Toniolo and Linder, Equation (7)

Percentage Accurate: 33.0% → 87.5%

Time: 19.3s

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: 33.0% 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: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+109}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-169}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{+88}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) + 2 \cdot t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))) (t_2 (* l (/ l x))))
   (if (<= t -5e+109)
     (- t_1)
     (if (<= t -1.26e-169)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt (+ t_2 (- (* 2.0 (fma t t (/ t (/ x t)))) (/ l (/ x (- l))))))))
       (if (<= t 3.6e-179)
         (/ (* t (sqrt x)) l)
         (if (<= t 1.18e+88)
           (*
            t
            (/
             (sqrt 2.0)
             (sqrt (+ (* 2.0 (fma t t (* t (/ t x)))) (* 2.0 t_2)))))
           t_1))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = l * (l / x);
	double tmp;
	if (t <= -5e+109) {
		tmp = -t_1;
	} else if (t <= -1.26e-169) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + ((2.0 * fma(t, t, (t / (x / t)))) - (l / (x / -l))))));
	} else if (t <= 3.6e-179) {
		tmp = (t * sqrt(x)) / l;
	} else if (t <= 1.18e+88) {
		tmp = t * (sqrt(2.0) / sqrt(((2.0 * fma(t, t, (t * (t / x)))) + (2.0 * t_2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(l * Float64(l / x))
	tmp = 0.0
	if (t <= -5e+109)
		tmp = Float64(-t_1);
	elseif (t <= -1.26e-169)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(Float64(2.0 * fma(t, t, Float64(t / Float64(x / t)))) - Float64(l / Float64(x / Float64(-l))))))));
	elseif (t <= 3.6e-179)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	elseif (t <= 1.18e+88)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(2.0 * fma(t, t, Float64(t * Float64(t / x)))) + Float64(2.0 * t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e+109], (-t$95$1), If[LessEqual[t, -1.26e-169], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(N[(2.0 * N[(t * t + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l / N[(x / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e-179], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t, 1.18e+88], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(t * t + N[(t * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.0000000000000001e109

   1. Initial program 14.0%

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

      1. associate-*r/ 13.9%

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

      2. fma-neg 13.9%

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

      3. remove-double-neg 13.9%

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

      4. fma-neg 13.9%

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

      5. sub-neg 13.9%

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

      6. metadata-eval 13.9%

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

      7. remove-double-neg 13.9%

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

      8. +-commutative 13.9%

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

      9. fma-def 13.9%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in t around inf 5.4%

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

      1. associate-/l* 17.0%

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

      2. +-commutative 17.0%

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

      3. sub-neg 17.0%

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

      4. metadata-eval 17.0%

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

      5. +-commutative 17.0%

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

      6. unpow2 17.0%

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

   6. Simplified 17.0%

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

   7. Taylor expanded in t around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. +-commutative 97.8%

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

      3. sub-neg 97.8%

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

      4. metadata-eval 97.8%

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

   9. Simplified 97.8%

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

   if -5.0000000000000001e109 < t < -1.26e-169

   1. Initial program 53.7%

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

      1. associate-*l/ 53.7%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in x around inf 71.4%

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

      1. associate--l+ 71.4%

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

      2. unpow2 71.4%

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

      3. distribute-lft-out 71.4%

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

      4. unpow2 71.4%

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

      5. unpow2 71.4%

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

      6. associate-*r/ 71.4%

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

      7. mul-1-neg 71.4%

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

      8. +-commutative 71.4%

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

      9. unpow2 71.4%

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

      10. unpow2 71.4%

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

      11. fma-udef 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in t around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. mul-1-neg 71.4%

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

      3. unpow2 71.4%

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

      4. distribute-rgt-neg-in 71.4%

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

   9. Simplified 71.4%

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

       1. expm1-log1p-u 69.9%

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

       2. expm1-udef 38.1%

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

       3. associate-/l* 38.1%

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

       4. +-commutative 38.1%

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

       5. fma-def 38.1%

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

   11. Applied egg-rr 38.1%

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

       1. expm1-def 69.9%

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

       2. expm1-log1p 71.4%

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

       3. associate-/r/ 71.4%

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

       4. associate-/l* 71.4%

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

       5. associate-/l* 83.9%

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

   13. Simplified 83.9%

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

   if -1.26e-169 < t < 3.60000000000000007e-179

   1. Initial program 4.1%

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

      1. associate-*l/ 4.1%

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

   3. Simplified 4.1%

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

   4. Taylor expanded in x around inf 55.3%

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

      1. associate--l+ 55.3%

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

      2. unpow2 55.3%

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

      3. distribute-lft-out 55.3%

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

      4. unpow2 55.3%

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

      5. unpow2 55.3%

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

      6. associate-*r/ 55.3%

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

      7. mul-1-neg 55.3%

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

      8. +-commutative 55.3%

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

      9. unpow2 55.3%

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

      10. unpow2 55.3%

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

      11. fma-udef 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in t around 0 55.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 55.0%

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

      2. metadata-eval 55.0%

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

      3. distribute-rgt1-in 55.0%

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

      4. metadata-eval 55.0%

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

      5. unpow2 55.0%

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

   9. Simplified 55.0%

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

   10. Taylor expanded in l around 0 42.4%

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 47.8%

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

   12. Applied egg-rr 47.8%

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

   if 3.60000000000000007e-179 < t < 1.1799999999999999e88

   1. Initial program 47.7%

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

      1. associate-*l/ 47.7%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in x around inf 76.1%

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

      1. associate--l+ 76.1%

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

      2. unpow2 76.1%

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

      3. distribute-lft-out 76.1%

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

      4. unpow2 76.1%

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

      5. unpow2 76.1%

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

      6. associate-*r/ 76.1%

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

      7. mul-1-neg 76.1%

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

      8. +-commutative 76.1%

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

      9. unpow2 76.1%

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

      10. unpow2 76.1%

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

      11. fma-udef 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in t around 0 76.1%

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

      1. associate-*r/ 76.1%

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

      2. mul-1-neg 76.1%

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

      3. unpow2 76.1%

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

      4. distribute-rgt-neg-in 76.1%

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

   9. Simplified 76.1%

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

       1. expm1-log1p-u 74.5%

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

       2. expm1-udef 36.7%

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

       3. associate-/l* 36.7%

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

       4. +-commutative 36.7%

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

       5. fma-def 36.7%

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

   11. Applied egg-rr 36.7%

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

       1. expm1-def 74.5%

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

       2. expm1-log1p 76.1%

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

       3. +-commutative 76.1%

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

       4. distribute-rgt-neg-out 76.1%

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

       5. unpow2 76.1%

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

       6. mul-1-neg 76.1%

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

       7. associate-*r/ 76.1%

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

       8. cancel-sign-sub-inv 76.1%

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

       9. metadata-eval 76.1%

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

       10. *-lft-identity 76.1%

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

       11. associate-+l+ 76.1%

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

   13. Simplified 84.3%

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

   if 1.1799999999999999e88 < t

   1. Initial program 15.6%

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

      1. associate-*r/ 15.5%

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

      2. fma-neg 15.5%

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

      3. remove-double-neg 15.5%

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

      4. fma-neg 15.5%

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

      5. sub-neg 15.5%

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

      6. metadata-eval 15.5%

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

      7. remove-double-neg 15.5%

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

      8. +-commutative 15.5%

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

      9. fma-def 15.5%

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

   3. Simplified 15.5%

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

   4. Taylor expanded in t around inf 5.1%

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

      1. associate-/l* 20.2%

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

      2. +-commutative 20.2%

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

      3. sub-neg 20.2%

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

      4. metadata-eval 20.2%

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

      5. +-commutative 20.2%

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

      6. unpow2 20.2%

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

   6. Simplified 20.2%

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

   7. Taylor expanded in t around 0 98.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+109}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-169}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\ell \cdot \frac{\ell}{x} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{+88}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -7 \cdot 10^{+109}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-179}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt (+ (* 2.0 (fma t t (* t (/ t x)))) (* 2.0 (* l (/ l x))))))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -7e+109)
     (- t_2)
     (if (<= t -1.7e-172)
       t_1
       (if (<= t 3.2e-179)
         (/ (* t (sqrt x)) l)
         (if (<= t 1.1e+88) t_1 t_2))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / sqrt(((2.0 * fma(t, t, (t * (t / x)))) + (2.0 * (l * (l / x))))));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -7e+109) {
		tmp = -t_2;
	} else if (t <= -1.7e-172) {
		tmp = t_1;
	} else if (t <= 3.2e-179) {
		tmp = (t * sqrt(x)) / l;
	} else if (t <= 1.1e+88) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(2.0 * fma(t, t, Float64(t * Float64(t / x)))) + Float64(2.0 * Float64(l * Float64(l / x)))))))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -7e+109)
		tmp = Float64(-t_2);
	elseif (t <= -1.7e-172)
		tmp = t_1;
	elseif (t <= 3.2e-179)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	elseif (t <= 1.1e+88)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(t * t + N[(t * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7e+109], (-t$95$2), If[LessEqual[t, -1.7e-172], t$95$1, If[LessEqual[t, 3.2e-179], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t, 1.1e+88], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.99999999999999966e109

   1. Initial program 14.0%

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

      1. associate-*r/ 13.9%

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

      2. fma-neg 13.9%

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

      3. remove-double-neg 13.9%

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

      4. fma-neg 13.9%

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

      5. sub-neg 13.9%

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

      6. metadata-eval 13.9%

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

      7. remove-double-neg 13.9%

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

      8. +-commutative 13.9%

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

      9. fma-def 13.9%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in t around inf 5.4%

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

      1. associate-/l* 17.0%

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

      2. +-commutative 17.0%

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

      3. sub-neg 17.0%

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

      4. metadata-eval 17.0%

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

      5. +-commutative 17.0%

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

      6. unpow2 17.0%

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

   6. Simplified 17.0%

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

   7. Taylor expanded in t around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. +-commutative 97.8%

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

      3. sub-neg 97.8%

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

      4. metadata-eval 97.8%

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

   9. Simplified 97.8%

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

   if -6.99999999999999966e109 < t < -1.6999999999999999e-172 or 3.2000000000000001e-179 < t < 1.10000000000000004e88

   1. Initial program 50.8%

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

      1. associate-*l/ 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in x around inf 73.7%

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

      1. associate--l+ 73.7%

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

      2. unpow2 73.7%

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

      3. distribute-lft-out 73.7%

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

      4. unpow2 73.7%

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

      5. unpow2 73.7%

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

      6. associate-*r/ 73.7%

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

      7. mul-1-neg 73.7%

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

      8. +-commutative 73.7%

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

      9. unpow2 73.7%

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

      10. unpow2 73.7%

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

      11. fma-udef 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in t around 0 73.7%

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

      1. associate-*r/ 73.7%

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

      2. mul-1-neg 73.7%

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

      3. unpow2 73.7%

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

      4. distribute-rgt-neg-in 73.7%

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

   9. Simplified 73.7%

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

       1. expm1-log1p-u 72.1%

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

       2. expm1-udef 37.4%

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

       3. associate-/l* 37.4%

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

       4. +-commutative 37.4%

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

       5. fma-def 37.4%

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

   11. Applied egg-rr 37.4%

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

       1. expm1-def 72.1%

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

       2. expm1-log1p 73.7%

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

       3. +-commutative 73.7%

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

       4. distribute-rgt-neg-out 73.7%

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

       5. unpow2 73.7%

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

       6. mul-1-neg 73.7%

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

       7. associate-*r/ 73.7%

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

       8. cancel-sign-sub-inv 73.7%

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

       9. metadata-eval 73.7%

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

       10. *-lft-identity 73.7%

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

       11. associate-+l+ 73.7%

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

   13. Simplified 84.1%

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

   if -1.6999999999999999e-172 < t < 3.2000000000000001e-179

   1. Initial program 4.1%

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

      1. associate-*l/ 4.1%

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

   3. Simplified 4.1%

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

   4. Taylor expanded in x around inf 55.3%

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

      1. associate--l+ 55.3%

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

      2. unpow2 55.3%

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

      3. distribute-lft-out 55.3%

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

      4. unpow2 55.3%

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

      5. unpow2 55.3%

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

      6. associate-*r/ 55.3%

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

      7. mul-1-neg 55.3%

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

      8. +-commutative 55.3%

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

      9. unpow2 55.3%

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

      10. unpow2 55.3%

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

      11. fma-udef 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in t around 0 55.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 55.0%

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

      2. metadata-eval 55.0%

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

      3. distribute-rgt1-in 55.0%

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

      4. metadata-eval 55.0%

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

      5. unpow2 55.0%

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

   9. Simplified 55.0%

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

   10. Taylor expanded in l around 0 42.4%

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 47.8%

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

   12. Applied egg-rr 47.8%

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

   if 1.10000000000000004e88 < t

   1. Initial program 15.6%

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

      1. associate-*r/ 15.5%

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

      2. fma-neg 15.5%

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

      3. remove-double-neg 15.5%

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

      4. fma-neg 15.5%

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

      5. sub-neg 15.5%

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

      6. metadata-eval 15.5%

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

      7. remove-double-neg 15.5%

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

      8. +-commutative 15.5%

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

      9. fma-def 15.5%

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

   3. Simplified 15.5%

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

   4. Taylor expanded in t around inf 5.1%

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

      1. associate-/l* 20.2%

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

      2. +-commutative 20.2%

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

      3. sub-neg 20.2%

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

      4. metadata-eval 20.2%

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

      5. +-commutative 20.2%

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

      6. unpow2 20.2%

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

   6. Simplified 20.2%

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

   7. Taylor expanded in t around 0 98.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+109}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-172}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-179}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+88}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{-20}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -1.58 \cdot 10^{-139}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 80000000000:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_2 (/ (* l l) x))
        (t_3
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt (+ t_2 (+ t_2 (* 2.0 (+ (* t t) (/ (* t t) x))))))))))
   (if (<= t -1.75e-20)
     (- t_1)
     (if (<= t -1.58e-139)
       t_3
       (if (<= t -8.2e-184)
         -1.0
         (if (<= t 1.5e-155)
           (/ (* t (sqrt x)) l)
           (if (<= t 80000000000.0) t_3 t_1)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = t * (sqrt(2.0) / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -1.75e-20) {
		tmp = -t_1;
	} else if (t <= -1.58e-139) {
		tmp = t_3;
	} else if (t <= -8.2e-184) {
		tmp = -1.0;
	} else if (t <= 1.5e-155) {
		tmp = (t * sqrt(x)) / l;
	} else if (t <= 80000000000.0) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    t_2 = (l * l) / x
    t_3 = t * (sqrt(2.0d0) / sqrt((t_2 + (t_2 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    if (t <= (-1.75d-20)) then
        tmp = -t_1
    else if (t <= (-1.58d-139)) then
        tmp = t_3
    else if (t <= (-8.2d-184)) then
        tmp = -1.0d0
    else if (t <= 1.5d-155) then
        tmp = (t * sqrt(x)) / l
    else if (t <= 80000000000.0d0) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = t * (Math.sqrt(2.0) / Math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -1.75e-20) {
		tmp = -t_1;
	} else if (t <= -1.58e-139) {
		tmp = t_3;
	} else if (t <= -8.2e-184) {
		tmp = -1.0;
	} else if (t <= 1.5e-155) {
		tmp = (t * Math.sqrt(x)) / l;
	} else if (t <= 80000000000.0) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	t_2 = (l * l) / x
	t_3 = t * (math.sqrt(2.0) / math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))))
	tmp = 0
	if t <= -1.75e-20:
		tmp = -t_1
	elif t <= -1.58e-139:
		tmp = t_3
	elif t <= -8.2e-184:
		tmp = -1.0
	elif t <= 1.5e-155:
		tmp = (t * math.sqrt(x)) / l
	elif t <= 80000000000.0:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))))
	tmp = 0.0
	if (t <= -1.75e-20)
		tmp = Float64(-t_1);
	elseif (t <= -1.58e-139)
		tmp = t_3;
	elseif (t <= -8.2e-184)
		tmp = -1.0;
	elseif (t <= 1.5e-155)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	elseif (t <= 80000000000.0)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	t_2 = (l * l) / x;
	t_3 = t * (sqrt(2.0) / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	tmp = 0.0;
	if (t <= -1.75e-20)
		tmp = -t_1;
	elseif (t <= -1.58e-139)
		tmp = t_3;
	elseif (t <= -8.2e-184)
		tmp = -1.0;
	elseif (t <= 1.5e-155)
		tmp = (t * sqrt(x)) / l;
	elseif (t <= 80000000000.0)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.75e-20], (-t$95$1), If[LessEqual[t, -1.58e-139], t$95$3, If[LessEqual[t, -8.2e-184], -1.0, If[LessEqual[t, 1.5e-155], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t, 80000000000.0], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.75000000000000002e-20

   1. Initial program 37.8%

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

      1. associate-*r/ 37.7%

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

      2. fma-neg 37.7%

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

      3. remove-double-neg 37.7%

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

      4. fma-neg 37.7%

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

      5. sub-neg 37.7%

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

      6. metadata-eval 37.7%

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

      7. remove-double-neg 37.7%

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

      8. +-commutative 37.7%

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

      9. fma-def 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in t around inf 28.6%

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

      1. associate-/l* 40.2%

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

      2. +-commutative 40.2%

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

      3. sub-neg 40.2%

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

      4. metadata-eval 40.2%

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

      5. +-commutative 40.2%

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

      6. unpow2 40.2%

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

   6. Simplified 40.2%

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

   7. Taylor expanded in t around -inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. +-commutative 87.6%

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

      3. sub-neg 87.6%

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

      4. metadata-eval 87.6%

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

   9. Simplified 87.6%

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

   if -1.75000000000000002e-20 < t < -1.57999999999999993e-139 or 1.49999999999999992e-155 < t < 8e10

   1. Initial program 42.8%

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

      1. associate-*l/ 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in x around inf 81.4%

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

      1. associate--l+ 81.4%

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

      2. unpow2 81.4%

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

      3. distribute-lft-out 81.4%

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

      4. unpow2 81.4%

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

      5. unpow2 81.4%

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

      6. associate-*r/ 81.4%

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

      7. mul-1-neg 81.4%

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

      8. +-commutative 81.4%

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

      9. unpow2 81.4%

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

      10. unpow2 81.4%

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

      11. fma-udef 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in t around 0 81.4%

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

      1. associate-*r/ 81.4%

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

      2. mul-1-neg 81.4%

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

      3. unpow2 81.4%

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

      4. distribute-rgt-neg-in 81.4%

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

   9. Simplified 81.4%

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

   if -1.57999999999999993e-139 < t < -8.2e-184

   1. Initial program 28.2%

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

      1. associate-*l/ 28.0%

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

   3. Simplified 28.0%

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

   4. Taylor expanded in t around -inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. *-commutative 87.1%

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

      3. distribute-rgt-neg-in 87.1%

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

      4. +-commutative 87.1%

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

      5. sub-neg 87.1%

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

      6. metadata-eval 87.1%

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

      7. +-commutative 87.1%

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

      8. *-commutative 87.1%

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

      9. distribute-rgt-neg-in 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in x around inf 88.0%

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

   if -8.2e-184 < t < 1.49999999999999992e-155

   1. Initial program 3.8%

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

      1. associate-*l/ 3.9%

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

   3. Simplified 3.9%

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

   4. Taylor expanded in x around inf 53.0%

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

      1. associate--l+ 53.0%

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

      2. unpow2 53.0%

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

      3. distribute-lft-out 53.0%

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

      4. unpow2 53.0%

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

      5. unpow2 53.0%

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

      6. associate-*r/ 53.0%

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

      7. mul-1-neg 53.0%

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

      8. +-commutative 53.0%

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

      9. unpow2 53.0%

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

      10. unpow2 53.0%

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

      11. fma-udef 53.0%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in t around 0 52.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 52.6%

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

      2. metadata-eval 52.6%

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

      3. distribute-rgt1-in 52.6%

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

      4. metadata-eval 52.6%

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

      5. unpow2 52.6%

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

   9. Simplified 52.6%

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

   10. Taylor expanded in l around 0 39.6%

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 46.2%

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

   12. Applied egg-rr 46.2%

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

   if 8e10 < t

   1. Initial program 29.0%

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

      1. associate-*r/ 29.0%

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

      2. fma-neg 29.0%

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

      3. remove-double-neg 29.0%

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

      4. fma-neg 29.0%

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

      5. sub-neg 29.0%

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

      6. metadata-eval 29.0%

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

      7. remove-double-neg 29.0%

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

      8. +-commutative 29.0%

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

      9. fma-def 29.0%

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

   3. Simplified 29.0%

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

   4. Taylor expanded in t around inf 17.8%

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

      1. associate-/l* 33.9%

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

      2. +-commutative 33.9%

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

      3. sub-neg 33.9%

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

      4. metadata-eval 33.9%

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

      5. +-commutative 33.9%

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

      6. unpow2 33.9%

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

   6. Simplified 33.9%

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

   7. Taylor expanded in t around 0 94.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-20}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.58 \cdot 10^{-139}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 80000000000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 79.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-184}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.75e-184)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (<= t 2.5e-173) (/ (* t (sqrt x)) l) (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.75e-184) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 2.5e-173) {
		tmp = (t * sqrt(x)) / l;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.75d-184)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if (t <= 2.5d-173) then
        tmp = (t * sqrt(x)) / l
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.75e-184) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 2.5e-173) {
		tmp = (t * Math.sqrt(x)) / l;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.75e-184:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif t <= 2.5e-173:
		tmp = (t * math.sqrt(x)) / l
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.75e-184)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif (t <= 2.5e-173)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.75e-184)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif (t <= 2.5e-173)
		tmp = (t * sqrt(x)) / l;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.75e-184], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e-173], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.74999999999999991e-184

   1. Initial program 37.2%

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

      1. associate-*l/ 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in t around -inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. *-commutative 80.6%

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

      3. distribute-rgt-neg-in 80.6%

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

      4. +-commutative 80.6%

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

      5. sub-neg 80.6%

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

      6. metadata-eval 80.6%

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

      7. +-commutative 80.6%

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

      8. *-commutative 80.6%

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

      9. distribute-rgt-neg-in 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in x around inf 80.5%

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

      1. associate-*r/ 80.5%

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

      2. metadata-eval 80.5%

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

      3. unpow2 80.5%

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

   9. Simplified 80.5%

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

   if -1.74999999999999991e-184 < t < 2.5000000000000001e-173

   1. Initial program 4.0%

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

      1. associate-*l/ 4.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in x around inf 57.4%

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

      1. associate--l+ 57.4%

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

      2. unpow2 57.4%

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

      3. distribute-lft-out 57.4%

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

      4. unpow2 57.4%

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

      5. unpow2 57.4%

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

      6. associate-*r/ 57.4%

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

      7. mul-1-neg 57.4%

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

      8. +-commutative 57.4%

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

      9. unpow2 57.4%

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

      10. unpow2 57.4%

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

      11. fma-udef 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around 0 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 57.2%

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

      2. metadata-eval 57.2%

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

      3. distribute-rgt1-in 57.2%

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

      4. metadata-eval 57.2%

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

      5. unpow2 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in l around 0 44.6%

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 50.0%

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

   12. Applied egg-rr 50.0%

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

   if 2.5000000000000001e-173 < t

   1. Initial program 32.8%

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

      1. associate-*r/ 32.8%

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

      2. fma-neg 32.8%

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

      3. remove-double-neg 32.8%

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

      4. fma-neg 32.8%

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

      5. sub-neg 32.8%

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

      6. metadata-eval 32.8%

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

      7. remove-double-neg 32.8%

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

      8. +-commutative 32.8%

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

      9. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around inf 31.3%

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

      1. associate-/l* 31.1%

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

      2. +-commutative 31.1%

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

      3. sub-neg 31.1%

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

      4. metadata-eval 31.1%

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

      5. +-commutative 31.1%

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

      6. unpow2 31.1%

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

   6. Simplified 31.1%

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

   7. Taylor expanded in t around 0 82.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-184}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 5: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -7 \cdot 10^{-176}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.92 \cdot 10^{-178}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -7e-176) (- t_1) (if (<= t 1.92e-178) (/ (* t (sqrt x)) l) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -7e-176) {
		tmp = -t_1;
	} else if (t <= 1.92e-178) {
		tmp = (t * sqrt(x)) / l;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-7d-176)) then
        tmp = -t_1
    else if (t <= 1.92d-178) then
        tmp = (t * sqrt(x)) / l
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -7e-176) {
		tmp = -t_1;
	} else if (t <= 1.92e-178) {
		tmp = (t * Math.sqrt(x)) / l;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -7e-176:
		tmp = -t_1
	elif t <= 1.92e-178:
		tmp = (t * math.sqrt(x)) / l
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -7e-176)
		tmp = Float64(-t_1);
	elseif (t <= 1.92e-178)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -7e-176)
		tmp = -t_1;
	elseif (t <= 1.92e-178)
		tmp = (t * sqrt(x)) / l;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7e-176], (-t$95$1), If[LessEqual[t, 1.92e-178], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -7e-176

   1. Initial program 37.2%

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

      1. associate-*r/ 37.1%

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

      2. fma-neg 37.1%

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

      3. remove-double-neg 37.1%

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

      4. fma-neg 37.1%

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

      5. sub-neg 37.1%

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

      6. metadata-eval 37.1%

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

      7. remove-double-neg 37.1%

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

      8. +-commutative 37.1%

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

      9. fma-def 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in t around inf 34.5%

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

      1. associate-/l* 35.9%

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

      2. +-commutative 35.9%

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

      3. sub-neg 35.9%

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

      4. metadata-eval 35.9%

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

      5. +-commutative 35.9%

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

      6. unpow2 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in t around -inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. +-commutative 81.1%

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

      3. sub-neg 81.1%

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

      4. metadata-eval 81.1%

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

   9. Simplified 81.1%

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

   if -7e-176 < t < 1.92e-178

   1. Initial program 4.0%

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

      1. associate-*l/ 4.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in x around inf 57.4%

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

      1. associate--l+ 57.4%

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

      2. unpow2 57.4%

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

      3. distribute-lft-out 57.4%

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

      4. unpow2 57.4%

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

      5. unpow2 57.4%

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

      6. associate-*r/ 57.4%

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

      7. mul-1-neg 57.4%

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

      8. +-commutative 57.4%

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

      9. unpow2 57.4%

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

      10. unpow2 57.4%

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

      11. fma-udef 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around 0 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 57.2%

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

      2. metadata-eval 57.2%

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

      3. distribute-rgt1-in 57.2%

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

      4. metadata-eval 57.2%

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

      5. unpow2 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in l around 0 44.6%

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 50.0%

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

   12. Applied egg-rr 50.0%

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

   if 1.92e-178 < t

   1. Initial program 32.8%

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

      1. associate-*r/ 32.8%

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

      2. fma-neg 32.8%

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

      3. remove-double-neg 32.8%

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

      4. fma-neg 32.8%

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

      5. sub-neg 32.8%

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

      6. metadata-eval 32.8%

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

      7. remove-double-neg 32.8%

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

      8. +-commutative 32.8%

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

      9. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around inf 31.3%

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

      1. associate-/l* 31.1%

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

      2. +-commutative 31.1%

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

      3. sub-neg 31.1%

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

      4. metadata-eval 31.1%

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

      5. +-commutative 31.1%

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

      6. unpow2 31.1%

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

   6. Simplified 31.1%

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

   7. Taylor expanded in t around 0 82.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-176}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.92 \cdot 10^{-178}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 6: 78.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{-182}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_1\right)\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_1\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -6.6e-182)
     (+ (/ 1.0 x) (- -1.0 t_1))
     (if (<= t 2.95e-176) (* (sqrt x) (/ t l)) (+ (+ 1.0 t_1) (/ -1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -6.6e-182) {
		tmp = (1.0 / x) + (-1.0 - t_1);
	} else if (t <= 2.95e-176) {
		tmp = sqrt(x) * (t / l);
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-6.6d-182)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - t_1)
    else if (t <= 2.95d-176) then
        tmp = sqrt(x) * (t / l)
    else
        tmp = (1.0d0 + t_1) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -6.6e-182) {
		tmp = (1.0 / x) + (-1.0 - t_1);
	} else if (t <= 2.95e-176) {
		tmp = Math.sqrt(x) * (t / l);
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -6.6e-182:
		tmp = (1.0 / x) + (-1.0 - t_1)
	elif t <= 2.95e-176:
		tmp = math.sqrt(x) * (t / l)
	else:
		tmp = (1.0 + t_1) + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -6.6e-182)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - t_1));
	elseif (t <= 2.95e-176)
		tmp = Float64(sqrt(x) * Float64(t / l));
	else
		tmp = Float64(Float64(1.0 + t_1) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -6.6e-182)
		tmp = (1.0 / x) + (-1.0 - t_1);
	elseif (t <= 2.95e-176)
		tmp = sqrt(x) * (t / l);
	else
		tmp = (1.0 + t_1) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.6e-182], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.95e-176], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.59999999999999991e-182

   1. Initial program 37.2%

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

      1. associate-*l/ 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in t around -inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. *-commutative 80.6%

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

      3. distribute-rgt-neg-in 80.6%

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

      4. +-commutative 80.6%

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

      5. sub-neg 80.6%

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

      6. metadata-eval 80.6%

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

      7. +-commutative 80.6%

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

      8. *-commutative 80.6%

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

      9. distribute-rgt-neg-in 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in x around inf 80.5%

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

      1. associate-*r/ 80.5%

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

      2. metadata-eval 80.5%

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

      3. unpow2 80.5%

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

   9. Simplified 80.5%

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

   if -6.59999999999999991e-182 < t < 2.9499999999999998e-176

   1. Initial program 4.0%

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

      1. associate-*l/ 4.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in x around inf 57.4%

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

      1. associate--l+ 57.4%

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

      2. unpow2 57.4%

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

      3. distribute-lft-out 57.4%

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

      4. unpow2 57.4%

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

      5. unpow2 57.4%

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

      6. associate-*r/ 57.4%

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

      7. mul-1-neg 57.4%

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

      8. +-commutative 57.4%

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

      9. unpow2 57.4%

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

      10. unpow2 57.4%

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

      11. fma-udef 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around 0 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 57.2%

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

      2. metadata-eval 57.2%

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

      3. distribute-rgt1-in 57.2%

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

      4. metadata-eval 57.2%

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

      5. unpow2 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in l around 0 44.6%

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

   if 2.9499999999999998e-176 < t

   1. Initial program 32.8%

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

      1. associate-*r/ 32.8%

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

      2. fma-neg 32.8%

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

      3. remove-double-neg 32.8%

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

      4. fma-neg 32.8%

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

      5. sub-neg 32.8%

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

      6. metadata-eval 32.8%

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

      7. remove-double-neg 32.8%

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

      8. +-commutative 32.8%

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

      9. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around inf 31.3%

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

      1. associate-/l* 31.1%

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

      2. +-commutative 31.1%

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

      3. sub-neg 31.1%

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

      4. metadata-eval 31.1%

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

      5. +-commutative 31.1%

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

      6. unpow2 31.1%

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

   6. Simplified 31.1%

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

   7. Taylor expanded in x around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. metadata-eval 81.5%

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

      3. unpow2 81.5%

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

   9. Simplified 81.5%

      \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) - \frac{1}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-182}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 7: 79.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_1\right)\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-174}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_1\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -1.15e-176)
     (+ (/ 1.0 x) (- -1.0 t_1))
     (if (<= t 2.05e-174) (* t (/ (sqrt x) l)) (+ (+ 1.0 t_1) (/ -1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -1.15e-176) {
		tmp = (1.0 / x) + (-1.0 - t_1);
	} else if (t <= 2.05e-174) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-1.15d-176)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - t_1)
    else if (t <= 2.05d-174) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = (1.0d0 + t_1) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -1.15e-176) {
		tmp = (1.0 / x) + (-1.0 - t_1);
	} else if (t <= 2.05e-174) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -1.15e-176:
		tmp = (1.0 / x) + (-1.0 - t_1)
	elif t <= 2.05e-174:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = (1.0 + t_1) + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -1.15e-176)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - t_1));
	elseif (t <= 2.05e-174)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = Float64(Float64(1.0 + t_1) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -1.15e-176)
		tmp = (1.0 / x) + (-1.0 - t_1);
	elseif (t <= 2.05e-174)
		tmp = t * (sqrt(x) / l);
	else
		tmp = (1.0 + t_1) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.15e-176], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e-174], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1500000000000001e-176

   1. Initial program 37.2%

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

      1. associate-*l/ 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in t around -inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. *-commutative 80.6%

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

      3. distribute-rgt-neg-in 80.6%

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

      4. +-commutative 80.6%

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

      5. sub-neg 80.6%

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

      6. metadata-eval 80.6%

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

      7. +-commutative 80.6%

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

      8. *-commutative 80.6%

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

      9. distribute-rgt-neg-in 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in x around inf 80.5%

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

      1. associate-*r/ 80.5%

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

      2. metadata-eval 80.5%

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

      3. unpow2 80.5%

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

   9. Simplified 80.5%

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

   if -1.1500000000000001e-176 < t < 2.05e-174

   1. Initial program 4.0%

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

      1. associate-*l/ 4.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in x around inf 57.4%

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

      1. associate--l+ 57.4%

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

      2. unpow2 57.4%

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

      3. distribute-lft-out 57.4%

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

      4. unpow2 57.4%

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

      5. unpow2 57.4%

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

      6. associate-*r/ 57.4%

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

      7. mul-1-neg 57.4%

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

      8. +-commutative 57.4%

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

      9. unpow2 57.4%

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

      10. unpow2 57.4%

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

      11. fma-udef 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around 0 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 57.2%

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

      2. metadata-eval 57.2%

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

      3. distribute-rgt1-in 57.2%

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

      4. metadata-eval 57.2%

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

      5. unpow2 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in l around 0 49.8%

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

       1. associate-*l/ 49.9%

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

       2. *-lft-identity 49.9%

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

   12. Simplified 49.9%

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

   if 2.05e-174 < t

   1. Initial program 32.8%

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

      1. associate-*r/ 32.8%

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

      2. fma-neg 32.8%

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

      3. remove-double-neg 32.8%

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

      4. fma-neg 32.8%

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

      5. sub-neg 32.8%

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

      6. metadata-eval 32.8%

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

      7. remove-double-neg 32.8%

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

      8. +-commutative 32.8%

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

      9. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around inf 31.3%

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

      1. associate-/l* 31.1%

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

      2. +-commutative 31.1%

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

      3. sub-neg 31.1%

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

      4. metadata-eval 31.1%

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

      5. +-commutative 31.1%

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

      6. unpow2 31.1%

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

   6. Simplified 31.1%

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

   7. Taylor expanded in x around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. metadata-eval 81.5%

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

      3. unpow2 81.5%

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

   9. Simplified 81.5%

      \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) - \frac{1}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-174}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 8: 79.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_1\right)\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-176}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_1\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -7.2e-176)
     (+ (/ 1.0 x) (- -1.0 t_1))
     (if (<= t 2.75e-176) (/ (* t (sqrt x)) l) (+ (+ 1.0 t_1) (/ -1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -7.2e-176) {
		tmp = (1.0 / x) + (-1.0 - t_1);
	} else if (t <= 2.75e-176) {
		tmp = (t * sqrt(x)) / l;
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-7.2d-176)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - t_1)
    else if (t <= 2.75d-176) then
        tmp = (t * sqrt(x)) / l
    else
        tmp = (1.0d0 + t_1) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -7.2e-176) {
		tmp = (1.0 / x) + (-1.0 - t_1);
	} else if (t <= 2.75e-176) {
		tmp = (t * Math.sqrt(x)) / l;
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -7.2e-176:
		tmp = (1.0 / x) + (-1.0 - t_1)
	elif t <= 2.75e-176:
		tmp = (t * math.sqrt(x)) / l
	else:
		tmp = (1.0 + t_1) + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -7.2e-176)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - t_1));
	elseif (t <= 2.75e-176)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	else
		tmp = Float64(Float64(1.0 + t_1) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -7.2e-176)
		tmp = (1.0 / x) + (-1.0 - t_1);
	elseif (t <= 2.75e-176)
		tmp = (t * sqrt(x)) / l;
	else
		tmp = (1.0 + t_1) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e-176], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.75e-176], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.2000000000000005e-176

   1. Initial program 37.2%

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

      1. associate-*l/ 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in t around -inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. *-commutative 80.6%

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

      3. distribute-rgt-neg-in 80.6%

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

      4. +-commutative 80.6%

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

      5. sub-neg 80.6%

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

      6. metadata-eval 80.6%

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

      7. +-commutative 80.6%

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

      8. *-commutative 80.6%

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

      9. distribute-rgt-neg-in 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in x around inf 80.5%

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

      1. associate-*r/ 80.5%

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

      2. metadata-eval 80.5%

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

      3. unpow2 80.5%

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

   9. Simplified 80.5%

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

   if -7.2000000000000005e-176 < t < 2.75e-176

   1. Initial program 4.0%

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

      1. associate-*l/ 4.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in x around inf 57.4%

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

      1. associate--l+ 57.4%

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

      2. unpow2 57.4%

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

      3. distribute-lft-out 57.4%

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

      4. unpow2 57.4%

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

      5. unpow2 57.4%

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

      6. associate-*r/ 57.4%

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

      7. mul-1-neg 57.4%

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

      8. +-commutative 57.4%

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

      9. unpow2 57.4%

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

      10. unpow2 57.4%

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

      11. fma-udef 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around 0 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 57.2%

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

      2. metadata-eval 57.2%

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

      3. distribute-rgt1-in 57.2%

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

      4. metadata-eval 57.2%

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

      5. unpow2 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in l around 0 44.6%

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 50.0%

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

   12. Applied egg-rr 50.0%

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

   if 2.75e-176 < t

   1. Initial program 32.8%

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

      1. associate-*r/ 32.8%

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

      2. fma-neg 32.8%

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

      3. remove-double-neg 32.8%

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

      4. fma-neg 32.8%

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

      5. sub-neg 32.8%

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

      6. metadata-eval 32.8%

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

      7. remove-double-neg 32.8%

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

      8. +-commutative 32.8%

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

      9. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around inf 31.3%

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

      1. associate-/l* 31.1%

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

      2. +-commutative 31.1%

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

      3. sub-neg 31.1%

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

      4. metadata-eval 31.1%

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

      5. +-commutative 31.1%

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

      6. unpow2 31.1%

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

   6. Simplified 31.1%

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

   7. Taylor expanded in x around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. metadata-eval 81.5%

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

      3. unpow2 81.5%

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

   9. Simplified 81.5%

      \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) - \frac{1}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-176}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 9: 75.6% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) (+ -1.0 (/ 1.0 x)) (+ (+ 1.0 (/ 0.5 (* x x))) (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1d-309)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = (1.0d0 + (0.5d0 / (x * x))) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-309], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 31.0%

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

      1. associate-*l/ 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around -inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. *-commutative 72.0%

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

      3. distribute-rgt-neg-in 72.0%

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

      4. +-commutative 72.0%

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

      5. sub-neg 72.0%

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

      6. metadata-eval 72.0%

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

      7. +-commutative 72.0%

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

      8. *-commutative 72.0%

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

      9. distribute-rgt-neg-in 72.0%

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

   6. Simplified 72.0%

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

   7. Taylor expanded in x around inf 71.8%

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

   if -1.000000000000002e-309 < t

   1. Initial program 28.1%

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

      1. associate-*r/ 28.0%

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

      2. fma-neg 28.0%

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

      3. remove-double-neg 28.0%

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

      4. fma-neg 28.0%

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

      5. sub-neg 28.0%

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

      6. metadata-eval 28.0%

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

      7. remove-double-neg 28.0%

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

      8. +-commutative 28.0%

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

      9. fma-def 28.0%

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

   3. Simplified 28.0%

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

   4. Taylor expanded in t around inf 26.2%

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

      1. associate-/l* 26.0%

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

      2. +-commutative 26.0%

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

      3. sub-neg 26.0%

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

      4. metadata-eval 26.0%

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

      5. +-commutative 26.0%

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

      6. unpow2 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in x around inf 72.2%

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

      1. associate-*r/ 72.2%

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

      2. metadata-eval 72.2%

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

      3. unpow2 72.2%

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

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 10: 75.8% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_1\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -1e-309) (+ (/ 1.0 x) (- -1.0 t_1)) (+ (+ 1.0 t_1) (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -1e-309) {
		tmp = (1.0 / x) + (-1.0 - t_1);
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-1d-309)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - t_1)
    else
        tmp = (1.0d0 + t_1) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -1e-309) {
		tmp = (1.0 / x) + (-1.0 - t_1);
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -1e-309:
		tmp = (1.0 / x) + (-1.0 - t_1)
	else:
		tmp = (1.0 + t_1) + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - t_1));
	else
		tmp = Float64(Float64(1.0 + t_1) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = (1.0 / x) + (-1.0 - t_1);
	else
		tmp = (1.0 + t_1) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e-309], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 31.0%

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

      1. associate-*l/ 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around -inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. *-commutative 72.0%

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

      3. distribute-rgt-neg-in 72.0%

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

      4. +-commutative 72.0%

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

      5. sub-neg 72.0%

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

      6. metadata-eval 72.0%

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

      7. +-commutative 72.0%

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

      8. *-commutative 72.0%

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

      9. distribute-rgt-neg-in 72.0%

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

   6. Simplified 72.0%

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

   7. Taylor expanded in x around inf 71.9%

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

      1. associate-*r/ 71.9%

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

      2. metadata-eval 71.9%

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

      3. unpow2 71.9%

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

   9. Simplified 71.9%

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

   if -1.000000000000002e-309 < t

   1. Initial program 28.1%

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

      1. associate-*r/ 28.0%

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

      2. fma-neg 28.0%

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

      3. remove-double-neg 28.0%

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

      4. fma-neg 28.0%

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

      5. sub-neg 28.0%

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

      6. metadata-eval 28.0%

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

      7. remove-double-neg 28.0%

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

      8. +-commutative 28.0%

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

      9. fma-def 28.0%

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

   3. Simplified 28.0%

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

   4. Taylor expanded in t around inf 26.2%

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

      1. associate-/l* 26.0%

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

      2. +-commutative 26.0%

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

      3. sub-neg 26.0%

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

      4. metadata-eval 26.0%

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

      5. +-commutative 26.0%

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

      6. unpow2 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in x around inf 72.2%

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

      1. associate-*r/ 72.2%

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

      2. metadata-eval 72.2%

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

      3. unpow2 72.2%

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

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 11: 75.3% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) -1.0 (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1d-309)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-309], -1.0, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 31.0%

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

      1. associate-*l/ 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around -inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. *-commutative 72.0%

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

      3. distribute-rgt-neg-in 72.0%

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

      4. +-commutative 72.0%

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

      5. sub-neg 72.0%

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

      6. metadata-eval 72.0%

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

      7. +-commutative 72.0%

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

      8. *-commutative 72.0%

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

      9. distribute-rgt-neg-in 72.0%

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

   6. Simplified 72.0%

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

   7. Taylor expanded in x around inf 71.5%

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

   if -1.000000000000002e-309 < t

   1. Initial program 28.1%

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

      1. associate-*r/ 28.0%

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

      2. fma-neg 28.0%

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

      3. remove-double-neg 28.0%

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

      4. fma-neg 28.0%

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

      5. sub-neg 28.0%

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

      6. metadata-eval 28.0%

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

      7. remove-double-neg 28.0%

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

      8. +-commutative 28.0%

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

      9. fma-def 28.0%

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

   3. Simplified 28.0%

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

   4. Taylor expanded in t around inf 26.2%

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

      1. associate-/l* 26.0%

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

      2. +-commutative 26.0%

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

      3. sub-neg 26.0%

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

      4. metadata-eval 26.0%

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

      5. +-commutative 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \]

      6. unpow2 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \]

   6. Simplified 26.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \]

   7. Taylor expanded in x around inf 71.7%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12: 75.6% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) (+ -1.0 (/ 1.0 x)) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1d-309)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-309], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 31.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 31.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 31.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around -inf 72.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \cdot t \]
   5. Step-by-step derivation

      1. mul-1-neg 72.0%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

      2. *-commutative 72.0%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \cdot t \]

      3. distribute-rgt-neg-in 72.0%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)}} \cdot t \]

      4. +-commutative 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      5. sub-neg 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      6. metadata-eval 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      7. +-commutative 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      8. *-commutative 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-\color{blue}{t \cdot \sqrt{2}}\right)} \cdot t \]

      9. distribute-rgt-neg-in 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

   6. Simplified 72.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

   7. Taylor expanded in x around inf 71.8%

      \[\leadsto \color{blue}{\frac{1}{x} - 1} \]

   if -1.000000000000002e-309 < t

   1. Initial program 28.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 28.0%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 28.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   4. Taylor expanded in t around inf 26.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \]

      2. +-commutative 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \]

      3. sub-neg 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \]

      4. metadata-eval 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \]

      5. +-commutative 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \]

      6. unpow2 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \]

   6. Simplified 26.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \]

   7. Taylor expanded in x around inf 71.7%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 13: 75.0% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -1e-309) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1d-309)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
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

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
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

NOTE: l should be positive before calling this function
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 31.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 31.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 31.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around -inf 72.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \cdot t \]
   5. Step-by-step derivation

      1. mul-1-neg 72.0%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

      2. *-commutative 72.0%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \cdot t \]

      3. distribute-rgt-neg-in 72.0%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)}} \cdot t \]

      4. +-commutative 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      5. sub-neg 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      6. metadata-eval 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      7. +-commutative 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      8. *-commutative 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-\color{blue}{t \cdot \sqrt{2}}\right)} \cdot t \]

      9. distribute-rgt-neg-in 72.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

   6. Simplified 72.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

   7. Taylor expanded in x around inf 71.5%

      \[\leadsto \color{blue}{-1} \]

   if -1.000000000000002e-309 < t

   1. Initial program 28.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 28.0%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 28.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 28.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   4. Taylor expanded in t around inf 26.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \]

      2. +-commutative 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \]

      3. sub-neg 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \]

      4. metadata-eval 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \]

      5. +-commutative 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \]

      6. unpow2 26.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \]

   6. Simplified 26.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \]

   7. Taylor expanded in x around inf 70.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 38.2% accurate, 225.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ -1 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 -1.0)

C

l = abs(l);
double code(double x, double l, double t) {
	return -1.0;
}

Fortran

NOTE: l should be positive before calling this function
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

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return -1.0;
}

Python

l = abs(l)
def code(x, l, t):
	return -1.0

Julia

l = abs(l)
function code(x, l, t)
	return -1.0
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := -1.0

Derivation

1. Initial program 29.5%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 29.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 29.5%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

4. Taylor expanded in t around -inf 36.3%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \cdot t \]
5. Step-by-step derivation

   1. mul-1-neg 36.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   2. *-commutative 36.3%

      \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \cdot t \]

   3. distribute-rgt-neg-in 36.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)}} \cdot t \]

   4. +-commutative 36.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

   5. sub-neg 36.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

   6. metadata-eval 36.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

   7. +-commutative 36.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

   8. *-commutative 36.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-\color{blue}{t \cdot \sqrt{2}}\right)} \cdot t \]

   9. distribute-rgt-neg-in 36.3%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

6. Simplified 36.3%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

7. Taylor expanded in x around inf 36.1%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 36.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023230 
(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)))))