Toniolo and Linder, Equation (7)

Percentage Accurate: 32.9% → 87.7%

Time: 34.5s

Alternatives: 19

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: 32.9% 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: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{\frac{2}{t_1 + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, t_1\right)\right)}}\\ t_3 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_3}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{t_3}}\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-225}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* l (/ l x)))
        (t_2
         (*
          t
          (sqrt
           (/ 2.0 (+ t_1 (fma 2.0 (/ t (/ x t)) (fma 2.0 (* t t) t_1)))))))
        (t_3 (+ 2.0 (/ 4.0 x))))
   (if (<= t -7.6e+82)
     (- (+ (/ 1.0 x) -1.0) (/ 0.5 (* x x)))
     (if (<= t -4.9e-154)
       t_2
       (if (<= t -3.5e-308)
         (*
          t
          (/
           (sqrt 2.0)
           (- (fma t (sqrt t_3) (* (* (/ l x) (/ l t)) (sqrt (/ 1.0 t_3)))))))
         (if (<= t 1.8e-225)
           (* t (/ (sqrt 2.0) (* l (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
           (if (<= t 9.5e-151)
             (*
              t
              (/
               (sqrt 2.0)
               (fma
                0.5
                (/ (* 2.0 (fma 2.0 (* t t) (* l l))) (* (sqrt 2.0) (* t x)))
                (* t (sqrt 2.0)))))
             (if (<= t 4.5e+43) t_2 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = l * (l / x);
	double t_2 = t * sqrt((2.0 / (t_1 + fma(2.0, (t / (x / t)), fma(2.0, (t * t), t_1)))));
	double t_3 = 2.0 + (4.0 / x);
	double tmp;
	if (t <= -7.6e+82) {
		tmp = ((1.0 / x) + -1.0) - (0.5 / (x * x));
	} else if (t <= -4.9e-154) {
		tmp = t_2;
	} else if (t <= -3.5e-308) {
		tmp = t * (sqrt(2.0) / -fma(t, sqrt(t_3), (((l / x) * (l / t)) * sqrt((1.0 / t_3)))));
	} else if (t <= 1.8e-225) {
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t <= 9.5e-151) {
		tmp = t * (sqrt(2.0) / fma(0.5, ((2.0 * fma(2.0, (t * t), (l * l))) / (sqrt(2.0) * (t * x))), (t * sqrt(2.0))));
	} else if (t <= 4.5e+43) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(l * Float64(l / x))
	t_2 = Float64(t * sqrt(Float64(2.0 / Float64(t_1 + fma(2.0, Float64(t / Float64(x / t)), fma(2.0, Float64(t * t), t_1))))))
	t_3 = Float64(2.0 + Float64(4.0 / x))
	tmp = 0.0
	if (t <= -7.6e+82)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) - Float64(0.5 / Float64(x * x)));
	elseif (t <= -4.9e-154)
		tmp = t_2;
	elseif (t <= -3.5e-308)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-fma(t, sqrt(t_3), Float64(Float64(Float64(l / x) * Float64(l / t)) * sqrt(Float64(1.0 / t_3)))))));
	elseif (t <= 1.8e-225)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t <= 9.5e-151)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(2.0 * fma(2.0, Float64(t * t), Float64(l * l))) / Float64(sqrt(2.0) * Float64(t * x))), Float64(t * sqrt(2.0)))));
	elseif (t <= 4.5e+43)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[N[(2.0 / N[(t$95$1 + N[(2.0 * N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.6e+82], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.9e-154], t$95$2, If[LessEqual[t, -3.5e-308], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-N[(t * N[Sqrt[t$95$3], $MachinePrecision] + N[(N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8e-225], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-151], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(2.0 * N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e+43], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -7.60000000000000067e82

   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 91.1%

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

      1. mul-1-neg 91.1%

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

      2. *-commutative 91.1%

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

      3. distribute-rgt-neg-in 91.1%

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

      4. +-commutative 91.1%

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

      5. sub-neg 91.1%

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

      6. metadata-eval 91.1%

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

      7. +-commutative 91.1%

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

      8. distribute-rgt-neg-in 91.1%

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

   6. Simplified 91.1%

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

   7. Taylor expanded in x around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. metadata-eval 91.4%

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

      3. unpow2 91.4%

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

   9. Simplified 91.4%

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

   10. Taylor expanded in x around 0 91.4%

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

       1. associate--r+ 91.5%

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

       2. sub-neg 91.5%

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

       3. metadata-eval 91.5%

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

       4. associate-*r/ 91.5%

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

       5. metadata-eval 91.5%

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

       6. unpow2 91.5%

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

   12. Simplified 91.5%

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

   if -7.60000000000000067e82 < t < -4.89999999999999997e-154 or 9.4999999999999996e-151 < t < 4.5e43

   1. Initial program 61.6%

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

      1. associate-*l/ 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in x around inf 83.0%

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

      1. sub-neg 83.0%

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

      2. fma-def 83.0%

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

      3. unpow2 83.0%

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

      4. fma-def 83.0%

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

      5. unpow2 83.0%

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

      6. unpow2 83.0%

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

      7. mul-1-neg 83.0%

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

      8. remove-double-neg 83.0%

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

      9. fma-def 83.0%

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

      10. unpow2 83.0%

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

      11. unpow2 83.0%

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

   6. Simplified 83.0%

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

   7. Taylor expanded in t around 0 82.6%

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

      1. unpow2 82.6%

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

   9. Simplified 82.6%

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

       1. sqrt-undiv 82.7%

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

       2. associate-/l* 82.7%

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

       3. associate-/l* 82.7%

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

       4. associate-/r/ 82.7%

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

       5. associate-/l* 93.3%

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

       6. associate-/r/ 93.3%

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

   11. Applied egg-rr 93.3%

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

   if -4.89999999999999997e-154 < t < -3.5e-308

   1. Initial program 2.4%

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

      1. associate-*l/ 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in x around inf 38.6%

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

      1. sub-neg 38.6%

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

      2. fma-def 38.6%

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

      3. unpow2 38.6%

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

      4. fma-def 38.6%

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

      5. unpow2 38.6%

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

      6. unpow2 38.6%

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

      7. mul-1-neg 38.6%

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

      8. remove-double-neg 38.6%

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

      9. fma-def 38.6%

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

      10. unpow2 38.6%

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

      11. unpow2 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in t around -inf 79.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}\right) + -1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)}} \cdot t \]
   8. Step-by-step derivation

      1. distribute-lft-out 79.5%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}} + \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)}} \cdot t \]

      2. mul-1-neg 79.5%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}} + \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)}} \cdot t \]

      3. fma-def 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\mathsf{fma}\left(t, \sqrt{2 + 4 \cdot \frac{1}{x}}, \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)}} \cdot t \]

      4. associate-*r/ 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}, \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      5. metadata-eval 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{\color{blue}{4}}{x}}, \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      6. unpow2 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \frac{\color{blue}{\ell \cdot \ell}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      7. *-commutative 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \frac{\ell \cdot \ell}{\color{blue}{x \cdot t}} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      8. times-frac 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \color{blue}{\left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      9. associate-*r/ 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{2 + \color{blue}{\frac{4 \cdot 1}{x}}}}\right)} \cdot t \]

      10. metadata-eval 79.5%

          \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{2 + \frac{\color{blue}{4}}{x}}}\right)} \cdot t \]

   9. Simplified 79.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{2 + \frac{4}{x}}}\right)}} \cdot t \]

   if -3.5e-308 < t < 1.80000000000000005e-225

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in x around inf 73.1%

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

      1. sub-neg 73.1%

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

      2. fma-def 73.1%

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

      3. unpow2 73.1%

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

      4. fma-def 73.1%

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

      5. unpow2 73.1%

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

      6. unpow2 73.1%

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

      7. mul-1-neg 73.1%

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

      8. remove-double-neg 73.1%

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

      9. fma-def 73.1%

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

      10. unpow2 73.1%

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

      11. unpow2 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in t around 0 82.1%

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

      1. associate-*l* 82.1%

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

   9. Simplified 82.1%

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

   if 1.80000000000000005e-225 < t < 9.4999999999999996e-151

   1. Initial program 7.8%

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

      1. associate-*l/ 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in x around inf 70.0%

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

      1. fma-def 70.0%

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

      2. cancel-sign-sub-inv 70.0%

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

      3. metadata-eval 70.0%

         \[\leadsto \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \color{blue}{1} \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)} \cdot t \]

      4. distribute-rgt1-in 70.0%

         \[\leadsto \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\color{blue}{\left(1 + 1\right) \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{t \cdot \left(x \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)} \cdot t \]

      5. metadata-eval 70.0%

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

      6. fma-def 70.0%

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

      7. unpow2 70.0%

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

      8. unpow2 70.0%

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

      9. associate-*r* 70.0%

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

   6. Simplified 70.0%

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

   if 4.5e43 < t

   1. Initial program 48.1%

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

      1. associate-*l/ 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in t around inf 94.9%

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

   5. Taylor expanded in t around 0 95.3%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{2 + \frac{4}{x}}}\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-225}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 2: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{\frac{2}{t_1 + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, t_1\right)\right)}}\\ t_3 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_3}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{t_3}}\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* l (/ l x)))
        (t_2
         (*
          t
          (sqrt
           (/ 2.0 (+ t_1 (fma 2.0 (/ t (/ x t)) (fma 2.0 (* t t) t_1)))))))
        (t_3 (+ 2.0 (/ 4.0 x))))
   (if (<= t -5e+82)
     (- (+ (/ 1.0 x) -1.0) (/ 0.5 (* x x)))
     (if (<= t -4.9e-154)
       t_2
       (if (<= t -2.8e-308)
         (*
          t
          (/
           (sqrt 2.0)
           (- (fma t (sqrt t_3) (* (* (/ l x) (/ l t)) (sqrt (/ 1.0 t_3)))))))
         (if (<= t 4.5e+43) t_2 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = l * (l / x);
	double t_2 = t * sqrt((2.0 / (t_1 + fma(2.0, (t / (x / t)), fma(2.0, (t * t), t_1)))));
	double t_3 = 2.0 + (4.0 / x);
	double tmp;
	if (t <= -5e+82) {
		tmp = ((1.0 / x) + -1.0) - (0.5 / (x * x));
	} else if (t <= -4.9e-154) {
		tmp = t_2;
	} else if (t <= -2.8e-308) {
		tmp = t * (sqrt(2.0) / -fma(t, sqrt(t_3), (((l / x) * (l / t)) * sqrt((1.0 / t_3)))));
	} else if (t <= 4.5e+43) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(l * Float64(l / x))
	t_2 = Float64(t * sqrt(Float64(2.0 / Float64(t_1 + fma(2.0, Float64(t / Float64(x / t)), fma(2.0, Float64(t * t), t_1))))))
	t_3 = Float64(2.0 + Float64(4.0 / x))
	tmp = 0.0
	if (t <= -5e+82)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) - Float64(0.5 / Float64(x * x)));
	elseif (t <= -4.9e-154)
		tmp = t_2;
	elseif (t <= -2.8e-308)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-fma(t, sqrt(t_3), Float64(Float64(Float64(l / x) * Float64(l / t)) * sqrt(Float64(1.0 / t_3)))))));
	elseif (t <= 4.5e+43)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[N[(2.0 / N[(t$95$1 + N[(2.0 * N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e+82], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.9e-154], t$95$2, If[LessEqual[t, -2.8e-308], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-N[(t * N[Sqrt[t$95$3], $MachinePrecision] + N[(N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e+43], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.00000000000000015e82

   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 91.1%

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

      1. mul-1-neg 91.1%

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

      2. *-commutative 91.1%

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

      3. distribute-rgt-neg-in 91.1%

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

      4. +-commutative 91.1%

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

      5. sub-neg 91.1%

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

      6. metadata-eval 91.1%

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

      7. +-commutative 91.1%

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

      8. distribute-rgt-neg-in 91.1%

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

   6. Simplified 91.1%

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

   7. Taylor expanded in x around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. metadata-eval 91.4%

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

      3. unpow2 91.4%

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

   9. Simplified 91.4%

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

   10. Taylor expanded in x around 0 91.4%

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

       1. associate--r+ 91.5%

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

       2. sub-neg 91.5%

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

       3. metadata-eval 91.5%

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

       4. associate-*r/ 91.5%

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

       5. metadata-eval 91.5%

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

       6. unpow2 91.5%

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

   12. Simplified 91.5%

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

   if -5.00000000000000015e82 < t < -4.89999999999999997e-154 or -2.79999999999999984e-308 < t < 4.5e43

   1. Initial program 47.9%

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

      1. associate-*l/ 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in x around inf 78.3%

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

      1. sub-neg 78.3%

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

      2. fma-def 78.3%

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

      3. unpow2 78.3%

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

      4. fma-def 78.3%

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

      5. unpow2 78.3%

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

      6. unpow2 78.3%

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

      7. mul-1-neg 78.3%

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

      8. remove-double-neg 78.3%

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

      9. fma-def 78.3%

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

      10. unpow2 78.3%

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

      11. unpow2 78.3%

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

   6. Simplified 78.3%

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

   7. Taylor expanded in t around 0 77.7%

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

      1. unpow2 77.7%

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

   9. Simplified 77.7%

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

       1. sqrt-undiv 77.6%

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

       2. associate-/l* 77.6%

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

       3. associate-/l* 77.6%

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

       4. associate-/r/ 77.6%

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

       5. associate-/l* 86.4%

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

       6. associate-/r/ 86.3%

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

   11. Applied egg-rr 86.3%

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

   if -4.89999999999999997e-154 < t < -2.79999999999999984e-308

   1. Initial program 2.4%

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

      1. associate-*l/ 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in x around inf 38.6%

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

      1. sub-neg 38.6%

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

      2. fma-def 38.6%

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

      3. unpow2 38.6%

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

      4. fma-def 38.6%

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

      5. unpow2 38.6%

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

      6. unpow2 38.6%

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

      7. mul-1-neg 38.6%

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

      8. remove-double-neg 38.6%

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

      9. fma-def 38.6%

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

      10. unpow2 38.6%

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

      11. unpow2 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in t around -inf 79.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}\right) + -1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)}} \cdot t \]
   8. Step-by-step derivation

      1. distribute-lft-out 79.5%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}} + \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)}} \cdot t \]

      2. mul-1-neg 79.5%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}} + \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)}} \cdot t \]

      3. fma-def 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\mathsf{fma}\left(t, \sqrt{2 + 4 \cdot \frac{1}{x}}, \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)}} \cdot t \]

      4. associate-*r/ 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}, \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      5. metadata-eval 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{\color{blue}{4}}{x}}, \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      6. unpow2 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \frac{\color{blue}{\ell \cdot \ell}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      7. *-commutative 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \frac{\ell \cdot \ell}{\color{blue}{x \cdot t}} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      8. times-frac 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \color{blue}{\left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)} \cdot \sqrt{\frac{1}{2 + 4 \cdot \frac{1}{x}}}\right)} \cdot t \]

      9. associate-*r/ 79.5%

         \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{2 + \color{blue}{\frac{4 \cdot 1}{x}}}}\right)} \cdot t \]

      10. metadata-eval 79.5%

          \[\leadsto \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{2 + \frac{\color{blue}{4}}{x}}}\right)} \cdot t \]

   9. Simplified 79.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{2 + \frac{4}{x}}}\right)}} \cdot t \]

   if 4.5e43 < t

   1. Initial program 48.1%

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

      1. associate-*l/ 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in t around inf 94.9%

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

   5. Taylor expanded in t around 0 95.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right) \cdot \sqrt{\frac{1}{2 + \frac{4}{x}}}\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 3: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{1}{x} + -1\\ t_2 := \ell \cdot \frac{\ell}{x}\\ t_3 := t \cdot \sqrt{\frac{2}{t_2 + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, t_2\right)\right)}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+82}:\\ \;\;\;\;t_1 - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-156}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+43}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ (/ 1.0 x) -1.0))
        (t_2 (* l (/ l x)))
        (t_3
         (*
          t
          (sqrt
           (/ 2.0 (+ t_2 (fma 2.0 (/ t (/ x t)) (fma 2.0 (* t t) t_2))))))))
   (if (<= t -2e+82)
     (- t_1 (/ 0.5 (* x x)))
     (if (<= t -1.8e-156)
       t_3
       (if (<= t -3.7e-308)
         t_1
         (if (<= t 2e+43) t_3 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (1.0 / x) + -1.0;
	double t_2 = l * (l / x);
	double t_3 = t * sqrt((2.0 / (t_2 + fma(2.0, (t / (x / t)), fma(2.0, (t * t), t_2)))));
	double tmp;
	if (t <= -2e+82) {
		tmp = t_1 - (0.5 / (x * x));
	} else if (t <= -1.8e-156) {
		tmp = t_3;
	} else if (t <= -3.7e-308) {
		tmp = t_1;
	} else if (t <= 2e+43) {
		tmp = t_3;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(1.0 / x) + -1.0)
	t_2 = Float64(l * Float64(l / x))
	t_3 = Float64(t * sqrt(Float64(2.0 / Float64(t_2 + fma(2.0, Float64(t / Float64(x / t)), fma(2.0, Float64(t * t), t_2))))))
	tmp = 0.0
	if (t <= -2e+82)
		tmp = Float64(t_1 - Float64(0.5 / Float64(x * x)));
	elseif (t <= -1.8e-156)
		tmp = t_3;
	elseif (t <= -3.7e-308)
		tmp = t_1;
	elseif (t <= 2e+43)
		tmp = t_3;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Sqrt[N[(2.0 / N[(t$95$2 + N[(2.0 * N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+82], N[(t$95$1 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.8e-156], t$95$3, If[LessEqual[t, -3.7e-308], t$95$1, If[LessEqual[t, 2e+43], t$95$3, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.9999999999999999e82

   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 91.1%

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

      1. mul-1-neg 91.1%

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

      2. *-commutative 91.1%

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

      3. distribute-rgt-neg-in 91.1%

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

      4. +-commutative 91.1%

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

      5. sub-neg 91.1%

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

      6. metadata-eval 91.1%

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

      7. +-commutative 91.1%

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

      8. distribute-rgt-neg-in 91.1%

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

   6. Simplified 91.1%

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

   7. Taylor expanded in x around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. metadata-eval 91.4%

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

      3. unpow2 91.4%

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

   9. Simplified 91.4%

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

   10. Taylor expanded in x around 0 91.4%

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

       1. associate--r+ 91.5%

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

       2. sub-neg 91.5%

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

       3. metadata-eval 91.5%

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

       4. associate-*r/ 91.5%

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

       5. metadata-eval 91.5%

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

       6. unpow2 91.5%

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

   12. Simplified 91.5%

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

   if -1.9999999999999999e82 < t < -1.79999999999999999e-156 or -3.70000000000000006e-308 < t < 2.00000000000000003e43

   1. Initial program 46.8%

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

      1. associate-*l/ 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in x around inf 78.7%

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

      1. sub-neg 78.7%

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

      2. fma-def 78.7%

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

      3. unpow2 78.7%

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

      4. fma-def 78.7%

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

      5. unpow2 78.7%

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

      6. unpow2 78.7%

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

      7. mul-1-neg 78.7%

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

      8. remove-double-neg 78.7%

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

      9. fma-def 78.7%

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

      10. unpow2 78.7%

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

      11. unpow2 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in t around 0 78.2%

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

      1. unpow2 78.2%

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

   9. Simplified 78.2%

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

       1. sqrt-undiv 78.1%

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

       2. associate-/l* 78.1%

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

       3. associate-/l* 78.1%

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

       4. associate-/r/ 78.1%

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

       5. associate-/l* 86.7%

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

       6. associate-/r/ 86.6%

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

   11. Applied egg-rr 86.6%

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

   if -1.79999999999999999e-156 < t < -3.70000000000000006e-308

   1. Initial program 2.6%

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

      1. associate-*l/ 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in t around -inf 58.9%

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

      1. mul-1-neg 58.9%

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

      2. *-commutative 58.9%

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

      3. distribute-rgt-neg-in 58.9%

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

      4. +-commutative 58.9%

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

      5. sub-neg 58.9%

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

      6. metadata-eval 58.9%

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

      7. +-commutative 58.9%

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

      8. distribute-rgt-neg-in 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in x around inf 59.3%

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

   if 2.00000000000000003e43 < t

   1. Initial program 48.1%

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

      1. associate-*l/ 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in t around inf 94.9%

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

   5. Taylor expanded in t around 0 95.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 4: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-253}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\left(\frac{1}{x} + \frac{1}{x \cdot x}\right) + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -1.55e-253)
     (- t_1)
     (if (<= t 4e-127)
       (/
        (* t (sqrt 2.0))
        (* l (sqrt (+ (+ (/ 1.0 x) (/ 1.0 (* x x))) (/ 1.0 (+ x -1.0))))))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -1.55e-253) {
		tmp = -t_1;
	} else if (t <= 4e-127) {
		tmp = (t * sqrt(2.0)) / (l * sqrt((((1.0 / x) + (1.0 / (x * x))) + (1.0 / (x + -1.0)))));
	} 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)) / (1.0d0 + x)))
    if (t <= (-1.55d-253)) then
        tmp = -t_1
    else if (t <= 4d-127) then
        tmp = (t * sqrt(2.0d0)) / (l * sqrt((((1.0d0 / x) + (1.0d0 / (x * x))) + (1.0d0 / (x + (-1.0d0))))))
    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) / (1.0 + x)));
	double tmp;
	if (t <= -1.55e-253) {
		tmp = -t_1;
	} else if (t <= 4e-127) {
		tmp = (t * Math.sqrt(2.0)) / (l * Math.sqrt((((1.0 / x) + (1.0 / (x * x))) + (1.0 / (x + -1.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -1.55e-253:
		tmp = -t_1
	elif t <= 4e-127:
		tmp = (t * math.sqrt(2.0)) / (l * math.sqrt((((1.0 / x) + (1.0 / (x * x))) + (1.0 / (x + -1.0)))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -1.55e-253)
		tmp = Float64(-t_1);
	elseif (t <= 4e-127)
		tmp = Float64(Float64(t * sqrt(2.0)) / Float64(l * sqrt(Float64(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x * x))) + Float64(1.0 / Float64(x + -1.0))))));
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -1.55e-253)
		tmp = -t_1;
	elseif (t <= 4e-127)
		tmp = (t * sqrt(2.0)) / (l * sqrt((((1.0 / x) + (1.0 / (x * x))) + (1.0 / (x + -1.0)))));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.55e-253], (-t$95$1), If[LessEqual[t, 4e-127], N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.54999999999999998e-253

   1. Initial program 38.6%

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

      1. associate-*l/ 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in t around -inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. *-commutative 77.4%

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

      3. distribute-rgt-neg-in 77.4%

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

      4. +-commutative 77.4%

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

      5. sub-neg 77.4%

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

      6. metadata-eval 77.4%

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

      7. +-commutative 77.4%

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

      8. distribute-rgt-neg-in 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in t around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. sub-neg 77.7%

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

      3. metadata-eval 77.7%

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

      4. +-commutative 77.7%

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

   9. Simplified 77.7%

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

   if -1.54999999999999998e-253 < t < 4.0000000000000001e-127

   1. Initial program 4.7%

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

      1. associate-*l/ 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in l around inf 5.9%

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

      1. associate--l+ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

      4. +-commutative 29.5%

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

      5. sub-neg 29.5%

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

      6. metadata-eval 29.5%

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

      7. +-commutative 29.5%

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

   6. Simplified 29.5%

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

      1. associate-*l/ 29.5%

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

      2. *-commutative 29.5%

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

      3. +-commutative 29.5%

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

      4. sub-neg 29.5%

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

      5. +-commutative 29.5%

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

      6. metadata-eval 29.5%

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

      7. +-commutative 29.5%

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

   8. Applied egg-rr 29.5%

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

   9. Taylor expanded in x around inf 43.7%

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

       1. unpow2 43.7%

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

   11. Simplified 43.7%

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

   if 4.0000000000000001e-127 < t

   1. Initial program 59.1%

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

      1. associate-*l/ 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in t around inf 88.7%

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

   5. Taylor expanded in t around 0 89.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-253}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\left(\frac{1}{x} + \frac{1}{x \cdot x}\right) + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 5: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-253}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-129}:\\ \;\;\;\;\frac{t}{\ell} \cdot \frac{\sqrt{2}}{\sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -2.4e-253)
     (- t_1)
     (if (<= t 2.85e-129)
       (* (/ t l) (/ (sqrt 2.0) (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x))))))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -2.4e-253) {
		tmp = -t_1;
	} else if (t <= 2.85e-129) {
		tmp = (t / l) * (sqrt(2.0) / sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} 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)) / (1.0d0 + x)))
    if (t <= (-2.4d-253)) then
        tmp = -t_1
    else if (t <= 2.85d-129) then
        tmp = (t / l) * (sqrt(2.0d0) / sqrt(((2.0d0 / x) + (2.0d0 / (x * x)))))
    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) / (1.0 + x)));
	double tmp;
	if (t <= -2.4e-253) {
		tmp = -t_1;
	} else if (t <= 2.85e-129) {
		tmp = (t / l) * (Math.sqrt(2.0) / Math.sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -2.4e-253:
		tmp = -t_1
	elif t <= 2.85e-129:
		tmp = (t / l) * (math.sqrt(2.0) / math.sqrt(((2.0 / x) + (2.0 / (x * x)))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -2.4e-253)
		tmp = Float64(-t_1);
	elseif (t <= 2.85e-129)
		tmp = Float64(Float64(t / l) * Float64(sqrt(2.0) / sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))))));
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -2.4e-253)
		tmp = -t_1;
	elseif (t <= 2.85e-129)
		tmp = (t / l) * (sqrt(2.0) / sqrt(((2.0 / x) + (2.0 / (x * x)))));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.4e-253], (-t$95$1), If[LessEqual[t, 2.85e-129], N[(N[(t / l), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.40000000000000009e-253

   1. Initial program 38.6%

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

      1. associate-*l/ 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in t around -inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. *-commutative 77.4%

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

      3. distribute-rgt-neg-in 77.4%

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

      4. +-commutative 77.4%

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

      5. sub-neg 77.4%

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

      6. metadata-eval 77.4%

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

      7. +-commutative 77.4%

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

      8. distribute-rgt-neg-in 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in t around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. sub-neg 77.7%

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

      3. metadata-eval 77.7%

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

      4. +-commutative 77.7%

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

   9. Simplified 77.7%

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

   if -2.40000000000000009e-253 < t < 2.85e-129

   1. Initial program 4.7%

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

      1. associate-*l/ 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in l around inf 5.9%

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

      1. associate--l+ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

      4. +-commutative 29.5%

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

      5. sub-neg 29.5%

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

      6. metadata-eval 29.5%

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

      7. +-commutative 29.5%

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

   6. Simplified 29.5%

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

      1. associate-*l/ 29.5%

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

      2. *-commutative 29.5%

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

      3. +-commutative 29.5%

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

      4. sub-neg 29.5%

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

      5. +-commutative 29.5%

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

      6. metadata-eval 29.5%

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

      7. +-commutative 29.5%

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

   8. Applied egg-rr 29.5%

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

      1. times-frac 29.3%

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

      2. associate-+l+ 5.6%

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

   10. Applied egg-rr 5.6%

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

   11. Taylor expanded in x around inf 41.6%

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

       1. associate-*r/ 41.6%

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

       2. metadata-eval 41.6%

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

       3. associate-*r/ 41.6%

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

       4. metadata-eval 41.6%

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

       5. unpow2 41.6%

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

   13. Simplified 41.6%

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

   if 2.85e-129 < t

   1. Initial program 59.1%

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

      1. associate-*l/ 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in t around inf 88.7%

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

   5. Taylor expanded in t around 0 89.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-253}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-129}:\\ \;\;\;\;\frac{t}{\ell} \cdot \frac{\sqrt{2}}{\sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 6: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{-308}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-129}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -3.7e-308)
     (- t_1)
     (if (<= t 5.4e-129)
       (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0)))))))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -3.7e-308) {
		tmp = -t_1;
	} else if (t <= 5.4e-129) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	} 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)) / (1.0d0 + x)))
    if (t <= (-3.7d-308)) then
        tmp = -t_1
    else if (t <= 5.4d-129) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0)))))))
    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) / (1.0 + x)));
	double tmp;
	if (t <= -3.7e-308) {
		tmp = -t_1;
	} else if (t <= 5.4e-129) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -3.7e-308:
		tmp = -t_1
	elif t <= 5.4e-129:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -3.7e-308)
		tmp = Float64(-t_1);
	elseif (t <= 5.4e-129)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0)))))));
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -3.7e-308)
		tmp = -t_1;
	elseif (t <= 5.4e-129)
		tmp = t * (sqrt(2.0) / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.7e-308], (-t$95$1), If[LessEqual[t, 5.4e-129], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.70000000000000006e-308

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around -inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. +-commutative 75.1%

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

      5. sub-neg 75.1%

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

      6. metadata-eval 75.1%

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

      7. +-commutative 75.1%

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

      8. distribute-rgt-neg-in 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in t around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

      4. +-commutative 75.4%

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

   9. Simplified 75.4%

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

   if -3.70000000000000006e-308 < t < 5.39999999999999998e-129

   1. Initial program 5.1%

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

      1. associate-*l/ 5.1%

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

   3. Simplified 5.1%

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

   4. Taylor expanded in l around inf 4.4%

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

      1. associate--l+ 31.1%

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

      2. sub-neg 31.1%

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

      3. metadata-eval 31.1%

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

      4. +-commutative 31.1%

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

      5. sub-neg 31.1%

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

      6. metadata-eval 31.1%

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

      7. +-commutative 31.1%

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

   6. Simplified 31.1%

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

   7. Taylor expanded in x around inf 50.2%

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

   if 5.39999999999999998e-129 < t

   1. Initial program 59.1%

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

      1. associate-*l/ 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in t around inf 88.7%

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

   5. Taylor expanded in t around 0 89.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-308}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-129}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 7: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-253}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-129}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -1.55e-253)
     (- t_1)
     (if (<= t 4.4e-129)
       (/ (* t (sqrt 2.0)) (* l (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0))))))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -1.55e-253) {
		tmp = -t_1;
	} else if (t <= 4.4e-129) {
		tmp = (t * sqrt(2.0)) / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	} 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)) / (1.0d0 + x)))
    if (t <= (-1.55d-253)) then
        tmp = -t_1
    else if (t <= 4.4d-129) then
        tmp = (t * sqrt(2.0d0)) / (l * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0))))))
    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) / (1.0 + x)));
	double tmp;
	if (t <= -1.55e-253) {
		tmp = -t_1;
	} else if (t <= 4.4e-129) {
		tmp = (t * Math.sqrt(2.0)) / (l * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -1.55e-253:
		tmp = -t_1
	elif t <= 4.4e-129:
		tmp = (t * math.sqrt(2.0)) / (l * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0)))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -1.55e-253)
		tmp = Float64(-t_1);
	elseif (t <= 4.4e-129)
		tmp = Float64(Float64(t * sqrt(2.0)) / Float64(l * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0))))));
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -1.55e-253)
		tmp = -t_1;
	elseif (t <= 4.4e-129)
		tmp = (t * sqrt(2.0)) / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.55e-253], (-t$95$1), If[LessEqual[t, 4.4e-129], N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.54999999999999998e-253

   1. Initial program 38.6%

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

      1. associate-*l/ 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in t around -inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. *-commutative 77.4%

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

      3. distribute-rgt-neg-in 77.4%

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

      4. +-commutative 77.4%

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

      5. sub-neg 77.4%

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

      6. metadata-eval 77.4%

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

      7. +-commutative 77.4%

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

      8. distribute-rgt-neg-in 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in t around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. sub-neg 77.7%

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

      3. metadata-eval 77.7%

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

      4. +-commutative 77.7%

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

   9. Simplified 77.7%

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

   if -1.54999999999999998e-253 < t < 4.40000000000000006e-129

   1. Initial program 4.7%

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

      1. associate-*l/ 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in l around inf 5.9%

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

      1. associate--l+ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

      4. +-commutative 29.5%

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

      5. sub-neg 29.5%

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

      6. metadata-eval 29.5%

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

      7. +-commutative 29.5%

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

   6. Simplified 29.5%

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

      1. associate-*l/ 29.5%

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

      2. *-commutative 29.5%

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

      3. +-commutative 29.5%

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

      4. sub-neg 29.5%

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

      5. +-commutative 29.5%

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

      6. metadata-eval 29.5%

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

      7. +-commutative 29.5%

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

   8. Applied egg-rr 29.5%

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

   9. Taylor expanded in x around inf 43.5%

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

   if 4.40000000000000006e-129 < t

   1. Initial program 59.1%

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

      1. associate-*l/ 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in t around inf 88.7%

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

   5. Taylor expanded in t around 0 89.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-253}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-129}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 8: 78.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{-253}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-129}:\\ \;\;\;\;\frac{t}{\ell} \cdot \frac{\sqrt{2}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -1.25e-253)
     (- t_1)
     (if (<= t 2.85e-129) (* (/ t l) (/ (sqrt 2.0) (sqrt (/ 2.0 x)))) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -1.25e-253) {
		tmp = -t_1;
	} else if (t <= 2.85e-129) {
		tmp = (t / l) * (sqrt(2.0) / sqrt((2.0 / x)));
	} 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)) / (1.0d0 + x)))
    if (t <= (-1.25d-253)) then
        tmp = -t_1
    else if (t <= 2.85d-129) then
        tmp = (t / l) * (sqrt(2.0d0) / sqrt((2.0d0 / x)))
    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) / (1.0 + x)));
	double tmp;
	if (t <= -1.25e-253) {
		tmp = -t_1;
	} else if (t <= 2.85e-129) {
		tmp = (t / l) * (Math.sqrt(2.0) / Math.sqrt((2.0 / x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -1.25e-253:
		tmp = -t_1
	elif t <= 2.85e-129:
		tmp = (t / l) * (math.sqrt(2.0) / math.sqrt((2.0 / x)))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -1.25e-253)
		tmp = Float64(-t_1);
	elseif (t <= 2.85e-129)
		tmp = Float64(Float64(t / l) * Float64(sqrt(2.0) / sqrt(Float64(2.0 / x))));
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -1.25e-253)
		tmp = -t_1;
	elseif (t <= 2.85e-129)
		tmp = (t / l) * (sqrt(2.0) / sqrt((2.0 / x)));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.25e-253], (-t$95$1), If[LessEqual[t, 2.85e-129], N[(N[(t / l), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.24999999999999993e-253

   1. Initial program 38.6%

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

      1. associate-*l/ 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in t around -inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. *-commutative 77.4%

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

      3. distribute-rgt-neg-in 77.4%

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

      4. +-commutative 77.4%

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

      5. sub-neg 77.4%

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

      6. metadata-eval 77.4%

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

      7. +-commutative 77.4%

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

      8. distribute-rgt-neg-in 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in t around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. sub-neg 77.7%

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

      3. metadata-eval 77.7%

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

      4. +-commutative 77.7%

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

   9. Simplified 77.7%

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

   if -1.24999999999999993e-253 < t < 2.85e-129

   1. Initial program 4.7%

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

      1. associate-*l/ 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in l around inf 5.9%

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

      1. associate--l+ 29.5%

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

      2. sub-neg 29.5%

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

      3. metadata-eval 29.5%

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

      4. +-commutative 29.5%

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

      5. sub-neg 29.5%

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

      6. metadata-eval 29.5%

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

      7. +-commutative 29.5%

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

   6. Simplified 29.5%

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

      1. associate-*l/ 29.5%

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

      2. *-commutative 29.5%

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

      3. +-commutative 29.5%

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

      4. sub-neg 29.5%

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

      5. +-commutative 29.5%

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

      6. metadata-eval 29.5%

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

      7. +-commutative 29.5%

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

   8. Applied egg-rr 29.5%

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

      1. times-frac 29.3%

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

      2. associate-+l+ 5.6%

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

   10. Applied egg-rr 5.6%

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

   11. Taylor expanded in x around inf 41.5%

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

   if 2.85e-129 < t

   1. Initial program 59.1%

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

      1. associate-*l/ 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in t around inf 88.7%

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

   5. Taylor expanded in t around 0 89.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-253}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-129}:\\ \;\;\;\;\frac{t}{\ell} \cdot \frac{\sqrt{2}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 9: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-308}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-129}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(\frac{\ell}{x} \cdot \left(\ell + \frac{\ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -1.8e-308)
     (- t_1)
     (if (<= t 2.85e-129)
       (* t (sqrt (/ 2.0 (* 2.0 (* (/ l x) (+ l (/ l x)))))))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -1.8e-308) {
		tmp = -t_1;
	} else if (t <= 2.85e-129) {
		tmp = t * sqrt((2.0 / (2.0 * ((l / x) * (l + (l / x))))));
	} 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)) / (1.0d0 + x)))
    if (t <= (-1.8d-308)) then
        tmp = -t_1
    else if (t <= 2.85d-129) then
        tmp = t * sqrt((2.0d0 / (2.0d0 * ((l / x) * (l + (l / x))))))
    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) / (1.0 + x)));
	double tmp;
	if (t <= -1.8e-308) {
		tmp = -t_1;
	} else if (t <= 2.85e-129) {
		tmp = t * Math.sqrt((2.0 / (2.0 * ((l / x) * (l + (l / x))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -1.8e-308:
		tmp = -t_1
	elif t <= 2.85e-129:
		tmp = t * math.sqrt((2.0 / (2.0 * ((l / x) * (l + (l / x))))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -1.8e-308)
		tmp = Float64(-t_1);
	elseif (t <= 2.85e-129)
		tmp = Float64(t * sqrt(Float64(2.0 / Float64(2.0 * Float64(Float64(l / x) * Float64(l + Float64(l / x)))))));
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -1.8e-308)
		tmp = -t_1;
	elseif (t <= 2.85e-129)
		tmp = t * sqrt((2.0 / (2.0 * ((l / x) * (l + (l / x))))));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.8e-308], (-t$95$1), If[LessEqual[t, 2.85e-129], N[(t * N[Sqrt[N[(2.0 / N[(2.0 * N[(N[(l / x), $MachinePrecision] * N[(l + N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.7999999999999999e-308

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around -inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. +-commutative 75.1%

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

      5. sub-neg 75.1%

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

      6. metadata-eval 75.1%

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

      7. +-commutative 75.1%

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

      8. distribute-rgt-neg-in 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in t around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

      4. +-commutative 75.4%

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

   9. Simplified 75.4%

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

   if -1.7999999999999999e-308 < t < 2.85e-129

   1. Initial program 5.1%

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

      1. associate-*l/ 5.1%

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

   3. Simplified 5.1%

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

      1. expm1-log1p-u 4.8%

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

      2. expm1-udef 4.8%

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

   5. Applied egg-rr 10.0%

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

      1. expm1-def 10.1%

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

      2. expm1-log1p 10.3%

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

      3. metadata-eval 10.3%

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

      4. sub-neg 10.3%

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

      5. associate-/l* 4.3%

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

      6. sub-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

   7. Simplified 4.3%

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

   8. Taylor expanded in t around 0 4.3%

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

      1. sub-neg 4.3%

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

      2. metadata-eval 4.3%

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

      3. unpow2 4.3%

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

   10. Simplified 4.3%

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

   11. Taylor expanded in x around inf 50.1%

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

       1. associate--l+ 50.1%

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

       2. unpow2 50.1%

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

       3. associate-/l* 50.2%

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

       4. unpow2 50.2%

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

       5. unpow2 50.2%

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

       6. distribute-lft-out 50.2%

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

       7. unpow2 50.2%

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

       8. associate-/l* 50.2%

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

       9. unpow2 50.2%

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

       10. unpow2 50.2%

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

   13. Simplified 50.2%

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

       1. *-un-lft-identity 50.2%

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

       2. associate-/r/ 50.2%

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

       3. cancel-sign-sub-inv 50.2%

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

       4. times-frac 50.2%

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

       5. metadata-eval 50.2%

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

       6. *-un-lft-identity 50.2%

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

       7. associate-/r/ 50.1%

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

       8. times-frac 63.4%

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

   15. Applied egg-rr 63.4%

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

       1. *-lft-identity 63.4%

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

       2. associate-+r+ 63.4%

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

       3. count-2 63.4%

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

       4. +-commutative 63.4%

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

       5. distribute-lft-out 63.1%

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

   17. Simplified 63.1%

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

   if 2.85e-129 < t

   1. Initial program 59.1%

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

      1. associate-*l/ 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in t around inf 88.7%

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

   5. Taylor expanded in t around 0 89.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-308}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-129}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(\frac{\ell}{x} \cdot \left(\ell + \frac{\ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 10: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-308}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-194}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -1.8e-308)
     (- t_1)
     (if (<= t 2.2e-194) (* t (sqrt (/ 2.0 (/ (* 2.0 (* l l)) x)))) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -1.8e-308) {
		tmp = -t_1;
	} else if (t <= 2.2e-194) {
		tmp = t * sqrt((2.0 / ((2.0 * (l * l)) / x)));
	} 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)) / (1.0d0 + x)))
    if (t <= (-1.8d-308)) then
        tmp = -t_1
    else if (t <= 2.2d-194) then
        tmp = t * sqrt((2.0d0 / ((2.0d0 * (l * l)) / x)))
    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) / (1.0 + x)));
	double tmp;
	if (t <= -1.8e-308) {
		tmp = -t_1;
	} else if (t <= 2.2e-194) {
		tmp = t * Math.sqrt((2.0 / ((2.0 * (l * l)) / x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -1.8e-308:
		tmp = -t_1
	elif t <= 2.2e-194:
		tmp = t * math.sqrt((2.0 / ((2.0 * (l * l)) / x)))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -1.8e-308)
		tmp = Float64(-t_1);
	elseif (t <= 2.2e-194)
		tmp = Float64(t * sqrt(Float64(2.0 / Float64(Float64(2.0 * Float64(l * l)) / x))));
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -1.8e-308)
		tmp = -t_1;
	elseif (t <= 2.2e-194)
		tmp = t * sqrt((2.0 / ((2.0 * (l * l)) / x)));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.8e-308], (-t$95$1), If[LessEqual[t, 2.2e-194], N[(t * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.7999999999999999e-308

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around -inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. +-commutative 75.1%

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

      5. sub-neg 75.1%

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

      6. metadata-eval 75.1%

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

      7. +-commutative 75.1%

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

      8. distribute-rgt-neg-in 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in t around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

      4. +-commutative 75.4%

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

   9. Simplified 75.4%

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

   if -1.7999999999999999e-308 < t < 2.2000000000000001e-194

   1. Initial program 1.7%

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

      1. associate-*l/ 1.7%

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

   3. Simplified 1.7%

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

      1. expm1-log1p-u 1.7%

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

      2. expm1-udef 1.7%

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

   5. Applied egg-rr 13.0%

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

      1. expm1-def 13.0%

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

      2. expm1-log1p 13.0%

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

      3. metadata-eval 13.0%

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

      4. sub-neg 13.0%

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

      5. associate-/l* 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in t around 0 6.7%

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

      1. sub-neg 6.7%

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

      2. metadata-eval 6.7%

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

      3. unpow2 6.7%

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

   10. Simplified 6.7%

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

   11. Taylor expanded in x around inf 72.5%

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

       1. cancel-sign-sub-inv 72.5%

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

       2. metadata-eval 72.5%

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

       3. distribute-rgt1-in 72.5%

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

       4. metadata-eval 72.5%

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

       5. unpow2 72.5%

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

   13. Simplified 72.5%

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

   if 2.2000000000000001e-194 < t

   1. Initial program 51.2%

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

      1. associate-*l/ 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in t around inf 82.0%

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

   5. Taylor expanded in t around 0 82.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-308}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-194}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 11: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3.35 \cdot 10^{-195}:\\ \;\;\;\;t \cdot \sqrt{\frac{x \cdot -2}{\left(\ell \cdot \ell\right) \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -3.6e-308)
     (- t_1)
     (if (<= t 3.35e-195) (* t (sqrt (/ (* x -2.0) (* (* l l) -2.0)))) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -3.6e-308) {
		tmp = -t_1;
	} else if (t <= 3.35e-195) {
		tmp = t * sqrt(((x * -2.0) / ((l * l) * -2.0)));
	} 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)) / (1.0d0 + x)))
    if (t <= (-3.6d-308)) then
        tmp = -t_1
    else if (t <= 3.35d-195) then
        tmp = t * sqrt(((x * (-2.0d0)) / ((l * l) * (-2.0d0))))
    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) / (1.0 + x)));
	double tmp;
	if (t <= -3.6e-308) {
		tmp = -t_1;
	} else if (t <= 3.35e-195) {
		tmp = t * Math.sqrt(((x * -2.0) / ((l * l) * -2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -3.6e-308:
		tmp = -t_1
	elif t <= 3.35e-195:
		tmp = t * math.sqrt(((x * -2.0) / ((l * l) * -2.0)))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -3.6e-308)
		tmp = Float64(-t_1);
	elseif (t <= 3.35e-195)
		tmp = Float64(t * sqrt(Float64(Float64(x * -2.0) / Float64(Float64(l * l) * -2.0))));
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -3.6e-308)
		tmp = -t_1;
	elseif (t <= 3.35e-195)
		tmp = t * sqrt(((x * -2.0) / ((l * l) * -2.0)));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.6e-308], (-t$95$1), If[LessEqual[t, 3.35e-195], N[(t * N[Sqrt[N[(N[(x * -2.0), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.5999999999999999e-308

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around -inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. +-commutative 75.1%

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

      5. sub-neg 75.1%

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

      6. metadata-eval 75.1%

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

      7. +-commutative 75.1%

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

      8. distribute-rgt-neg-in 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in t around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

      4. +-commutative 75.4%

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

   9. Simplified 75.4%

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

   if -3.5999999999999999e-308 < t < 3.3500000000000001e-195

   1. Initial program 1.7%

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

      1. associate-*l/ 1.7%

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

   3. Simplified 1.7%

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

      1. expm1-log1p-u 1.7%

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

      2. expm1-udef 1.7%

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

   5. Applied egg-rr 13.0%

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

      1. expm1-def 13.0%

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

      2. expm1-log1p 13.0%

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

      3. metadata-eval 13.0%

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

      4. sub-neg 13.0%

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

      5. associate-/l* 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in t around 0 6.7%

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

      1. sub-neg 6.7%

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

      2. metadata-eval 6.7%

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

      3. unpow2 6.7%

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

   10. Simplified 6.7%

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

   11. Taylor expanded in x around inf 61.7%

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

       1. associate--l+ 61.7%

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

       2. unpow2 61.7%

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

       3. associate-/l* 61.7%

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

       4. unpow2 61.7%

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

       5. unpow2 61.7%

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

       6. distribute-lft-out 61.7%

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

       7. unpow2 61.7%

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

       8. associate-/l* 61.7%

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

       9. unpow2 61.7%

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

       10. unpow2 61.7%

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

   13. Simplified 61.7%

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

   14. Taylor expanded in x around -inf 72.6%

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

       1. associate-*r/ 72.6%

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

       2. *-commutative 72.6%

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

       3. sub-neg 72.6%

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

       4. mul-1-neg 72.6%

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

       5. distribute-rgt-out 72.6%

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

       6. metadata-eval 72.6%

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

       7. unpow2 72.6%

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

   16. Simplified 72.6%

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

   if 3.3500000000000001e-195 < t

   1. Initial program 51.2%

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

      1. associate-*l/ 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in t around inf 82.0%

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

   5. Taylor expanded in t around 0 82.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 3.35 \cdot 10^{-195}:\\ \;\;\;\;t \cdot \sqrt{\frac{x \cdot -2}{\left(\ell \cdot \ell\right) \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 12: 76.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -5e-310) (- t_1) t_1)))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -5e-310) {
		tmp = -t_1;
	} 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)) / (1.0d0 + x)))
    if (t <= (-5d-310)) then
        tmp = -t_1
    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) / (1.0 + x)));
	double tmp;
	if (t <= -5e-310) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -5e-310:
		tmp = -t_1
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-t_1);
	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) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -t_1;
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -5e-310], (-t$95$1), t$95$1]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around -inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. +-commutative 75.1%

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

      5. sub-neg 75.1%

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

      6. metadata-eval 75.1%

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

      7. +-commutative 75.1%

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

      8. distribute-rgt-neg-in 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in t around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

      4. +-commutative 75.4%

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

   9. Simplified 75.4%

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

   if -4.999999999999985e-310 < t

   1. Initial program 44.8%

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

      1. associate-*l/ 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in t around inf 73.4%

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

   5. Taylor expanded in t around 0 73.7%

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

4. Final simplification 74.5%

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

Alternative 13: 76.3% accurate, 2.1× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310)
   (- (+ (/ 1.0 x) -1.0) (/ 0.5 (* x x)))
   (sqrt (/ (+ x -1.0) (+ 1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = ((1.0 / x) + -1.0) - (0.5 / (x * x));
	} else {
		tmp = sqrt(((x + -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 <= (-5d-310)) then
        tmp = ((1.0d0 / x) + (-1.0d0)) - (0.5d0 / (x * x))
    else
        tmp = sqrt(((x + (-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 <= -5e-310) {
		tmp = ((1.0 / x) + -1.0) - (0.5 / (x * x));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = ((1.0 / x) + -1.0) - (0.5 / (x * x))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) - Float64(0.5 / Float64(x * x)));
	else
		tmp = sqrt(Float64(Float64(x + -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 <= -5e-310)
		tmp = ((1.0 / x) + -1.0) - (0.5 / (x * x));
	else
		tmp = sqrt(((x + -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, -5e-310], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around -inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. +-commutative 75.1%

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

      5. sub-neg 75.1%

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

      6. metadata-eval 75.1%

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

      7. +-commutative 75.1%

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

      8. distribute-rgt-neg-in 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in x around inf 74.9%

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

      1. associate-*r/ 74.9%

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

      2. metadata-eval 74.9%

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

      3. unpow2 74.9%

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

   9. Simplified 74.9%

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

   10. Taylor expanded in x around 0 74.9%

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

       1. associate--r+ 74.9%

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

       2. sub-neg 74.9%

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

       3. metadata-eval 74.9%

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

       4. associate-*r/ 74.9%

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

       5. metadata-eval 74.9%

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

       6. unpow2 74.9%

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

   12. Simplified 74.9%

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

   if -4.999999999999985e-310 < t

   1. Initial program 44.8%

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

      1. associate-*l/ 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in t around inf 73.4%

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

   5. Taylor expanded in t around 0 73.7%

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

4. Final simplification 74.3%

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

Alternative 14: 76.0% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t 4.5e-305)
   (+ (/ 1.0 x) -1.0)
   (+ 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 <= 4.5e-305) {
		tmp = (1.0 / x) + -1.0;
	} 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 <= 4.5d-305) then
        tmp = (1.0d0 / x) + (-1.0d0)
    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 <= 4.5e-305) {
		tmp = (1.0 / x) + -1.0;
	} 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 <= 4.5e-305:
		tmp = (1.0 / x) + -1.0
	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 <= 4.5e-305)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	else
		tmp = Float64(1.0 + Float64(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 <= 4.5e-305)
		tmp = (1.0 / x) + -1.0;
	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, 4.5e-305], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.5000000000000002e-305

   1. Initial program 35.5%

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

      1. associate-*l/ 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in t around -inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. *-commutative 74.5%

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

      3. distribute-rgt-neg-in 74.5%

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

      4. +-commutative 74.5%

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

      5. sub-neg 74.5%

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

      6. metadata-eval 74.5%

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

      7. +-commutative 74.5%

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

      8. distribute-rgt-neg-in 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in x around inf 74.1%

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

   if 4.5000000000000002e-305 < t

   1. Initial program 45.1%

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

      1. associate-*l/ 45.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in t around inf 73.9%

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

   5. Taylor expanded in x around inf 73.7%

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

      1. associate--l+ 73.7%

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

      2. associate-*r/ 73.7%

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

      3. metadata-eval 73.7%

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

      4. unpow2 73.7%

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

   7. Simplified 73.7%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 15: 76.1% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

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 4.5e-305) (- (+ (/ 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 <= 4.5e-305) {
		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 <= 4.5d-305) 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 <= 4.5e-305) {
		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 <= 4.5e-305:
		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 <= 4.5e-305)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) - t_1);
	else
		tmp = Float64(1.0 + Float64(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 <= 4.5e-305)
		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, 4.5e-305], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(1.0 + N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 4.5000000000000002e-305

   1. Initial program 35.5%

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

      1. associate-*l/ 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in t around -inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. *-commutative 74.5%

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

      3. distribute-rgt-neg-in 74.5%

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

      4. +-commutative 74.5%

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

      5. sub-neg 74.5%

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

      6. metadata-eval 74.5%

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

      7. +-commutative 74.5%

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

      8. distribute-rgt-neg-in 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in x around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. metadata-eval 74.3%

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

      3. unpow2 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in x around 0 74.3%

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

       1. associate--r+ 74.3%

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

       2. sub-neg 74.3%

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

       3. metadata-eval 74.3%

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

       4. associate-*r/ 74.3%

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

       5. metadata-eval 74.3%

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

       6. unpow2 74.3%

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

   12. Simplified 74.3%

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

   if 4.5000000000000002e-305 < t

   1. Initial program 45.1%

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

      1. associate-*l/ 45.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in t around inf 73.9%

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

   5. Taylor expanded in x around inf 73.7%

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

      1. associate--l+ 73.7%

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

      2. associate-*r/ 73.7%

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

      3. metadata-eval 73.7%

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

      4. unpow2 73.7%

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

   7. Simplified 73.7%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 16: 75.7% accurate, 31.9× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) -1.0 (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], -1.0, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around -inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. +-commutative 75.1%

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

      5. sub-neg 75.1%

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

      6. metadata-eval 75.1%

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

      7. +-commutative 75.1%

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

      8. distribute-rgt-neg-in 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in x around inf 73.4%

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

   8. Taylor expanded in t around 0 73.7%

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

   if -4.999999999999985e-310 < t

   1. Initial program 44.8%

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

      1. associate-*l/ 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in t around inf 73.4%

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

   5. Taylor expanded in x around inf 72.8%

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

4. Final simplification 73.2%

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

Alternative 17: 76.0% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t 4.5e-305) (+ (/ 1.0 x) -1.0) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= 4.5e-305) {
		tmp = (1.0 / x) + -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 <= 4.5d-305) then
        tmp = (1.0d0 / x) + (-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 <= 4.5e-305) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= 4.5e-305:
		tmp = (1.0 / x) + -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 <= 4.5e-305)
		tmp = Float64(Float64(1.0 / x) + -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 <= 4.5e-305)
		tmp = (1.0 / x) + -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, 4.5e-305], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.5000000000000002e-305

   1. Initial program 35.5%

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

      1. associate-*l/ 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in t around -inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. *-commutative 74.5%

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

      3. distribute-rgt-neg-in 74.5%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-t \cdot \sqrt{2}\right)}} \cdot t \]

      4. +-commutative 74.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

      5. sub-neg 74.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

      6. metadata-eval 74.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

      7. +-commutative 74.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

      8. distribute-rgt-neg-in 74.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

   6. Simplified 74.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

   7. Taylor expanded in x around inf 74.1%

      \[\leadsto \color{blue}{\frac{1}{x} - 1} \]

   if 4.5000000000000002e-305 < t

   1. Initial program 45.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 45.2%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 73.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   5. Taylor expanded in x around inf 73.3%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 18: 75.4% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-310) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 35.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 35.9%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 35.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around -inf 75.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \cdot t \]
   5. Step-by-step derivation

      1. mul-1-neg 75.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

      2. *-commutative 75.1%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(t \cdot \sqrt{2}\right)}} \cdot t \]

      3. distribute-rgt-neg-in 75.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-t \cdot \sqrt{2}\right)}} \cdot t \]

      4. +-commutative 75.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

      5. sub-neg 75.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

      6. metadata-eval 75.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

      7. +-commutative 75.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

      8. distribute-rgt-neg-in 75.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

   6. Simplified 75.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

   7. Taylor expanded in x around inf 73.4%

      \[\leadsto \color{blue}{\frac{-1}{t}} \cdot t \]

   8. Taylor expanded in t around 0 73.7%

      \[\leadsto \color{blue}{-1} \]

   if -4.999999999999985e-310 < t

   1. Initial program 44.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 44.9%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 44.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 73.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   5. Taylor expanded in x around inf 71.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19: 39.2% 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 40.7%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 40.8%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

4. Taylor expanded in t around -inf 35.0%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \cdot t \]
5. Step-by-step derivation

   1. mul-1-neg 35.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   2. *-commutative 35.0%

      \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(t \cdot \sqrt{2}\right)}} \cdot t \]

   3. distribute-rgt-neg-in 35.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-t \cdot \sqrt{2}\right)}} \cdot t \]

   4. +-commutative 35.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

   5. sub-neg 35.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

   6. metadata-eval 35.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

   7. +-commutative 35.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-t \cdot \sqrt{2}\right)} \cdot t \]

   8. distribute-rgt-neg-in 35.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

6. Simplified 35.0%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}} \cdot t \]

7. Taylor expanded in x around inf 34.2%

   \[\leadsto \color{blue}{\frac{-1}{t}} \cdot t \]

8. Taylor expanded in t around 0 34.4%

   \[\leadsto \color{blue}{-1} \]

9. Final simplification 34.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023283 
(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)))))