Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 85.3%

Time: 12.7s

Alternatives: 11

Speedup: 85.0×

Specification

Math

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

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

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

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

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

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

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

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

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

Initial Program: 33.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

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

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

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

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

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

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

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

Alternative 1: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{{\left({2}^{0.25}\right)}^{2} \cdot t\_m}{\sqrt{\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 840000:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x} + t\_m \cdot t\_m, \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x} + \frac{l\_m \cdot l\_m}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 3.2e-253)
      (/
       (* (pow (pow 2.0 0.25) 2.0) t_m)
       (* (sqrt (/ (+ (+ (/ 2.0 x) (/ 2.0 (* x x))) 2.0) x)) l_m))
      (if (<= t_m 4.7e-161)
        (/ t_2 (fma (/ 0.5 (* (sqrt 2.0) x)) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (if (<= t_m 840000.0)
          (/
           t_2
           (sqrt
            (fma
             2.0
             (+ (/ (* t_m t_m) x) (* t_m t_m))
             (+ (/ (fma (* t_m t_m) 2.0 (* l_m l_m)) x) (/ (* l_m l_m) x)))))
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3.2e-253) {
		tmp = (pow(pow(2.0, 0.25), 2.0) * t_m) / (sqrt(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x)) * l_m);
	} else if (t_m <= 4.7e-161) {
		tmp = t_2 / fma((0.5 / (sqrt(2.0) * x)), (((l_m * l_m) / t_m) * 2.0), t_2);
	} else if (t_m <= 840000.0) {
		tmp = t_2 / sqrt(fma(2.0, (((t_m * t_m) / x) + (t_m * t_m)), ((fma((t_m * t_m), 2.0, (l_m * l_m)) / x) + ((l_m * l_m) / x))));
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 3.2e-253)
		tmp = Float64(Float64(((2.0 ^ 0.25) ^ 2.0) * t_m) / Float64(sqrt(Float64(Float64(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))) + 2.0) / x)) * l_m));
	elseif (t_m <= 4.7e-161)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(sqrt(2.0) * x)), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	elseif (t_m <= 840000.0)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(Float64(Float64(t_m * t_m) / x) + Float64(t_m * t_m)), Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)) / x) + Float64(Float64(l_m * l_m) / x)))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.2e-253], N[(N[(N[Power[N[Power[2.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.7e-161], N[(t$95$2 / N[(N[(0.5 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 840000.0], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.1999999999999997e-253

   1. Initial program 28.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.5%

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

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. sqr-pow N/A

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

       4. pow2 N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. lower-pow.f64 N/A

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

       9. metadata-eval 18.5

          \[\leadsto \frac{{\left({2}^{\color{blue}{0.25}}\right)}^{2} \cdot t}{\ell \cdot \sqrt{\frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) + 2}{x}}} \]

   12. Applied rewrites 18.5%

       \[\leadsto \frac{\color{blue}{{\left({2}^{0.25}\right)}^{2}} \cdot t}{\ell \cdot \sqrt{\frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) + 2}{x}}} \]

   if 3.1999999999999997e-253 < t < 4.7000000000000004e-161

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.6%

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

   if 4.7000000000000004e-161 < t < 8.4e5

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

   5. Applied rewrites 89.4%

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

   if 8.4e5 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.7

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

   5. Applied rewrites 93.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 840000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x} + t \cdot t, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}, -1, -2\right)}{-x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 840000:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x} + t\_m \cdot t\_m, \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x} + \frac{l\_m \cdot l\_m}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 3.2e-253)
      (/
       t_2
       (*
        (sqrt
         (/ (fma (/ (+ (+ (/ 2.0 x) (/ 2.0 (* x x))) 2.0) x) -1.0 -2.0) (- x)))
        l_m))
      (if (<= t_m 4.7e-161)
        (/ t_2 (fma (/ 0.5 (* (sqrt 2.0) x)) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (if (<= t_m 840000.0)
          (/
           t_2
           (sqrt
            (fma
             2.0
             (+ (/ (* t_m t_m) x) (* t_m t_m))
             (+ (/ (fma (* t_m t_m) 2.0 (* l_m l_m)) x) (/ (* l_m l_m) x)))))
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3.2e-253) {
		tmp = t_2 / (sqrt((fma(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x), -1.0, -2.0) / -x)) * l_m);
	} else if (t_m <= 4.7e-161) {
		tmp = t_2 / fma((0.5 / (sqrt(2.0) * x)), (((l_m * l_m) / t_m) * 2.0), t_2);
	} else if (t_m <= 840000.0) {
		tmp = t_2 / sqrt(fma(2.0, (((t_m * t_m) / x) + (t_m * t_m)), ((fma((t_m * t_m), 2.0, (l_m * l_m)) / x) + ((l_m * l_m) / x))));
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 3.2e-253)
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(Float64(Float64(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))) + 2.0) / x), -1.0, -2.0) / Float64(-x))) * l_m));
	elseif (t_m <= 4.7e-161)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(sqrt(2.0) * x)), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	elseif (t_m <= 840000.0)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(Float64(Float64(t_m * t_m) / x) + Float64(t_m * t_m)), Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)) / x) + Float64(Float64(l_m * l_m) / x)))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.2e-253], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision] * -1.0 + -2.0), $MachinePrecision] / (-x)), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.7e-161], N[(t$95$2 / N[(N[(0.5 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 840000.0], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.1999999999999997e-253

   1. Initial program 28.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 18.5%

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

   if 3.1999999999999997e-253 < t < 4.7000000000000004e-161

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.6%

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

   if 4.7000000000000004e-161 < t < 8.4e5

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

   5. Applied rewrites 89.4%

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

   if 8.4e5 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.7

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

   5. Applied rewrites 93.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}, -1, -2\right)}{-x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 840000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x} + t \cdot t, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}, -1, -2\right)}{-x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 5600:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, l\_m \cdot l\_m, \left(t\_m \cdot t\_m\right) \cdot 2\right) - \frac{-2 \cdot \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x}}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 3.2e-253)
      (/
       t_2
       (*
        (sqrt
         (/ (fma (/ (+ (+ (/ 2.0 x) (/ 2.0 (* x x))) 2.0) x) -1.0 -2.0) (- x)))
        l_m))
      (if (<= t_m 4.7e-161)
        (/ t_2 (fma (/ 0.5 (* (sqrt 2.0) x)) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (if (<= t_m 5600.0)
          (*
           (/
            (sqrt 2.0)
            (sqrt
             (-
              (fma 0.0 (* l_m l_m) (* (* t_m t_m) 2.0))
              (/ (* -2.0 (fma (* t_m t_m) 2.0 (* l_m l_m))) x))))
           t_m)
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3.2e-253) {
		tmp = t_2 / (sqrt((fma(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x), -1.0, -2.0) / -x)) * l_m);
	} else if (t_m <= 4.7e-161) {
		tmp = t_2 / fma((0.5 / (sqrt(2.0) * x)), (((l_m * l_m) / t_m) * 2.0), t_2);
	} else if (t_m <= 5600.0) {
		tmp = (sqrt(2.0) / sqrt((fma(0.0, (l_m * l_m), ((t_m * t_m) * 2.0)) - ((-2.0 * fma((t_m * t_m), 2.0, (l_m * l_m))) / x)))) * t_m;
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 3.2e-253)
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(Float64(Float64(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))) + 2.0) / x), -1.0, -2.0) / Float64(-x))) * l_m));
	elseif (t_m <= 4.7e-161)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(sqrt(2.0) * x)), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	elseif (t_m <= 5600.0)
		tmp = Float64(Float64(sqrt(2.0) / sqrt(Float64(fma(0.0, Float64(l_m * l_m), Float64(Float64(t_m * t_m) * 2.0)) - Float64(Float64(-2.0 * fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))) / x)))) * t_m);
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.2e-253], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision] * -1.0 + -2.0), $MachinePrecision] / (-x)), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.7e-161], N[(t$95$2 / N[(N[(0.5 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5600.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(0.0 * N[(l$95$m * l$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.1999999999999997e-253

   1. Initial program 28.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 18.5%

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

   if 3.1999999999999997e-253 < t < 4.7000000000000004e-161

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.6%

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

   if 4.7000000000000004e-161 < t < 5600

   1. Initial program 46.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 86.0%

      \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, \ell \cdot \ell, \left(t \cdot t\right) \cdot 2\right) - \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -2}{x}}}} \]

   if 5600 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.7

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

   5. Applied rewrites 93.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}, -1, -2\right)}{-x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 5600:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, \ell \cdot \ell, \left(t \cdot t\right) \cdot 2\right) - \frac{-2 \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 5600:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, l\_m \cdot l\_m, \left(t\_m \cdot t\_m\right) \cdot 2\right) - \frac{-2 \cdot \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x}}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 3.2e-253)
      (/ t_2 (* (sqrt (/ (+ (+ (/ 2.0 x) (/ 2.0 (* x x))) 2.0) x)) l_m))
      (if (<= t_m 4.7e-161)
        (/ t_2 (fma (/ 0.5 (* (sqrt 2.0) x)) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (if (<= t_m 5600.0)
          (*
           (/
            (sqrt 2.0)
            (sqrt
             (-
              (fma 0.0 (* l_m l_m) (* (* t_m t_m) 2.0))
              (/ (* -2.0 (fma (* t_m t_m) 2.0 (* l_m l_m))) x))))
           t_m)
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3.2e-253) {
		tmp = t_2 / (sqrt(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x)) * l_m);
	} else if (t_m <= 4.7e-161) {
		tmp = t_2 / fma((0.5 / (sqrt(2.0) * x)), (((l_m * l_m) / t_m) * 2.0), t_2);
	} else if (t_m <= 5600.0) {
		tmp = (sqrt(2.0) / sqrt((fma(0.0, (l_m * l_m), ((t_m * t_m) * 2.0)) - ((-2.0 * fma((t_m * t_m), 2.0, (l_m * l_m))) / x)))) * t_m;
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 3.2e-253)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))) + 2.0) / x)) * l_m));
	elseif (t_m <= 4.7e-161)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(sqrt(2.0) * x)), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	elseif (t_m <= 5600.0)
		tmp = Float64(Float64(sqrt(2.0) / sqrt(Float64(fma(0.0, Float64(l_m * l_m), Float64(Float64(t_m * t_m) * 2.0)) - Float64(Float64(-2.0 * fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))) / x)))) * t_m);
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.2e-253], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.7e-161], N[(t$95$2 / N[(N[(0.5 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5600.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(0.0 * N[(l$95$m * l$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.1999999999999997e-253

   1. Initial program 28.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.5%

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

   if 3.1999999999999997e-253 < t < 4.7000000000000004e-161

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 85.6%

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

   if 4.7000000000000004e-161 < t < 5600

   1. Initial program 46.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 86.0%

      \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, \ell \cdot \ell, \left(t \cdot t\right) \cdot 2\right) - \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -2}{x}}}} \]

   if 5600 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.7

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

   5. Applied rewrites 93.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 5600:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, \ell \cdot \ell, \left(t \cdot t\right) \cdot 2\right) - \frac{-2 \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 5600:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, l\_m \cdot l\_m, \left(t\_m \cdot t\_m\right) \cdot 2\right) - \frac{-2 \cdot \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x}}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 3.1e-220)
      (/ t_2 (* (sqrt (/ (+ (+ (/ 2.0 x) (/ 2.0 (* x x))) 2.0) x)) l_m))
      (if (<= t_m 4.7e-161)
        1.0
        (if (<= t_m 5600.0)
          (*
           (/
            (sqrt 2.0)
            (sqrt
             (-
              (fma 0.0 (* l_m l_m) (* (* t_m t_m) 2.0))
              (/ (* -2.0 (fma (* t_m t_m) 2.0 (* l_m l_m))) x))))
           t_m)
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3.1e-220) {
		tmp = t_2 / (sqrt(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x)) * l_m);
	} else if (t_m <= 4.7e-161) {
		tmp = 1.0;
	} else if (t_m <= 5600.0) {
		tmp = (sqrt(2.0) / sqrt((fma(0.0, (l_m * l_m), ((t_m * t_m) * 2.0)) - ((-2.0 * fma((t_m * t_m), 2.0, (l_m * l_m))) / x)))) * t_m;
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 3.1e-220)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))) + 2.0) / x)) * l_m));
	elseif (t_m <= 4.7e-161)
		tmp = 1.0;
	elseif (t_m <= 5600.0)
		tmp = Float64(Float64(sqrt(2.0) / sqrt(Float64(fma(0.0, Float64(l_m * l_m), Float64(Float64(t_m * t_m) * 2.0)) - Float64(Float64(-2.0 * fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))) / x)))) * t_m);
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.1e-220], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.7e-161], 1.0, If[LessEqual[t$95$m, 5600.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(0.0 * N[(l$95$m * l$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.10000000000000011e-220

   1. Initial program 27.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if 3.10000000000000011e-220 < t < 4.7000000000000004e-161

   1. Initial program 6.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 84.6

         \[\leadsto \sqrt{0.5} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.8%

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

   if 4.7000000000000004e-161 < t < 5600

   1. Initial program 46.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 86.0%

      \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, \ell \cdot \ell, \left(t \cdot t\right) \cdot 2\right) - \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -2}{x}}}} \]

   if 5600 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.7

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

   5. Applied rewrites 93.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5600:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(0, \ell \cdot \ell, \left(t \cdot t\right) \cdot 2\right) - \frac{-2 \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 3.1e-220)
      (/ t_2 (* (sqrt (/ (+ (+ (/ 2.0 x) (/ 2.0 (* x x))) 2.0) x)) l_m))
      (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3.1e-220) {
		tmp = t_2 / (sqrt(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x)) * l_m);
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * t_m
    if (t_m <= 3.1d-220) then
        tmp = t_2 / (sqrt(((((2.0d0 / x) + (2.0d0 / (x * x))) + 2.0d0) / x)) * l_m)
    else
        tmp = t_2 / (sqrt(((x - (-1.0d0)) / (x - 1.0d0))) * t_2)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3.1e-220) {
		tmp = t_2 / (Math.sqrt(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x)) * l_m);
	} else {
		tmp = t_2 / (Math.sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 3.1e-220:
		tmp = t_2 / (math.sqrt(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x)) * l_m)
	else:
		tmp = t_2 / (math.sqrt(((x - -1.0) / (x - 1.0))) * t_2)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 3.1e-220)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))) + 2.0) / x)) * l_m));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 3.1e-220)
		tmp = t_2 / (sqrt(((((2.0 / x) + (2.0 / (x * x))) + 2.0) / x)) * l_m);
	else
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.1e-220], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.10000000000000011e-220

   1. Initial program 27.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if 3.10000000000000011e-220 < t

   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. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.0

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

   5. Applied rewrites 85.0%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(\frac{2}{x} + \frac{2}{x \cdot x}\right) + 2}{x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.2 \cdot 10^{+221}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot l\_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= l_m 2.2e+221)
      (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))
      (/ t_2 (* (sqrt (/ 1.0 x)) (* (sqrt 2.0) l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (l_m <= 2.2e+221) {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	} else {
		tmp = t_2 / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * t_m
    if (l_m <= 2.2d+221) then
        tmp = t_2 / (sqrt(((x - (-1.0d0)) / (x - 1.0d0))) * t_2)
    else
        tmp = t_2 / (sqrt((1.0d0 / x)) * (sqrt(2.0d0) * l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (l_m <= 2.2e+221) {
		tmp = t_2 / (Math.sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	} else {
		tmp = t_2 / (Math.sqrt((1.0 / x)) * (Math.sqrt(2.0) * l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if l_m <= 2.2e+221:
		tmp = t_2 / (math.sqrt(((x - -1.0) / (x - 1.0))) * t_2)
	else:
		tmp = t_2 / (math.sqrt((1.0 / x)) * (math.sqrt(2.0) * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (l_m <= 2.2e+221)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(1.0 / x)) * Float64(sqrt(2.0) * l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (l_m <= 2.2e+221)
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	else
		tmp = t_2 / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 2.2e+221], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1999999999999999e221

   1. Initial program 34.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 38.1

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

   5. Applied rewrites 38.1%

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

   if 2.1999999999999999e221 < l

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 42.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.1

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

   7. Applied rewrites 2.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 89.4%

       \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.2 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t\_m} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot l\_m\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.2e+221)
    (* (/ (sqrt 2.0) (* (sqrt (* (/ (- -1.0 x) (- 1.0 x)) 2.0)) t_m)) t_m)
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 1.0 x)) (* (sqrt 2.0) l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.2e+221) {
		tmp = (sqrt(2.0) / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * t_m;
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 2.2d+221) then
        tmp = (sqrt(2.0d0) / (sqrt(((((-1.0d0) - x) / (1.0d0 - x)) * 2.0d0)) * t_m)) * t_m
    else
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((1.0d0 / x)) * (sqrt(2.0d0) * l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.2e+221) {
		tmp = (Math.sqrt(2.0) / (Math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * t_m;
	} else {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((1.0 / x)) * (Math.sqrt(2.0) * l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 2.2e+221:
		tmp = (math.sqrt(2.0) / (math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * t_m
	else:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((1.0 / x)) * (math.sqrt(2.0) * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.2e+221)
		tmp = Float64(Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(Float64(-1.0 - x) / Float64(1.0 - x)) * 2.0)) * t_m)) * t_m);
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(1.0 / x)) * Float64(sqrt(2.0) * l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 2.2e+221)
		tmp = (sqrt(2.0) / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * t_m;
	else
		tmp = (sqrt(2.0) * t_m) / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.2e+221], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.1999999999999999e221

   1. Initial program 34.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 38.1

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

   5. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 38.0

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

   7. Applied rewrites 38.0%

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

   if 2.1999999999999999e221 < l

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 42.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.1

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

   7. Applied rewrites 2.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 89.4%

       \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.2 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t\_m} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{2}{x}} \cdot l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.2e+221)
    (* (/ (sqrt 2.0) (* (sqrt (* (/ (- -1.0 x) (- 1.0 x)) 2.0)) t_m)) t_m)
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 2.0 x)) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.2e+221) {
		tmp = (sqrt(2.0) / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * t_m;
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 2.2d+221) then
        tmp = (sqrt(2.0d0) / (sqrt(((((-1.0d0) - x) / (1.0d0 - x)) * 2.0d0)) * t_m)) * t_m
    else
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((2.0d0 / x)) * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.2e+221) {
		tmp = (Math.sqrt(2.0) / (Math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * t_m;
	} else {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((2.0 / x)) * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 2.2e+221:
		tmp = (math.sqrt(2.0) / (math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * t_m
	else:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((2.0 / x)) * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.2e+221)
		tmp = Float64(Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(Float64(-1.0 - x) / Float64(1.0 - x)) * 2.0)) * t_m)) * t_m);
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(2.0 / x)) * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 2.2e+221)
		tmp = (sqrt(2.0) / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * t_m;
	else
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.2e+221], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.1999999999999999e221

   1. Initial program 34.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 38.1

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

   5. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 38.0

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

   7. Applied rewrites 38.0%

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

   if 2.1999999999999999e221 < l

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 42.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.1

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

   7. Applied rewrites 2.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 89.7%

       \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.2 \cdot 10^{+221}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{2}{x}} \cdot l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.2e+221) 1.0 (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 2.0 x)) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.2e+221) {
		tmp = 1.0;
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 2.2d+221) then
        tmp = 1.0d0
    else
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((2.0d0 / x)) * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.2e+221) {
		tmp = 1.0;
	} else {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((2.0 / x)) * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 2.2e+221:
		tmp = 1.0
	else:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((2.0 / x)) * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.2e+221)
		tmp = 1.0;
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(2.0 / x)) * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 2.2e+221)
		tmp = 1.0;
	else
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.2e+221], 1.0, N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.1999999999999999e221

   1. Initial program 34.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 37.4

         \[\leadsto \sqrt{0.5} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 37.4%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.9%

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

   if 2.1999999999999999e221 < l

   1. Initial program 0.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 42.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 2.1

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

   7. Applied rewrites 2.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 89.7%

       \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+221}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.8% accurate, 85.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 32.8%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 36.4

      \[\leadsto \sqrt{0.5} \cdot \color{blue}{\sqrt{2}} \]

5. Applied rewrites 36.4%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{2}} \]
6. Step-by-step derivation

7. Applied rewrites 37.0%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024282 
(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)))))