%

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

      6. metadata-eval 24.7%

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

      7. remove-double-neg 24.7%

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

      8. fma-def 24.7%

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

   3. Simplified 24.7%

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

   4. Taylor expanded in t around -inf 95.1%

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

      1. mul-1-neg 95.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 95.1%

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

      3. +-commutative 95.1%

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

      4. sub-neg 95.1%

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

      5. metadata-eval 95.1%

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

      6. +-commutative 95.1%

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

   6. Simplified 95.1%

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

      1. div-inv 95.1%

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

      2. +-commutative 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. associate-*r/ 95.1%

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

      2. *-rgt-identity 95.1%

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

      3. associate-/r* 95.1%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 95.1%

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

   10. Simplified 95.1%

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

   if -3e74 < t < -3.10000000000000007e-143 or -7.99999999999999965e-246 < t < 4e145

   1. Initial program 49.5%

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

      1. associate-*l/ 49.6%

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

   3. Simplified 49.6%

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

   5. Applied egg-rr 34.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 34.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 35.5%

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

      3. metadata-eval 35.5%

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

      4. sub-neg 35.5%

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

      5. associate-/l* 37.4%

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

      6. sub-neg 37.4%

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

      7. metadata-eval 37.4%

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

      8. +-commutative 37.4%

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

   7. Simplified 37.4%

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

   8. Taylor expanded in x around inf 76.2%

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

      1. distribute-lft-in 76.2%

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

      2. associate--l+ 76.2%

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

      3. unpow2 76.2%

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

      4. +-commutative 76.2%

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

      5. unpow2 76.2%

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

      6. unpow2 76.2%

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

      7. associate-*r/ 76.2%

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

   10. Simplified 76.2%

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

   11. Taylor expanded in t around 0 76.2%

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

       1. +-commutative 76.2%

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

       2. mul-1-neg 76.2%

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

       3. unsub-neg 76.2%

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

       4. *-commutative 76.2%

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

       5. unpow2 76.2%

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

       6. unpow2 76.2%

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

   13. Simplified 76.2%

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

       1. expm1-log1p-u 73.8%

          \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \left(\frac{t \cdot t}{x} \cdot -2 - \frac{\ell \cdot \ell}{x}\right)\right)}{2}}\right)\right)}} \]

       2. expm1-udef 42.8%

          \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \left(\frac{t \cdot t}{x} \cdot -2 - \frac{\ell \cdot \ell}{x}\right)\right)}{2}}\right)} - 1}} \]

   15. Applied egg-rr 50.3%

       \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right) \cdot 0.5}\right)} - 1}} \]
   16. Step-by-step derivation

       1. expm1-def 81.3%

          \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right) \cdot 0.5}\right)\right)}} \]

       2. expm1-log1p 84.4%

          \[\leadsto \frac{t}{\color{blue}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right) \cdot 0.5}}} \]

       3. *-commutative 84.4%

          \[\leadsto \frac{t}{\sqrt{\color{blue}{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right)}}} \]

       4. +-commutative 84.4%

          \[\leadsto \frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right)\right)}\right)}} \]

       5. associate-/r/ 84.4%

          \[\leadsto \frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{t}{\frac{x}{t}} \cdot -2\right)\right)\right)}} \]

       6. associate-*l/ 84.4%

          \[\leadsto \frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \color{blue}{\frac{t \cdot -2}{\frac{x}{t}}}\right)\right)\right)}} \]

   17. Simplified 84.4%

       \[\leadsto \frac{t}{\color{blue}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{t \cdot -2}{\frac{x}{t}}\right)\right)\right)}}} \]

   if -3.10000000000000007e-143 < t < -7.99999999999999965e-246

   1. Initial program 7.6%

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

      1. associate-/l* 7.5%

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

      2. fma-neg 7.5%

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

      3. remove-double-neg 7.5%

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

      4. fma-neg 7.5%

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

      5. sub-neg 7.5%

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

      6. metadata-eval 7.5%

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

      7. remove-double-neg 7.5%

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

      8. fma-def 7.5%

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

   3. Simplified 7.5%

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

   4. Taylor expanded in t around -inf 69.7%

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

      1. mul-1-neg 69.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 69.7%

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

      3. +-commutative 69.7%

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

      4. sub-neg 69.7%

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

      5. metadata-eval 69.7%

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

      6. +-commutative 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in x around inf 69.9%

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

   if 4e145 < t

   1. Initial program 4.2%

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

      1. associate-*l/ 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in t around inf 3.1%

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

      1. associate-/l* 5.4%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 5.4%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 5.4%

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

      4. metadata-eval 5.4%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 5.4%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in t around 0 94.0%

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

      1. associate-*r/ 94.0%

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

      2. *-rgt-identity 94.0%

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

      3. sub-neg 94.0%

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

      4. metadata-eval 94.0%

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

      5. +-commutative 94.0%

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

   9. Simplified 94.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{t \cdot -2}{\frac{x}{t}}\right)\right)\right)}}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-246}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\frac{t}{\sqrt{0.5 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) - \frac{t \cdot -2}{\frac{x}{t}}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \]

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot t}{x}\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-301}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+61}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + t_1\right) + \left(\ell \cdot \frac{\ell}{x} - -2 \cdot t_1\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* t t) x)))
   (if (<= t -2.6e-197)
     (/ 1.0 (- (sqrt (/ (+ 1.0 x) (+ x -1.0)))))
     (if (<= t -3.5e-297)
       (* (/ t l) (sqrt x))
       (if (<= t -4e-301)
         (* t (/ (sqrt 2.0) (* (sqrt (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x)))) (- t))))
         (if (<= t 1.45e+61)
           (/
            t
            (sqrt
             (/
              (+
               (/ (* l l) x)
               (+ (* 2.0 (+ (* t t) t_1)) (- (* l (/ l x)) (* -2.0 t_1))))
              2.0)))
           (* t (/ (sqrt (/ (+ x -1.0) (+ 1.0 x))) t))))))))

C

double code(double x, double l, double t) {
	double t_1 = (t * t) / x;
	double tmp;
	if (t <= -2.6e-197) {
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= -4e-301) {
		tmp = t * (sqrt(2.0) / (sqrt(((2.0 / x) + (2.0 + (2.0 / x)))) * -t));
	} else if (t <= 1.45e+61) {
		tmp = t / sqrt(((((l * l) / x) + ((2.0 * ((t * t) + t_1)) + ((l * (l / x)) - (-2.0 * t_1)))) / 2.0));
	} else {
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * t) / x
    if (t <= (-2.6d-197)) then
        tmp = 1.0d0 / -sqrt(((1.0d0 + x) / (x + (-1.0d0))))
    else if (t <= (-3.5d-297)) then
        tmp = (t / l) * sqrt(x)
    else if (t <= (-4d-301)) then
        tmp = t * (sqrt(2.0d0) / (sqrt(((2.0d0 / x) + (2.0d0 + (2.0d0 / x)))) * -t))
    else if (t <= 1.45d+61) then
        tmp = t / sqrt(((((l * l) / x) + ((2.0d0 * ((t * t) + t_1)) + ((l * (l / x)) - ((-2.0d0) * t_1)))) / 2.0d0))
    else
        tmp = t * (sqrt(((x + (-1.0d0)) / (1.0d0 + x))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = (t * t) / x;
	double tmp;
	if (t <= -2.6e-197) {
		tmp = 1.0 / -Math.sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = (t / l) * Math.sqrt(x);
	} else if (t <= -4e-301) {
		tmp = t * (Math.sqrt(2.0) / (Math.sqrt(((2.0 / x) + (2.0 + (2.0 / x)))) * -t));
	} else if (t <= 1.45e+61) {
		tmp = t / Math.sqrt(((((l * l) / x) + ((2.0 * ((t * t) + t_1)) + ((l * (l / x)) - (-2.0 * t_1)))) / 2.0));
	} else {
		tmp = t * (Math.sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (t * t) / x
	tmp = 0
	if t <= -2.6e-197:
		tmp = 1.0 / -math.sqrt(((1.0 + x) / (x + -1.0)))
	elif t <= -3.5e-297:
		tmp = (t / l) * math.sqrt(x)
	elif t <= -4e-301:
		tmp = t * (math.sqrt(2.0) / (math.sqrt(((2.0 / x) + (2.0 + (2.0 / x)))) * -t))
	elif t <= 1.45e+61:
		tmp = t / math.sqrt(((((l * l) / x) + ((2.0 * ((t * t) + t_1)) + ((l * (l / x)) - (-2.0 * t_1)))) / 2.0))
	else:
		tmp = t * (math.sqrt(((x + -1.0) / (1.0 + x))) / t)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(t * t) / x)
	tmp = 0.0
	if (t <= -2.6e-197)
		tmp = Float64(1.0 / Float64(-sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))));
	elseif (t <= -3.5e-297)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= -4e-301)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x)))) * Float64(-t))));
	elseif (t <= 1.45e+61)
		tmp = Float64(t / sqrt(Float64(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + t_1)) + Float64(Float64(l * Float64(l / x)) - Float64(-2.0 * t_1)))) / 2.0)));
	else
		tmp = Float64(t * Float64(sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (t * t) / x;
	tmp = 0.0;
	if (t <= -2.6e-197)
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	elseif (t <= -3.5e-297)
		tmp = (t / l) * sqrt(x);
	elseif (t <= -4e-301)
		tmp = t * (sqrt(2.0) / (sqrt(((2.0 / x) + (2.0 + (2.0 / x)))) * -t));
	elseif (t <= 1.45e+61)
		tmp = t / sqrt(((((l * l) / x) + ((2.0 * ((t * t) + t_1)) + ((l * (l / x)) - (-2.0 * t_1)))) / 2.0));
	else
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -2.6e-197], N[(1.0 / (-N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -3.5e-297], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4e-301], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+61], N[(t / N[Sqrt[N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.6000000000000001e-197

   1. Initial program 35.0%

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

      1. associate-/l* 34.9%

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

      2. fma-neg 34.9%

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

      3. remove-double-neg 34.9%

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

      4. fma-neg 34.9%

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

      5. sub-neg 34.9%

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

      6. metadata-eval 34.9%

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

      7. remove-double-neg 34.9%

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

      8. fma-def 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around -inf 82.9%

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

      1. mul-1-neg 82.9%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. sub-neg 82.9%

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

      5. metadata-eval 82.9%

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

      6. +-commutative 82.9%

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

   6. Simplified 82.9%

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

      1. div-inv 82.9%

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

      2. +-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

      3. associate-/r* 82.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 82.9%

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

   10. Simplified 82.9%

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

   if -2.6000000000000001e-197 < t < -3.4999999999999999e-297

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 1.9%

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

      3. metadata-eval 1.9%

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

      4. sub-neg 1.9%

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

      5. associate-/l* 1.2%

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

      6. sub-neg 1.2%

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

      7. metadata-eval 1.2%

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

      8. +-commutative 1.2%

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

   7. Simplified 1.2%

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

   8. Taylor expanded in x around inf 48.8%

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

      1. distribute-lft-in 48.8%

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

      2. associate--l+ 48.8%

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

      3. unpow2 48.8%

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

      4. +-commutative 48.8%

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

      5. unpow2 48.8%

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

      6. unpow2 48.8%

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

      7. associate-*r/ 48.8%

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

   10. Simplified 48.8%

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

   11. Taylor expanded in t around 0 48.8%

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

       1. +-commutative 48.8%

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

       2. mul-1-neg 48.8%

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

       3. unsub-neg 48.8%

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

       4. *-commutative 48.8%

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

       5. unpow2 48.8%

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

       6. unpow2 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in t around 0 35.4%

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

   if -3.4999999999999999e-297 < t < -4.00000000000000027e-301

   1. Initial program 3.1%

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

      1. associate-*l/ 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around inf 3.1%

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

      1. associate--l+ 3.1%

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

      2. unpow2 3.1%

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

      3. distribute-lft-out 3.1%

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

      4. unpow2 3.1%

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

      5. unpow2 3.1%

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

      6. associate-*r/ 3.1%

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

      7. mul-1-neg 3.1%

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

      8. unpow2 3.1%

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

      9. +-commutative 3.1%

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

      10. unpow2 3.1%

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

      11. fma-udef 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in t around -inf 85.7%

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

      1. mul-1-neg 85.7%

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

      2. *-commutative 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

      4. distribute-lft-in 85.7%

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

      5. metadata-eval 85.7%

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

      6. associate-*r/ 85.7%

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

      7. metadata-eval 85.7%

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

      8. associate-*r/ 85.7%

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

      9. metadata-eval 85.7%

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

   9. Simplified 85.7%

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

   if -4.00000000000000027e-301 < t < 1.45e61

   1. Initial program 40.7%

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

      1. associate-*l/ 40.6%

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

   3. Simplified 40.6%

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

   5. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 41.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 41.2%

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

      3. metadata-eval 41.2%

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

      4. sub-neg 41.2%

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

      5. associate-/l* 19.6%

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

      6. sub-neg 19.6%

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

      7. metadata-eval 19.6%

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

      8. +-commutative 19.6%

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

   7. Simplified 19.6%

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

   8. Taylor expanded in x around inf 80.6%

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

      1. distribute-lft-in 80.6%

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

      2. associate--l+ 80.6%

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

      3. unpow2 80.6%

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

      4. +-commutative 80.6%

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

      5. unpow2 80.6%

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

      6. unpow2 80.6%

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

      7. associate-*r/ 80.6%

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

   10. Simplified 80.6%

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

   11. Taylor expanded in t around 0 80.6%

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

       1. +-commutative 80.6%

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

       2. mul-1-neg 80.6%

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

       3. unsub-neg 80.6%

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

       4. *-commutative 80.6%

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

       5. unpow2 80.6%

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

       6. unpow2 80.6%

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

   13. Simplified 80.6%

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

       1. *-un-lft-identity 80.6%

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

       2. associate-/l* 80.6%

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

   15. Applied egg-rr 80.6%

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

       1. *-lft-identity 80.6%

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

       2. associate-/r/ 80.6%

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

   17. Simplified 80.6%

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

   if 1.45e61 < t

   1. Initial program 27.5%

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

      1. associate-*l/ 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in t around inf 6.1%

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

      1. associate-/l* 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 33.0%

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

      4. metadata-eval 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 33.0%

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

   6. Simplified 33.0%

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

   7. Taylor expanded in t around 0 94.3%

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

      1. associate-*r/ 94.4%

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

      2. *-rgt-identity 94.4%

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

      3. sub-neg 94.4%

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

      4. metadata-eval 94.4%

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

      5. +-commutative 94.4%

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

   9. Simplified 94.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-301}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+61}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \frac{\ell}{x} - -2 \cdot \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \]

Alternative 4: 80.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot t}{x}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + t_1\right) + \left(\ell \cdot \frac{\ell}{x} - -2 \cdot t_1\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* t t) x)))
   (if (<= t -5e-198)
     (/ 1.0 (- (sqrt (/ (+ 1.0 x) (+ x -1.0)))))
     (if (<= t -3.5e-297)
       (* (/ t l) (sqrt x))
       (if (<= t -5.2e-299)
         -1.0
         (if (<= t 4e+60)
           (/
            t
            (sqrt
             (/
              (+
               (/ (* l l) x)
               (+ (* 2.0 (+ (* t t) t_1)) (- (* l (/ l x)) (* -2.0 t_1))))
              2.0)))
           (* t (/ (sqrt (/ (+ x -1.0) (+ 1.0 x))) t))))))))

C

double code(double x, double l, double t) {
	double t_1 = (t * t) / x;
	double tmp;
	if (t <= -5e-198) {
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= -5.2e-299) {
		tmp = -1.0;
	} else if (t <= 4e+60) {
		tmp = t / sqrt(((((l * l) / x) + ((2.0 * ((t * t) + t_1)) + ((l * (l / x)) - (-2.0 * t_1)))) / 2.0));
	} else {
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * t) / x
    if (t <= (-5d-198)) then
        tmp = 1.0d0 / -sqrt(((1.0d0 + x) / (x + (-1.0d0))))
    else if (t <= (-3.5d-297)) then
        tmp = (t / l) * sqrt(x)
    else if (t <= (-5.2d-299)) then
        tmp = -1.0d0
    else if (t <= 4d+60) then
        tmp = t / sqrt(((((l * l) / x) + ((2.0d0 * ((t * t) + t_1)) + ((l * (l / x)) - ((-2.0d0) * t_1)))) / 2.0d0))
    else
        tmp = t * (sqrt(((x + (-1.0d0)) / (1.0d0 + x))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = (t * t) / x;
	double tmp;
	if (t <= -5e-198) {
		tmp = 1.0 / -Math.sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = (t / l) * Math.sqrt(x);
	} else if (t <= -5.2e-299) {
		tmp = -1.0;
	} else if (t <= 4e+60) {
		tmp = t / Math.sqrt(((((l * l) / x) + ((2.0 * ((t * t) + t_1)) + ((l * (l / x)) - (-2.0 * t_1)))) / 2.0));
	} else {
		tmp = t * (Math.sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (t * t) / x
	tmp = 0
	if t <= -5e-198:
		tmp = 1.0 / -math.sqrt(((1.0 + x) / (x + -1.0)))
	elif t <= -3.5e-297:
		tmp = (t / l) * math.sqrt(x)
	elif t <= -5.2e-299:
		tmp = -1.0
	elif t <= 4e+60:
		tmp = t / math.sqrt(((((l * l) / x) + ((2.0 * ((t * t) + t_1)) + ((l * (l / x)) - (-2.0 * t_1)))) / 2.0))
	else:
		tmp = t * (math.sqrt(((x + -1.0) / (1.0 + x))) / t)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(t * t) / x)
	tmp = 0.0
	if (t <= -5e-198)
		tmp = Float64(1.0 / Float64(-sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))));
	elseif (t <= -3.5e-297)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= -5.2e-299)
		tmp = -1.0;
	elseif (t <= 4e+60)
		tmp = Float64(t / sqrt(Float64(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + t_1)) + Float64(Float64(l * Float64(l / x)) - Float64(-2.0 * t_1)))) / 2.0)));
	else
		tmp = Float64(t * Float64(sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (t * t) / x;
	tmp = 0.0;
	if (t <= -5e-198)
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	elseif (t <= -3.5e-297)
		tmp = (t / l) * sqrt(x);
	elseif (t <= -5.2e-299)
		tmp = -1.0;
	elseif (t <= 4e+60)
		tmp = t / sqrt(((((l * l) / x) + ((2.0 * ((t * t) + t_1)) + ((l * (l / x)) - (-2.0 * t_1)))) / 2.0));
	else
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -5e-198], N[(1.0 / (-N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -3.5e-297], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.2e-299], -1.0, If[LessEqual[t, 4e+60], N[(t / N[Sqrt[N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4.9999999999999999e-198

   1. Initial program 35.0%

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

      1. associate-/l* 34.9%

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

      2. fma-neg 34.9%

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

      3. remove-double-neg 34.9%

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

      4. fma-neg 34.9%

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

      5. sub-neg 34.9%

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

      6. metadata-eval 34.9%

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

      7. remove-double-neg 34.9%

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

      8. fma-def 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around -inf 82.9%

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

      1. mul-1-neg 82.9%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. sub-neg 82.9%

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

      5. metadata-eval 82.9%

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

      6. +-commutative 82.9%

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

   6. Simplified 82.9%

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

      1. div-inv 82.9%

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

      2. +-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

      3. associate-/r* 82.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 82.9%

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

   10. Simplified 82.9%

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

   if -4.9999999999999999e-198 < t < -3.4999999999999999e-297

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 1.9%

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

      3. metadata-eval 1.9%

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

      4. sub-neg 1.9%

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

      5. associate-/l* 1.2%

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

      6. sub-neg 1.2%

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

      7. metadata-eval 1.2%

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

      8. +-commutative 1.2%

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

   7. Simplified 1.2%

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

   8. Taylor expanded in x around inf 48.8%

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

      1. distribute-lft-in 48.8%

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

      2. associate--l+ 48.8%

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

      3. unpow2 48.8%

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

      4. +-commutative 48.8%

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

      5. unpow2 48.8%

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

      6. unpow2 48.8%

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

      7. associate-*r/ 48.8%

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

   10. Simplified 48.8%

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

   11. Taylor expanded in t around 0 48.8%

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

       1. +-commutative 48.8%

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

       2. mul-1-neg 48.8%

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

       3. unsub-neg 48.8%

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

       4. *-commutative 48.8%

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

       5. unpow2 48.8%

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

       6. unpow2 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in t around 0 35.4%

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

   if -3.4999999999999999e-297 < t < -5.1999999999999998e-299

   1. Initial program 3.1%

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

      1. associate-/l* 3.1%

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

      2. fma-neg 3.1%

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

      3. remove-double-neg 3.1%

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

      4. fma-neg 3.1%

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

      5. sub-neg 3.1%

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

      6. metadata-eval 3.1%

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

      7. remove-double-neg 3.1%

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

      8. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in t around -inf 85.7%

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

      1. mul-1-neg 85.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 85.7%

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

      3. +-commutative 85.7%

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

      4. sub-neg 85.7%

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

      5. metadata-eval 85.7%

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

      6. +-commutative 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in x around inf 85.7%

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

   if -5.1999999999999998e-299 < t < 3.9999999999999998e60

   1. Initial program 40.7%

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

      1. associate-*l/ 40.6%

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

   3. Simplified 40.6%

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

   5. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 41.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 41.2%

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

      3. metadata-eval 41.2%

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

      4. sub-neg 41.2%

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

      5. associate-/l* 19.6%

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

      6. sub-neg 19.6%

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

      7. metadata-eval 19.6%

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

      8. +-commutative 19.6%

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

   7. Simplified 19.6%

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

   8. Taylor expanded in x around inf 80.6%

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

      1. distribute-lft-in 80.6%

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

      2. associate--l+ 80.6%

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

      3. unpow2 80.6%

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

      4. +-commutative 80.6%

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

      5. unpow2 80.6%

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

      6. unpow2 80.6%

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

      7. associate-*r/ 80.6%

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

   10. Simplified 80.6%

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

   11. Taylor expanded in t around 0 80.6%

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

       1. +-commutative 80.6%

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

       2. mul-1-neg 80.6%

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

       3. unsub-neg 80.6%

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

       4. *-commutative 80.6%

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

       5. unpow2 80.6%

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

       6. unpow2 80.6%

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

   13. Simplified 80.6%

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

       1. *-un-lft-identity 80.6%

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

       2. associate-/l* 80.6%

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

   15. Applied egg-rr 80.6%

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

       1. *-lft-identity 80.6%

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

       2. associate-/r/ 80.6%

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

   17. Simplified 80.6%

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

   if 3.9999999999999998e60 < t

   1. Initial program 27.5%

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

      1. associate-*l/ 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in t around inf 6.1%

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

      1. associate-/l* 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 33.0%

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

      4. metadata-eval 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 33.0%

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

   6. Simplified 33.0%

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

   7. Taylor expanded in t around 0 94.3%

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

      1. associate-*r/ 94.4%

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

      2. *-rgt-identity 94.4%

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

      3. sub-neg 94.4%

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

      4. metadata-eval 94.4%

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

      5. +-commutative 94.4%

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

   9. Simplified 94.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(\ell \cdot \frac{\ell}{x} - -2 \cdot \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \]

Alternative 5: 79.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x)))
   (if (<= t -3.5e-198)
     (/ 1.0 (- (sqrt (/ (+ 1.0 x) (+ x -1.0)))))
     (if (<= t -3.5e-297)
       (* (/ t l) (sqrt x))
       (if (<= t -3e-299)
         -1.0
         (if (<= t 9.5e+58)
           (/
            t
            (sqrt (/ (+ t_1 (+ t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))) 2.0)))
           (* t (/ (sqrt (/ (+ x -1.0) (+ 1.0 x))) t))))))))

C

double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double tmp;
	if (t <= -3.5e-198) {
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= -3e-299) {
		tmp = -1.0;
	} else if (t <= 9.5e+58) {
		tmp = t / sqrt(((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x))))) / 2.0));
	} else {
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * l) / x
    if (t <= (-3.5d-198)) then
        tmp = 1.0d0 / -sqrt(((1.0d0 + x) / (x + (-1.0d0))))
    else if (t <= (-3.5d-297)) then
        tmp = (t / l) * sqrt(x)
    else if (t <= (-3d-299)) then
        tmp = -1.0d0
    else if (t <= 9.5d+58) then
        tmp = t / sqrt(((t_1 + (t_1 + (2.0d0 * ((t * t) + ((t * t) / x))))) / 2.0d0))
    else
        tmp = t * (sqrt(((x + (-1.0d0)) / (1.0d0 + x))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double tmp;
	if (t <= -3.5e-198) {
		tmp = 1.0 / -Math.sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = (t / l) * Math.sqrt(x);
	} else if (t <= -3e-299) {
		tmp = -1.0;
	} else if (t <= 9.5e+58) {
		tmp = t / Math.sqrt(((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x))))) / 2.0));
	} else {
		tmp = t * (Math.sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (l * l) / x
	tmp = 0
	if t <= -3.5e-198:
		tmp = 1.0 / -math.sqrt(((1.0 + x) / (x + -1.0)))
	elif t <= -3.5e-297:
		tmp = (t / l) * math.sqrt(x)
	elif t <= -3e-299:
		tmp = -1.0
	elif t <= 9.5e+58:
		tmp = t / math.sqrt(((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x))))) / 2.0))
	else:
		tmp = t * (math.sqrt(((x + -1.0) / (1.0 + x))) / t)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -3.5e-198)
		tmp = Float64(1.0 / Float64(-sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))));
	elseif (t <= -3.5e-297)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= -3e-299)
		tmp = -1.0;
	elseif (t <= 9.5e+58)
		tmp = Float64(t / sqrt(Float64(Float64(t_1 + Float64(t_1 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))) / 2.0)));
	else
		tmp = Float64(t * Float64(sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (l * l) / x;
	tmp = 0.0;
	if (t <= -3.5e-198)
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	elseif (t <= -3.5e-297)
		tmp = (t / l) * sqrt(x);
	elseif (t <= -3e-299)
		tmp = -1.0;
	elseif (t <= 9.5e+58)
		tmp = t / sqrt(((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x))))) / 2.0));
	else
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -3.5e-198], N[(1.0 / (-N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -3.5e-297], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3e-299], -1.0, If[LessEqual[t, 9.5e+58], N[(t / N[Sqrt[N[(N[(t$95$1 + N[(t$95$1 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -3.50000000000000025e-198

   1. Initial program 35.0%

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

      1. associate-/l* 34.9%

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

      2. fma-neg 34.9%

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

      3. remove-double-neg 34.9%

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

      4. fma-neg 34.9%

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

      5. sub-neg 34.9%

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

      6. metadata-eval 34.9%

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

      7. remove-double-neg 34.9%

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

      8. fma-def 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around -inf 82.9%

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

      1. mul-1-neg 82.9%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. sub-neg 82.9%

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

      5. metadata-eval 82.9%

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

      6. +-commutative 82.9%

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

   6. Simplified 82.9%

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

      1. div-inv 82.9%

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

      2. +-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

      3. associate-/r* 82.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 82.9%

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

   10. Simplified 82.9%

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

   if -3.50000000000000025e-198 < t < -3.4999999999999999e-297

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 1.9%

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

      3. metadata-eval 1.9%

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

      4. sub-neg 1.9%

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

      5. associate-/l* 1.2%

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

      6. sub-neg 1.2%

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

      7. metadata-eval 1.2%

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

      8. +-commutative 1.2%

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

   7. Simplified 1.2%

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

   8. Taylor expanded in x around inf 48.8%

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

      1. distribute-lft-in 48.8%

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

      2. associate--l+ 48.8%

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

      3. unpow2 48.8%

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

      4. +-commutative 48.8%

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

      5. unpow2 48.8%

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

      6. unpow2 48.8%

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

      7. associate-*r/ 48.8%

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

   10. Simplified 48.8%

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

   11. Taylor expanded in t around 0 48.8%

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

       1. +-commutative 48.8%

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

       2. mul-1-neg 48.8%

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

       3. unsub-neg 48.8%

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

       4. *-commutative 48.8%

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

       5. unpow2 48.8%

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

       6. unpow2 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in t around 0 35.4%

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

   if -3.4999999999999999e-297 < t < -2.99999999999999984e-299

   1. Initial program 3.1%

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

      1. associate-/l* 3.1%

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

      2. fma-neg 3.1%

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

      3. remove-double-neg 3.1%

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

      4. fma-neg 3.1%

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

      5. sub-neg 3.1%

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

      6. metadata-eval 3.1%

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

      7. remove-double-neg 3.1%

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

      8. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in t around -inf 85.7%

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

      1. mul-1-neg 85.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 85.7%

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

      3. +-commutative 85.7%

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

      4. sub-neg 85.7%

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

      5. metadata-eval 85.7%

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

      6. +-commutative 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in x around inf 85.7%

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

   if -2.99999999999999984e-299 < t < 9.5000000000000002e58

   1. Initial program 40.7%

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

      1. associate-*l/ 40.6%

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

   3. Simplified 40.6%

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

   5. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 41.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 41.2%

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

      3. metadata-eval 41.2%

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

      4. sub-neg 41.2%

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

      5. associate-/l* 19.6%

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

      6. sub-neg 19.6%

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

      7. metadata-eval 19.6%

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

      8. +-commutative 19.6%

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

   7. Simplified 19.6%

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

   8. Taylor expanded in x around inf 80.6%

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

      1. distribute-lft-in 80.6%

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

      2. associate--l+ 80.6%

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

      3. unpow2 80.6%

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

      4. +-commutative 80.6%

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

      5. unpow2 80.6%

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

      6. unpow2 80.6%

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

      7. associate-*r/ 80.6%

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

   10. Simplified 80.6%

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

   11. Taylor expanded in t around 0 79.9%

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

       1. associate-*r/ 79.9%

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

       2. neg-mul-1 79.9%

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

       3. unpow2 79.9%

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

       4. distribute-rgt-neg-in 79.9%

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

   13. Simplified 79.9%

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

   if 9.5000000000000002e58 < t

   1. Initial program 27.5%

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

      1. associate-*l/ 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in t around inf 6.1%

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

      1. associate-/l* 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 33.0%

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

      4. metadata-eval 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 33.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 33.0%

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

   6. Simplified 33.0%

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

   7. Taylor expanded in t around 0 94.3%

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

      1. associate-*r/ 94.4%

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

      2. *-rgt-identity 94.4%

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

      3. sub-neg 94.4%

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

      4. metadata-eval 94.4%

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

      5. +-commutative 94.4%

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

   9. Simplified 94.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \]

Alternative 6: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-279}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right)\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (/ t l) (sqrt x))))
   (if (<= t -8.2e-199)
     (/ 1.0 (- (sqrt (/ (+ 1.0 x) (+ x -1.0)))))
     (if (<= t -3.5e-297)
       t_1
       (if (<= t 5e-309)
         -1.0
         (if (<= t 1.4e-279)
           (* t (- (/ 1.0 t) (/ (/ 1.0 t) x)))
           (if (<= t 7.8e-152)
             t_1
             (* t (/ (sqrt (/ (+ x -1.0) (+ 1.0 x))) t)))))))))

C

double code(double x, double l, double t) {
	double t_1 = (t / l) * sqrt(x);
	double tmp;
	if (t <= -8.2e-199) {
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = t_1;
	} else if (t <= 5e-309) {
		tmp = -1.0;
	} else if (t <= 1.4e-279) {
		tmp = t * ((1.0 / t) - ((1.0 / t) / x));
	} else if (t <= 7.8e-152) {
		tmp = t_1;
	} else {
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / l) * sqrt(x)
    if (t <= (-8.2d-199)) then
        tmp = 1.0d0 / -sqrt(((1.0d0 + x) / (x + (-1.0d0))))
    else if (t <= (-3.5d-297)) then
        tmp = t_1
    else if (t <= 5d-309) then
        tmp = -1.0d0
    else if (t <= 1.4d-279) then
        tmp = t * ((1.0d0 / t) - ((1.0d0 / t) / x))
    else if (t <= 7.8d-152) then
        tmp = t_1
    else
        tmp = t * (sqrt(((x + (-1.0d0)) / (1.0d0 + x))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = (t / l) * Math.sqrt(x);
	double tmp;
	if (t <= -8.2e-199) {
		tmp = 1.0 / -Math.sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = t_1;
	} else if (t <= 5e-309) {
		tmp = -1.0;
	} else if (t <= 1.4e-279) {
		tmp = t * ((1.0 / t) - ((1.0 / t) / x));
	} else if (t <= 7.8e-152) {
		tmp = t_1;
	} else {
		tmp = t * (Math.sqrt(((x + -1.0) / (1.0 + x))) / t);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (t / l) * math.sqrt(x)
	tmp = 0
	if t <= -8.2e-199:
		tmp = 1.0 / -math.sqrt(((1.0 + x) / (x + -1.0)))
	elif t <= -3.5e-297:
		tmp = t_1
	elif t <= 5e-309:
		tmp = -1.0
	elif t <= 1.4e-279:
		tmp = t * ((1.0 / t) - ((1.0 / t) / x))
	elif t <= 7.8e-152:
		tmp = t_1
	else:
		tmp = t * (math.sqrt(((x + -1.0) / (1.0 + x))) / t)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(t / l) * sqrt(x))
	tmp = 0.0
	if (t <= -8.2e-199)
		tmp = Float64(1.0 / Float64(-sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))));
	elseif (t <= -3.5e-297)
		tmp = t_1;
	elseif (t <= 5e-309)
		tmp = -1.0;
	elseif (t <= 1.4e-279)
		tmp = Float64(t * Float64(Float64(1.0 / t) - Float64(Float64(1.0 / t) / x)));
	elseif (t <= 7.8e-152)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (t / l) * sqrt(x);
	tmp = 0.0;
	if (t <= -8.2e-199)
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	elseif (t <= -3.5e-297)
		tmp = t_1;
	elseif (t <= 5e-309)
		tmp = -1.0;
	elseif (t <= 1.4e-279)
		tmp = t * ((1.0 / t) - ((1.0 / t) / x));
	elseif (t <= 7.8e-152)
		tmp = t_1;
	else
		tmp = t * (sqrt(((x + -1.0) / (1.0 + x))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.2e-199], N[(1.0 / (-N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -3.5e-297], t$95$1, If[LessEqual[t, 5e-309], -1.0, If[LessEqual[t, 1.4e-279], N[(t * N[(N[(1.0 / t), $MachinePrecision] - N[(N[(1.0 / t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.8e-152], t$95$1, N[(t * N[(N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -8.20000000000000043e-199

   1. Initial program 35.0%

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

      1. associate-/l* 34.9%

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

      2. fma-neg 34.9%

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

      3. remove-double-neg 34.9%

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

      4. fma-neg 34.9%

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

      5. sub-neg 34.9%

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

      6. metadata-eval 34.9%

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

      7. remove-double-neg 34.9%

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

      8. fma-def 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around -inf 82.9%

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

      1. mul-1-neg 82.9%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. sub-neg 82.9%

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

      5. metadata-eval 82.9%

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

      6. +-commutative 82.9%

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

   6. Simplified 82.9%

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

      1. div-inv 82.9%

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

      2. +-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

      3. associate-/r* 82.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 82.9%

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

   10. Simplified 82.9%

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

   if -8.20000000000000043e-199 < t < -3.4999999999999999e-297 or 1.4e-279 < t < 7.8000000000000008e-152

   1. Initial program 5.3%

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

      1. associate-*l/ 5.3%

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

   3. Simplified 5.3%

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

   5. Applied egg-rr 4.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 4.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 5.8%

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

      3. metadata-eval 5.8%

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

      4. sub-neg 5.8%

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

      5. associate-/l* 1.3%

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

      6. sub-neg 1.3%

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

      7. metadata-eval 1.3%

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

      8. +-commutative 1.3%

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

   7. Simplified 1.3%

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

   8. Taylor expanded in x around inf 63.4%

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

      1. distribute-lft-in 63.4%

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

      2. associate--l+ 63.4%

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

      3. unpow2 63.4%

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

      4. +-commutative 63.4%

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

      5. unpow2 63.4%

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

      6. unpow2 63.4%

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

      7. associate-*r/ 63.4%

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

   10. Simplified 63.4%

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

   11. Taylor expanded in t around 0 63.4%

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

       1. +-commutative 63.4%

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

       2. mul-1-neg 63.4%

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

       3. unsub-neg 63.4%

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

       4. *-commutative 63.4%

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

       5. unpow2 63.4%

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

       6. unpow2 63.4%

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

   13. Simplified 63.4%

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

   14. Taylor expanded in t around 0 31.9%

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

   if -3.4999999999999999e-297 < t < 4.9999999999999995e-309

   1. Initial program 3.1%

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

      1. associate-/l* 3.1%

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

      2. fma-neg 3.1%

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

      3. remove-double-neg 3.1%

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

      4. fma-neg 3.1%

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

      5. sub-neg 3.1%

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

      6. metadata-eval 3.1%

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

      7. remove-double-neg 3.1%

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

      8. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in t around -inf 85.7%

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

      1. mul-1-neg 85.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 85.7%

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

      3. +-commutative 85.7%

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

      4. sub-neg 85.7%

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

      5. metadata-eval 85.7%

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

      6. +-commutative 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in x around inf 85.7%

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

   if 4.9999999999999995e-309 < t < 1.4e-279

   1. Initial program 2.5%

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

      1. associate-*l/ 2.5%

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

   3. Simplified 2.5%

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

   4. Taylor expanded in t around inf 2.8%

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

      1. associate-/l* 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 2.8%

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

      4. metadata-eval 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in x around inf 80.6%

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

      1. associate-/r* 80.6%

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

   9. Simplified 80.6%

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

   if 7.8000000000000008e-152 < t

   1. Initial program 37.0%

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

      1. associate-*l/ 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in t around inf 30.4%

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

      1. associate-/l* 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 34.0%

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

      4. metadata-eval 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 34.0%

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

   6. Simplified 34.0%

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

   7. Taylor expanded in t around 0 84.7%

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

      1. associate-*r/ 84.7%

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

      2. *-rgt-identity 84.7%

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

      3. sub-neg 84.7%

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

      4. metadata-eval 84.7%

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

      5. +-commutative 84.7%

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

   9. Simplified 84.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-279}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right)\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{1 + x}}}{t}\\ \end{array} \]

Alternative 7: 75.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\ell} \cdot \sqrt{x}\\ t_2 := \frac{1}{t} - \frac{\frac{1}{t}}{x}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-279}:\\ \;\;\;\;t \cdot t_2\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t_2 + \frac{0.5}{t \cdot \left(x \cdot x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (/ t l) (sqrt x))) (t_2 (- (/ 1.0 t) (/ (/ 1.0 t) x))))
   (if (<= t -7.5e-197)
     (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
     (if (<= t -3.5e-297)
       t_1
       (if (<= t 5e-309)
         -1.0
         (if (<= t 1.9e-279)
           (* t t_2)
           (if (<= t 1.05e-151) t_1 (* t (+ t_2 (/ 0.5 (* t (* x x))))))))))))

C

double code(double x, double l, double t) {
	double t_1 = (t / l) * sqrt(x);
	double t_2 = (1.0 / t) - ((1.0 / t) / x);
	double tmp;
	if (t <= -7.5e-197) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= -3.5e-297) {
		tmp = t_1;
	} else if (t <= 5e-309) {
		tmp = -1.0;
	} else if (t <= 1.9e-279) {
		tmp = t * t_2;
	} else if (t <= 1.05e-151) {
		tmp = t_1;
	} else {
		tmp = t * (t_2 + (0.5 / (t * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t / l) * sqrt(x)
    t_2 = (1.0d0 / t) - ((1.0d0 / t) / x)
    if (t <= (-7.5d-197)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if (t <= (-3.5d-297)) then
        tmp = t_1
    else if (t <= 5d-309) then
        tmp = -1.0d0
    else if (t <= 1.9d-279) then
        tmp = t * t_2
    else if (t <= 1.05d-151) then
        tmp = t_1
    else
        tmp = t * (t_2 + (0.5d0 / (t * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = (t / l) * Math.sqrt(x);
	double t_2 = (1.0 / t) - ((1.0 / t) / x);
	double tmp;
	if (t <= -7.5e-197) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= -3.5e-297) {
		tmp = t_1;
	} else if (t <= 5e-309) {
		tmp = -1.0;
	} else if (t <= 1.9e-279) {
		tmp = t * t_2;
	} else if (t <= 1.05e-151) {
		tmp = t_1;
	} else {
		tmp = t * (t_2 + (0.5 / (t * (x * x))));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (t / l) * math.sqrt(x)
	t_2 = (1.0 / t) - ((1.0 / t) / x)
	tmp = 0
	if t <= -7.5e-197:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif t <= -3.5e-297:
		tmp = t_1
	elif t <= 5e-309:
		tmp = -1.0
	elif t <= 1.9e-279:
		tmp = t * t_2
	elif t <= 1.05e-151:
		tmp = t_1
	else:
		tmp = t * (t_2 + (0.5 / (t * (x * x))))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(t / l) * sqrt(x))
	t_2 = Float64(Float64(1.0 / t) - Float64(Float64(1.0 / t) / x))
	tmp = 0.0
	if (t <= -7.5e-197)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif (t <= -3.5e-297)
		tmp = t_1;
	elseif (t <= 5e-309)
		tmp = -1.0;
	elseif (t <= 1.9e-279)
		tmp = Float64(t * t_2);
	elseif (t <= 1.05e-151)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(t_2 + Float64(0.5 / Float64(t * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (t / l) * sqrt(x);
	t_2 = (1.0 / t) - ((1.0 / t) / x);
	tmp = 0.0;
	if (t <= -7.5e-197)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif (t <= -3.5e-297)
		tmp = t_1;
	elseif (t <= 5e-309)
		tmp = -1.0;
	elseif (t <= 1.9e-279)
		tmp = t * t_2;
	elseif (t <= 1.05e-151)
		tmp = t_1;
	else
		tmp = t * (t_2 + (0.5 / (t * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / t), $MachinePrecision] - N[(N[(1.0 / t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.5e-197], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.5e-297], t$95$1, If[LessEqual[t, 5e-309], -1.0, If[LessEqual[t, 1.9e-279], N[(t * t$95$2), $MachinePrecision], If[LessEqual[t, 1.05e-151], t$95$1, N[(t * N[(t$95$2 + N[(0.5 / N[(t * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -7.5e-197

   1. Initial program 35.0%

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

      1. associate-/l* 34.9%

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

      2. fma-neg 34.9%

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

      3. remove-double-neg 34.9%

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

      4. fma-neg 34.9%

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

      5. sub-neg 34.9%

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

      6. metadata-eval 34.9%

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

      7. remove-double-neg 34.9%

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

      8. fma-def 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around -inf 82.9%

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

      1. mul-1-neg 82.9%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. sub-neg 82.9%

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

      5. metadata-eval 82.9%

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

      6. +-commutative 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in x around inf 82.8%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 82.8%

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

      2. metadata-eval 82.8%

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

      3. unpow2 82.8%

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

   9. Simplified 82.8%

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

   if -7.5e-197 < t < -3.4999999999999999e-297 or 1.90000000000000016e-279 < t < 1.04999999999999995e-151

   1. Initial program 5.3%

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

      1. associate-*l/ 5.3%

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

   3. Simplified 5.3%

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

   5. Applied egg-rr 4.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 4.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 5.8%

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

      3. metadata-eval 5.8%

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

      4. sub-neg 5.8%

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

      5. associate-/l* 1.3%

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

      6. sub-neg 1.3%

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

      7. metadata-eval 1.3%

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

      8. +-commutative 1.3%

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

   7. Simplified 1.3%

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

   8. Taylor expanded in x around inf 63.4%

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

      1. distribute-lft-in 63.4%

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

      2. associate--l+ 63.4%

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

      3. unpow2 63.4%

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

      4. +-commutative 63.4%

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

      5. unpow2 63.4%

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

      6. unpow2 63.4%

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

      7. associate-*r/ 63.4%

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

   10. Simplified 63.4%

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

   11. Taylor expanded in t around 0 63.4%

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

       1. +-commutative 63.4%

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

       2. mul-1-neg 63.4%

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

       3. unsub-neg 63.4%

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

       4. *-commutative 63.4%

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

       5. unpow2 63.4%

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

       6. unpow2 63.4%

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

   13. Simplified 63.4%

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

   14. Taylor expanded in t around 0 31.9%

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

   if -3.4999999999999999e-297 < t < 4.9999999999999995e-309

   1. Initial program 3.1%

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

      1. associate-/l* 3.1%

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

      2. fma-neg 3.1%

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

      3. remove-double-neg 3.1%

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

      4. fma-neg 3.1%

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

      5. sub-neg 3.1%

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

      6. metadata-eval 3.1%

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

      7. remove-double-neg 3.1%

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

      8. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in t around -inf 85.7%

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

      1. mul-1-neg 85.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 85.7%

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

      3. +-commutative 85.7%

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

      4. sub-neg 85.7%

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

      5. metadata-eval 85.7%

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

      6. +-commutative 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in x around inf 85.7%

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

   if 4.9999999999999995e-309 < t < 1.90000000000000016e-279

   1. Initial program 2.5%

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

      1. associate-*l/ 2.5%

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

   3. Simplified 2.5%

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

   4. Taylor expanded in t around inf 2.8%

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

      1. associate-/l* 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 2.8%

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

      4. metadata-eval 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in x around inf 80.6%

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

      1. associate-/r* 80.6%

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

   9. Simplified 80.6%

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

   if 1.04999999999999995e-151 < t

   1. Initial program 37.0%

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

      1. associate-*l/ 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in t around inf 30.4%

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

      1. associate-/l* 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 34.0%

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

      4. metadata-eval 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 34.0%

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

   6. Simplified 34.0%

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

   7. Taylor expanded in x around inf 84.5%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{t \cdot {x}^{2}} + \frac{1}{t}\right) - \frac{1}{t \cdot x}\right)} \cdot t \]
   8. Step-by-step derivation

      1. associate--l+ 84.5%

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

      2. associate-*r/ 84.5%

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

      3. metadata-eval 84.5%

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

      4. unpow2 84.5%

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

      5. associate-/r* 84.5%

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

   9. Simplified 84.5%

      \[\leadsto \color{blue}{\left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right)\right)} \cdot t \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-279}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-151}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right) + \frac{0.5}{t \cdot \left(x \cdot x\right)}\right)\\ \end{array} \]

Alternative 8: 75.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\ell} \cdot \sqrt{x}\\ t_2 := \frac{1}{t} - \frac{\frac{1}{t}}{x}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-199}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-279}:\\ \;\;\;\;t \cdot t_2\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t_2 + \frac{0.5}{t \cdot \left(x \cdot x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (/ t l) (sqrt x))) (t_2 (- (/ 1.0 t) (/ (/ 1.0 t) x))))
   (if (<= t -9.5e-199)
     (/ 1.0 (- (sqrt (/ (+ 1.0 x) (+ x -1.0)))))
     (if (<= t -3.5e-297)
       t_1
       (if (<= t 5e-309)
         -1.0
         (if (<= t 1.55e-279)
           (* t t_2)
           (if (<= t 1.36e-151) t_1 (* t (+ t_2 (/ 0.5 (* t (* x x))))))))))))

C

double code(double x, double l, double t) {
	double t_1 = (t / l) * sqrt(x);
	double t_2 = (1.0 / t) - ((1.0 / t) / x);
	double tmp;
	if (t <= -9.5e-199) {
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = t_1;
	} else if (t <= 5e-309) {
		tmp = -1.0;
	} else if (t <= 1.55e-279) {
		tmp = t * t_2;
	} else if (t <= 1.36e-151) {
		tmp = t_1;
	} else {
		tmp = t * (t_2 + (0.5 / (t * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t / l) * sqrt(x)
    t_2 = (1.0d0 / t) - ((1.0d0 / t) / x)
    if (t <= (-9.5d-199)) then
        tmp = 1.0d0 / -sqrt(((1.0d0 + x) / (x + (-1.0d0))))
    else if (t <= (-3.5d-297)) then
        tmp = t_1
    else if (t <= 5d-309) then
        tmp = -1.0d0
    else if (t <= 1.55d-279) then
        tmp = t * t_2
    else if (t <= 1.36d-151) then
        tmp = t_1
    else
        tmp = t * (t_2 + (0.5d0 / (t * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = (t / l) * Math.sqrt(x);
	double t_2 = (1.0 / t) - ((1.0 / t) / x);
	double tmp;
	if (t <= -9.5e-199) {
		tmp = 1.0 / -Math.sqrt(((1.0 + x) / (x + -1.0)));
	} else if (t <= -3.5e-297) {
		tmp = t_1;
	} else if (t <= 5e-309) {
		tmp = -1.0;
	} else if (t <= 1.55e-279) {
		tmp = t * t_2;
	} else if (t <= 1.36e-151) {
		tmp = t_1;
	} else {
		tmp = t * (t_2 + (0.5 / (t * (x * x))));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (t / l) * math.sqrt(x)
	t_2 = (1.0 / t) - ((1.0 / t) / x)
	tmp = 0
	if t <= -9.5e-199:
		tmp = 1.0 / -math.sqrt(((1.0 + x) / (x + -1.0)))
	elif t <= -3.5e-297:
		tmp = t_1
	elif t <= 5e-309:
		tmp = -1.0
	elif t <= 1.55e-279:
		tmp = t * t_2
	elif t <= 1.36e-151:
		tmp = t_1
	else:
		tmp = t * (t_2 + (0.5 / (t * (x * x))))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(t / l) * sqrt(x))
	t_2 = Float64(Float64(1.0 / t) - Float64(Float64(1.0 / t) / x))
	tmp = 0.0
	if (t <= -9.5e-199)
		tmp = Float64(1.0 / Float64(-sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))));
	elseif (t <= -3.5e-297)
		tmp = t_1;
	elseif (t <= 5e-309)
		tmp = -1.0;
	elseif (t <= 1.55e-279)
		tmp = Float64(t * t_2);
	elseif (t <= 1.36e-151)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(t_2 + Float64(0.5 / Float64(t * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (t / l) * sqrt(x);
	t_2 = (1.0 / t) - ((1.0 / t) / x);
	tmp = 0.0;
	if (t <= -9.5e-199)
		tmp = 1.0 / -sqrt(((1.0 + x) / (x + -1.0)));
	elseif (t <= -3.5e-297)
		tmp = t_1;
	elseif (t <= 5e-309)
		tmp = -1.0;
	elseif (t <= 1.55e-279)
		tmp = t * t_2;
	elseif (t <= 1.36e-151)
		tmp = t_1;
	else
		tmp = t * (t_2 + (0.5 / (t * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / t), $MachinePrecision] - N[(N[(1.0 / t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.5e-199], N[(1.0 / (-N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -3.5e-297], t$95$1, If[LessEqual[t, 5e-309], -1.0, If[LessEqual[t, 1.55e-279], N[(t * t$95$2), $MachinePrecision], If[LessEqual[t, 1.36e-151], t$95$1, N[(t * N[(t$95$2 + N[(0.5 / N[(t * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -9.5000000000000005e-199

   1. Initial program 35.0%

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

      1. associate-/l* 34.9%

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

      2. fma-neg 34.9%

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

      3. remove-double-neg 34.9%

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

      4. fma-neg 34.9%

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

      5. sub-neg 34.9%

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

      6. metadata-eval 34.9%

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

      7. remove-double-neg 34.9%

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

      8. fma-def 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around -inf 82.9%

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

      1. mul-1-neg 82.9%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. sub-neg 82.9%

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

      5. metadata-eval 82.9%

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

      6. +-commutative 82.9%

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

   6. Simplified 82.9%

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

      1. div-inv 82.9%

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

      2. +-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. associate-*r/ 82.9%

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

      2. *-rgt-identity 82.9%

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

      3. associate-/r* 82.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 82.9%

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

   10. Simplified 82.9%

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

   if -9.5000000000000005e-199 < t < -3.4999999999999999e-297 or 1.55e-279 < t < 1.35999999999999994e-151

   1. Initial program 5.3%

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

      1. associate-*l/ 5.3%

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

   3. Simplified 5.3%

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

   5. Applied egg-rr 4.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 4.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}\right)\right)} \]

      2. expm1-log1p 5.8%

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

      3. metadata-eval 5.8%

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

      4. sub-neg 5.8%

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

      5. associate-/l* 1.3%

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

      6. sub-neg 1.3%

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

      7. metadata-eval 1.3%

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

      8. +-commutative 1.3%

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

   7. Simplified 1.3%

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

   8. Taylor expanded in x around inf 63.4%

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

      1. distribute-lft-in 63.4%

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

      2. associate--l+ 63.4%

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

      3. unpow2 63.4%

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

      4. +-commutative 63.4%

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

      5. unpow2 63.4%

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

      6. unpow2 63.4%

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

      7. associate-*r/ 63.4%

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

   10. Simplified 63.4%

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

   11. Taylor expanded in t around 0 63.4%

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

       1. +-commutative 63.4%

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

       2. mul-1-neg 63.4%

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

       3. unsub-neg 63.4%

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

       4. *-commutative 63.4%

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

       5. unpow2 63.4%

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

       6. unpow2 63.4%

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

   13. Simplified 63.4%

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

   14. Taylor expanded in t around 0 31.9%

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

   if -3.4999999999999999e-297 < t < 4.9999999999999995e-309

   1. Initial program 3.1%

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

      1. associate-/l* 3.1%

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

      2. fma-neg 3.1%

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

      3. remove-double-neg 3.1%

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

      4. fma-neg 3.1%

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

      5. sub-neg 3.1%

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

      6. metadata-eval 3.1%

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

      7. remove-double-neg 3.1%

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

      8. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in t around -inf 85.7%

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

      1. mul-1-neg 85.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 85.7%

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

      3. +-commutative 85.7%

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

      4. sub-neg 85.7%

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

      5. metadata-eval 85.7%

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

      6. +-commutative 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in x around inf 85.7%

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

   if 4.9999999999999995e-309 < t < 1.55e-279

   1. Initial program 2.5%

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

      1. associate-*l/ 2.5%

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

   3. Simplified 2.5%

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

   4. Taylor expanded in t around inf 2.8%

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

      1. associate-/l* 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 2.8%

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

      4. metadata-eval 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 2.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in x around inf 80.6%

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

      1. associate-/r* 80.6%

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

   9. Simplified 80.6%

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

   if 1.35999999999999994e-151 < t

   1. Initial program 37.0%

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

      1. associate-*l/ 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in t around inf 30.4%

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

      1. associate-/l* 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 34.0%

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

      4. metadata-eval 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 34.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 34.0%

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

   6. Simplified 34.0%

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

   7. Taylor expanded in x around inf 84.5%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{t \cdot {x}^{2}} + \frac{1}{t}\right) - \frac{1}{t \cdot x}\right)} \cdot t \]
   8. Step-by-step derivation

      1. associate--l+ 84.5%

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

      2. associate-*r/ 84.5%

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

      3. metadata-eval 84.5%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{t \cdot {x}^{2}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right) \cdot t \]

      4. unpow2 84.5%

         \[\leadsto \left(\frac{0.5}{t \cdot \color{blue}{\left(x \cdot x\right)}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right) \cdot t \]

      5. associate-/r* 84.5%

         \[\leadsto \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} - \color{blue}{\frac{\frac{1}{t}}{x}}\right)\right) \cdot t \]

   9. Simplified 84.5%

      \[\leadsto \color{blue}{\left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right)\right)} \cdot t \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-199}:\\ \;\;\;\;\frac{1}{-\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-279}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right)\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-151}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right) + \frac{0.5}{t \cdot \left(x \cdot x\right)}\right)\\ \end{array} \]

Alternative 9: 76.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right) + \frac{0.5}{t \cdot \left(x \cdot x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 5e-309)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (* t (+ (- (/ 1.0 t) (/ (/ 1.0 t) x)) (/ 0.5 (* t (* x x)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 5e-309) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = t * (((1.0 / t) - ((1.0 / t) / x)) + (0.5 / (t * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 5d-309) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else
        tmp = t * (((1.0d0 / t) - ((1.0d0 / t) / x)) + (0.5d0 / (t * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 5e-309) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = t * (((1.0 / t) - ((1.0 / t) / x)) + (0.5 / (t * (x * x))));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 5e-309:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	else:
		tmp = t * (((1.0 / t) - ((1.0 / t) / x)) + (0.5 / (t * (x * x))))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 5e-309)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(t * Float64(Float64(Float64(1.0 / t) - Float64(Float64(1.0 / t) / x)) + Float64(0.5 / Float64(t * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 5e-309)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	else
		tmp = t * (((1.0 / t) - ((1.0 / t) / x)) + (0.5 / (t * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 5e-309], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(N[(1.0 / t), $MachinePrecision] - N[(N[(1.0 / t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(t * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.9999999999999995e-309

   1. Initial program 30.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 30.8%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 30.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 77.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 77.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 77.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 77.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 77.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 77.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 77.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 77.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 77.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 77.0%

         \[\leadsto \frac{1}{x} - \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) \]

      2. metadata-eval 77.0%

         \[\leadsto \frac{1}{x} - \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) \]

      3. unpow2 77.0%

         \[\leadsto \frac{1}{x} - \left(1 + \frac{0.5}{\color{blue}{x \cdot x}}\right) \]

   9. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)} \]

   if 4.9999999999999995e-309 < t

   1. Initial program 33.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 33.6%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 27.9%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
   5. Step-by-step derivation

      1. associate-/l* 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

   6. Simplified 30.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

   7. Taylor expanded in x around inf 79.1%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{t \cdot {x}^{2}} + \frac{1}{t}\right) - \frac{1}{t \cdot x}\right)} \cdot t \]
   8. Step-by-step derivation

      1. associate--l+ 79.1%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{t \cdot {x}^{2}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right)} \cdot t \]

      2. associate-*r/ 79.1%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{t \cdot {x}^{2}}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right) \cdot t \]

      3. metadata-eval 79.1%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{t \cdot {x}^{2}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right) \cdot t \]

      4. unpow2 79.1%

         \[\leadsto \left(\frac{0.5}{t \cdot \color{blue}{\left(x \cdot x\right)}} + \left(\frac{1}{t} - \frac{1}{t \cdot x}\right)\right) \cdot t \]

      5. associate-/r* 79.1%

         \[\leadsto \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} - \color{blue}{\frac{\frac{1}{t}}{x}}\right)\right) \cdot t \]

   9. Simplified 79.1%

      \[\leadsto \color{blue}{\left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right)\right)} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\frac{1}{t} - \frac{\frac{1}{t}}{x}\right) + \frac{0.5}{t \cdot \left(x \cdot x\right)}\right)\\ \end{array} \]

Alternative 10: 75.9% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(--1\right) - \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 1.1e-301)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (/ 1.0 (- (- -1.0) (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 1.1e-301) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = 1.0 / (-(-1.0) - (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.1d-301) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else
        tmp = 1.0d0 / (-(-1.0d0) - ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 1.1e-301) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = 1.0 / (-(-1.0) - (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 1.1e-301:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	else:
		tmp = 1.0 / (-(-1.0) - (-1.0 / x))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 1.1e-301)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(1.0 / Float64(Float64(-(-1.0)) - Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 1.1e-301)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	else
		tmp = 1.0 / (-(-1.0) - (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 1.1e-301], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[((--1.0) - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e-301

   1. Initial program 30.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 30.6%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 76.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 76.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 76.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 76.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 76.5%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 76.5%

         \[\leadsto \frac{1}{x} - \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) \]

      2. metadata-eval 76.5%

         \[\leadsto \frac{1}{x} - \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) \]

      3. unpow2 76.5%

         \[\leadsto \frac{1}{x} - \left(1 + \frac{0.5}{\color{blue}{x \cdot x}}\right) \]

   9. Simplified 76.5%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)} \]

   if 1.1e-301 < t

   1. Initial program 33.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 33.8%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 1.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 1.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 1.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 1.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 1.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 1.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 1.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]
   7. Step-by-step derivation

      1. div-inv 1.7%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{1}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

      2. +-commutative 1.7%

         \[\leadsto \sqrt{2} \cdot \frac{1}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + -1}}}\right)} \]

   8. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{1}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 1.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot 1}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}} \]

      2. *-rgt-identity 1.7%

         \[\leadsto \frac{\color{blue}{\sqrt{2}}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)} \]

      3. associate-/r* 1.7%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 1.7%

         \[\leadsto \frac{\color{blue}{1}}{-\sqrt{\frac{x + 1}{x + -1}}} \]

   10. Simplified 1.7%

       \[\leadsto \color{blue}{\frac{1}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

   11. Taylor expanded in x around -inf 0.0%

       \[\leadsto \frac{1}{-\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{x}\right)}} \]
   12. Step-by-step derivation

       1. sub-neg 0.0%

          \[\leadsto \frac{1}{-\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \left(-\frac{1}{x}\right)\right)}} \]

       2. unpow2 0.0%

          \[\leadsto \frac{1}{-\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \left(-\frac{1}{x}\right)\right)} \]

       3. rem-square-sqrt 79.6%

          \[\leadsto \frac{1}{-\left(\color{blue}{-1} + \left(-\frac{1}{x}\right)\right)} \]

       4. distribute-neg-frac 79.6%

          \[\leadsto \frac{1}{-\left(-1 + \color{blue}{\frac{-1}{x}}\right)} \]

       5. metadata-eval 79.6%

          \[\leadsto \frac{1}{-\left(-1 + \frac{\color{blue}{-1}}{x}\right)} \]

   13. Simplified 79.6%

       \[\leadsto \frac{1}{-\color{blue}{\left(-1 + \frac{-1}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(--1\right) - \frac{-1}{x}}\\ \end{array} \]

Alternative 11: 75.8% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-301}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(--1\right) - \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 1.1e-301) (+ -1.0 (/ 1.0 x)) (/ 1.0 (- (- -1.0) (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 1.1e-301) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 / (-(-1.0) - (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.1d-301) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 / (-(-1.0d0) - ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 1.1e-301) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 / (-(-1.0) - (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 1.1e-301:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 / (-(-1.0) - (-1.0 / x))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 1.1e-301)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 / Float64(Float64(-(-1.0)) - Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 1.1e-301)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 / (-(-1.0) - (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 1.1e-301], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[((--1.0) - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e-301

   1. Initial program 30.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 30.6%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 76.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 76.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 76.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 76.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 76.3%

      \[\leadsto \color{blue}{\frac{1}{x} - 1} \]

   if 1.1e-301 < t

   1. Initial program 33.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 33.8%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 33.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 1.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 1.7%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 1.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 1.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 1.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 1.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 1.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]
   7. Step-by-step derivation

      1. div-inv 1.7%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{1}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

      2. +-commutative 1.7%

         \[\leadsto \sqrt{2} \cdot \frac{1}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + -1}}}\right)} \]

   8. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{1}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 1.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot 1}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}} \]

      2. *-rgt-identity 1.7%

         \[\leadsto \frac{\color{blue}{\sqrt{2}}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)} \]

      3. associate-/r* 1.7%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

      4. *-inverses 1.7%

         \[\leadsto \frac{\color{blue}{1}}{-\sqrt{\frac{x + 1}{x + -1}}} \]

   10. Simplified 1.7%

       \[\leadsto \color{blue}{\frac{1}{-\sqrt{\frac{x + 1}{x + -1}}}} \]

   11. Taylor expanded in x around -inf 0.0%

       \[\leadsto \frac{1}{-\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{x}\right)}} \]
   12. Step-by-step derivation

       1. sub-neg 0.0%

          \[\leadsto \frac{1}{-\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \left(-\frac{1}{x}\right)\right)}} \]

       2. unpow2 0.0%

          \[\leadsto \frac{1}{-\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \left(-\frac{1}{x}\right)\right)} \]

       3. rem-square-sqrt 79.6%

          \[\leadsto \frac{1}{-\left(\color{blue}{-1} + \left(-\frac{1}{x}\right)\right)} \]

       4. distribute-neg-frac 79.6%

          \[\leadsto \frac{1}{-\left(-1 + \color{blue}{\frac{-1}{x}}\right)} \]

       5. metadata-eval 79.6%

          \[\leadsto \frac{1}{-\left(-1 + \frac{\color{blue}{-1}}{x}\right)} \]

   13. Simplified 79.6%

       \[\leadsto \frac{1}{-\color{blue}{\left(-1 + \frac{-1}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-301}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(--1\right) - \frac{-1}{x}}\\ \end{array} \]

Alternative 12: 75.6% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-301}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 1.1e-301) (+ -1.0 (/ 1.0 x)) 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 1.1e-301) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	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
    real(8) :: tmp
    if (t <= 1.1d-301) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 1.1e-301) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 1.1e-301:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 1.1e-301)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 1.1e-301)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 1.1e-301], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e-301

   1. Initial program 30.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 30.6%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 30.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 76.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 76.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 76.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 76.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 76.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 76.3%

      \[\leadsto \color{blue}{\frac{1}{x} - 1} \]

   if 1.1e-301 < t

   1. Initial program 33.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 33.8%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 28.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
   5. Step-by-step derivation

      1. associate-/l* 30.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 30.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 30.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval 30.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 30.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 30.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

   6. Simplified 30.5%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

   7. Taylor expanded in x around inf 78.8%

      \[\leadsto \color{blue}{\frac{1}{t}} \cdot t \]

   8. Taylor expanded in t around 0 78.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-301}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 75.3% accurate, 73.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (if (<= t -1e-309) -1.0 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	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
    real(8) :: tmp
    if (t <= (-1d-309)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-309], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 30.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 30.8%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 30.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 30.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 77.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 77.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 77.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 77.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 77.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 77.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 77.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 77.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 76.5%

      \[\leadsto \color{blue}{-1} \]

   if -1.000000000000002e-309 < t

   1. Initial program 33.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 33.6%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 27.9%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
   5. Step-by-step derivation

      1. associate-/l* 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]

      2. +-commutative 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]

      3. sub-neg 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]

      4. metadata-eval 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]

      5. +-commutative 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]

      6. unpow2 30.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

   6. Simplified 30.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \cdot t \]

   7. Taylor expanded in x around inf 78.2%

      \[\leadsto \color{blue}{\frac{1}{t}} \cdot t \]

   8. Taylor expanded in t around 0 78.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 39.0% accurate, 225.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 -1.0)

C

double code(double x, double l, double t) {
	return -1.0;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

public static double code(double x, double l, double t) {
	return -1.0;
}

Python

def code(x, l, t):
	return -1.0

Julia

function code(x, l, t)
	return -1.0
end

MATLAB

function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

code[x_, l_, t_] := -1.0

Derivation

1. Initial program 32.1%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-/l* 32.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   2. fma-neg 32.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

   3. remove-double-neg 32.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

   4. fma-neg 32.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

   5. sub-neg 32.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

   6. metadata-eval 32.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

   7. remove-double-neg 32.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   8. fma-def 32.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

3. Simplified 32.1%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

4. Taylor expanded in t around -inf 41.8%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg 41.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

   2. distribute-rgt-neg-in 41.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

   3. +-commutative 41.8%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

   4. sub-neg 41.8%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

   5. metadata-eval 41.8%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

   6. +-commutative 41.8%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

6. Simplified 41.8%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

7. Taylor expanded in x around inf 41.4%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 41.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023260 
(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)))))