Toniolo and Linder, Equation (7)

Percentage Accurate: 32.4% → 85.4%

Time: 13.9s

Alternatives: 13

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: 32.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t\_m}{l\_m}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{t\_3 \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, \frac{\mathsf{fma}\left(t\_3, -2, \frac{\frac{t\_3}{x} + \mathsf{fma}\left(2, t\_3, \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{l\_m \cdot l\_m}{x}\right)\right)}{-x}\right)}{-x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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_3 (fma (* t_m t_m) 2.0 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 2.05e-219)
      (/ (* (sqrt (- x 1.0)) t_m) l_m)
      (if (<= t_m 2.7e-165)
        (/ t_2 (fma (/ (* t_3 2.0) (* (* (sqrt 2.0) x) t_m)) 0.5 t_2))
        (if (<= t_m 1.05e+68)
          (/
           t_2
           (sqrt
            (fma
             (* 2.0 t_m)
             t_m
             (/
              (fma
               t_3
               -2.0
               (/
                (+
                 (/ t_3 x)
                 (fma 2.0 t_3 (fma (/ (* t_m t_m) x) 2.0 (/ (* l_m l_m) x))))
                (- x)))
              (- 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 t_3 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double tmp;
	if (t_m <= 2.05e-219) {
		tmp = (sqrt((x - 1.0)) * t_m) / l_m;
	} else if (t_m <= 2.7e-165) {
		tmp = t_2 / fma(((t_3 * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else if (t_m <= 1.05e+68) {
		tmp = t_2 / sqrt(fma((2.0 * t_m), t_m, (fma(t_3, -2.0, (((t_3 / x) + fma(2.0, t_3, fma(((t_m * t_m) / x), 2.0, ((l_m * l_m) / x)))) / -x)) / -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)
	t_3 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 2.05e-219)
		tmp = Float64(Float64(sqrt(Float64(x - 1.0)) * t_m) / l_m);
	elseif (t_m <= 2.7e-165)
		tmp = Float64(t_2 / fma(Float64(Float64(t_3 * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	elseif (t_m <= 1.05e+68)
		tmp = Float64(t_2 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(fma(t_3, -2.0, Float64(Float64(Float64(t_3 / x) + fma(2.0, t_3, fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(l_m * l_m) / x)))) / Float64(-x))) / Float64(-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]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-219], N[(N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 2.7e-165], N[(t$95$2 / N[(N[(N[(t$95$3 * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+68], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(t$95$3 * -2.0 + N[(N[(N[(t$95$3 / x), $MachinePrecision] + N[(2.0 * t$95$3 + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] / (-x)), $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 < 2.05e-219

   1. Initial program 28.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 17.2%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 18.5%

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

   if 2.05e-219 < t < 2.6999999999999998e-165

   1. Initial program 2.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 69.1%

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

   if 2.6999999999999998e-165 < t < 1.05e68

   1. Initial program 49.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 84.1%

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

   if 1.05e68 < t

   1. Initial program 26.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t}{\ell}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2 \cdot t, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), -2, \frac{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\ell \cdot \ell}{x}\right)\right)}{-x}\right)}{-x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t\_m}{l\_m}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{t\_3 \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_3, \frac{t\_3}{x}\right) + \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{l\_m \cdot l\_m}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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_3 (fma (* t_m t_m) 2.0 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 2.05e-219)
      (/ (* (sqrt (- x 1.0)) t_m) l_m)
      (if (<= t_m 2.7e-165)
        (/ t_2 (fma (/ (* t_3 2.0) (* (* (sqrt 2.0) x) t_m)) 0.5 t_2))
        (if (<= t_m 1.05e+68)
          (/
           t_2
           (sqrt
            (fma
             (* 2.0 t_m)
             t_m
             (/
              (+
               (fma 2.0 t_3 (/ t_3 x))
               (fma (/ (* t_m t_m) x) 2.0 (/ (* l_m l_m) x)))
              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 t_3 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double tmp;
	if (t_m <= 2.05e-219) {
		tmp = (sqrt((x - 1.0)) * t_m) / l_m;
	} else if (t_m <= 2.7e-165) {
		tmp = t_2 / fma(((t_3 * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else if (t_m <= 1.05e+68) {
		tmp = t_2 / sqrt(fma((2.0 * t_m), t_m, ((fma(2.0, t_3, (t_3 / x)) + fma(((t_m * t_m) / x), 2.0, ((l_m * l_m) / x))) / 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)
	t_3 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 2.05e-219)
		tmp = Float64(Float64(sqrt(Float64(x - 1.0)) * t_m) / l_m);
	elseif (t_m <= 2.7e-165)
		tmp = Float64(t_2 / fma(Float64(Float64(t_3 * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	elseif (t_m <= 1.05e+68)
		tmp = Float64(t_2 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(Float64(fma(2.0, t_3, Float64(t_3 / x)) + fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(l_m * l_m) / x))) / 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]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-219], N[(N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 2.7e-165], N[(t$95$2 / N[(N[(N[(t$95$3 * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+68], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(N[(2.0 * t$95$3 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 2.05e-219

   1. Initial program 28.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 17.2%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 18.5%

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

   if 2.05e-219 < t < 2.6999999999999998e-165

   1. Initial program 2.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 69.1%

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

   if 2.6999999999999998e-165 < t < 1.05e68

   1. Initial program 49.1%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 84.1%

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

   if 1.05e68 < t

   1. Initial program 26.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t}{\ell}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2 \cdot t, t, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}\right) + \mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\ell \cdot \ell}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t\_m}{l\_m}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{t\_3 \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x} + t\_m \cdot t\_m, \frac{t\_3}{x} + \frac{l\_m \cdot l\_m}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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_3 (fma (* t_m t_m) 2.0 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 2.05e-219)
      (/ (* (sqrt (- x 1.0)) t_m) l_m)
      (if (<= t_m 2.7e-165)
        (/ t_2 (fma (/ (* t_3 2.0) (* (* (sqrt 2.0) x) t_m)) 0.5 t_2))
        (if (<= t_m 1.05e+68)
          (/
           t_2
           (sqrt
            (fma
             2.0
             (+ (/ (* t_m t_m) x) (* t_m t_m))
             (+ (/ t_3 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 t_3 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double tmp;
	if (t_m <= 2.05e-219) {
		tmp = (sqrt((x - 1.0)) * t_m) / l_m;
	} else if (t_m <= 2.7e-165) {
		tmp = t_2 / fma(((t_3 * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else if (t_m <= 1.05e+68) {
		tmp = t_2 / sqrt(fma(2.0, (((t_m * t_m) / x) + (t_m * t_m)), ((t_3 / 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)
	t_3 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 2.05e-219)
		tmp = Float64(Float64(sqrt(Float64(x - 1.0)) * t_m) / l_m);
	elseif (t_m <= 2.7e-165)
		tmp = Float64(t_2 / fma(Float64(Float64(t_3 * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	elseif (t_m <= 1.05e+68)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(Float64(Float64(t_m * t_m) / x) + Float64(t_m * t_m)), Float64(Float64(t_3 / 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]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-219], N[(N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 2.7e-165], N[(t$95$2 / N[(N[(N[(t$95$3 * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+68], 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[(t$95$3 / 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 < 2.05e-219

   1. Initial program 28.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 17.2%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 18.5%

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

   if 2.05e-219 < t < 2.6999999999999998e-165

   1. Initial program 2.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 69.1%

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

   if 2.6999999999999998e-165 < t < 1.05e68

   1. Initial program 49.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]
   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 84.1%

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

   if 1.05e68 < t

   1. Initial program 26.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t}{\ell}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x} + t \cdot t, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t\_m}{l\_m}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(2, \mathsf{fma}\left(t\_m, t\_m, \frac{l\_m \cdot l\_m}{x}\right), 4 \cdot \frac{t\_m \cdot t\_m}{x}\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 2.05e-219)
      (/ (* (sqrt (- x 1.0)) t_m) l_m)
      (if (<= t_m 7.5e-155)
        (/
         t_2
         (fma
          (/
           (* (fma (* t_m t_m) 2.0 (* l_m l_m)) 2.0)
           (* (* (sqrt 2.0) x) t_m))
          0.5
          t_2))
        (if (<= t_m 1.05e+68)
          (*
           (sqrt
            (/
             2.0
             (fma
              2.0
              (fma t_m t_m (/ (* l_m l_m) x))
              (* 4.0 (/ (* t_m t_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 <= 2.05e-219) {
		tmp = (sqrt((x - 1.0)) * t_m) / l_m;
	} else if (t_m <= 7.5e-155) {
		tmp = t_2 / fma(((fma((t_m * t_m), 2.0, (l_m * l_m)) * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else if (t_m <= 1.05e+68) {
		tmp = sqrt((2.0 / fma(2.0, fma(t_m, t_m, ((l_m * l_m) / x)), (4.0 * ((t_m * t_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 <= 2.05e-219)
		tmp = Float64(Float64(sqrt(Float64(x - 1.0)) * t_m) / l_m);
	elseif (t_m <= 7.5e-155)
		tmp = Float64(t_2 / fma(Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)) * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	elseif (t_m <= 1.05e+68)
		tmp = Float64(sqrt(Float64(2.0 / fma(2.0, fma(t_m, t_m, Float64(Float64(l_m * l_m) / x)), Float64(4.0 * Float64(Float64(t_m * t_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, 2.05e-219], N[(N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 7.5e-155], N[(t$95$2 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+68], N[(N[Sqrt[N[(2.0 / N[(2.0 * N[(t$95$m * t$95$m + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $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 < 2.05e-219

   1. Initial program 28.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 17.2%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 18.5%

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

   if 2.05e-219 < t < 7.5000000000000006e-155

   1. Initial program 2.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 60.3%

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

   if 7.5000000000000006e-155 < t < 1.05e68

   1. Initial program 50.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 50.2

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 84.0%

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

   if 1.05e68 < t

   1. Initial program 26.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t}{\ell}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell \cdot \ell}{x}\right), 4 \cdot \frac{t \cdot t}{x}\right)}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t\_m}{l\_m}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\frac{4}{x} + 2} \cdot t\_m} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(2, \mathsf{fma}\left(t\_m, t\_m, \frac{l\_m \cdot l\_m}{x}\right), 4 \cdot \frac{t\_m \cdot t\_m}{x}\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 4.2e-204)
      (/ (* (sqrt (- x 1.0)) t_m) l_m)
      (if (<= t_m 3.5e-177)
        (* (/ t_m (* (sqrt (+ (/ 4.0 x) 2.0)) t_m)) (sqrt 2.0))
        (if (<= t_m 1.05e+68)
          (*
           (sqrt
            (/
             2.0
             (fma
              2.0
              (fma t_m t_m (/ (* l_m l_m) x))
              (* 4.0 (/ (* t_m t_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 <= 4.2e-204) {
		tmp = (sqrt((x - 1.0)) * t_m) / l_m;
	} else if (t_m <= 3.5e-177) {
		tmp = (t_m / (sqrt(((4.0 / x) + 2.0)) * t_m)) * sqrt(2.0);
	} else if (t_m <= 1.05e+68) {
		tmp = sqrt((2.0 / fma(2.0, fma(t_m, t_m, ((l_m * l_m) / x)), (4.0 * ((t_m * t_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 <= 4.2e-204)
		tmp = Float64(Float64(sqrt(Float64(x - 1.0)) * t_m) / l_m);
	elseif (t_m <= 3.5e-177)
		tmp = Float64(Float64(t_m / Float64(sqrt(Float64(Float64(4.0 / x) + 2.0)) * t_m)) * sqrt(2.0));
	elseif (t_m <= 1.05e+68)
		tmp = Float64(sqrt(Float64(2.0 / fma(2.0, fma(t_m, t_m, Float64(Float64(l_m * l_m) / x)), Float64(4.0 * Float64(Float64(t_m * t_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, 4.2e-204], N[(N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 3.5e-177], N[(N[(t$95$m / N[(N[Sqrt[N[(N[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+68], N[(N[Sqrt[N[(2.0 / N[(2.0 * N[(t$95$m * t$95$m + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $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 < 4.20000000000000018e-204

   1. Initial program 27.5%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 16.9%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 18.1%

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

   if 4.20000000000000018e-204 < t < 3.5000000000000002e-177

   1. Initial program 1.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{t}{\sqrt{2 + 4 \cdot \frac{1}{x}} \cdot t} \cdot \sqrt{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.6%

       \[\leadsto \frac{t}{\sqrt{\frac{4}{x} + 2} \cdot t} \cdot \sqrt{2} \]

   if 3.5000000000000002e-177 < t < 1.05e68

   1. Initial program 48.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 48.4

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   7. Applied rewrites 66.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 82.8%

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

   if 1.05e68 < t

   1. Initial program 26.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t}{\ell}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{4}{x} + 2} \cdot t} \cdot \sqrt{2}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell \cdot \ell}{x}\right), 4 \cdot \frac{t \cdot t}{x}\right)}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{4}{x} + 2\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t\_m}{l\_m}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{t\_m}{\sqrt{t\_2} \cdot t\_m} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(t\_2, t\_m \cdot t\_m, \frac{l\_m \cdot l\_m}{x} \cdot 2\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

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 (+ (/ 4.0 x) 2.0)) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 4.2e-204)
      (/ (* (sqrt (- x 1.0)) t_m) l_m)
      (if (<= t_m 3.5e-177)
        (* (/ t_m (* (sqrt t_2) t_m)) (sqrt 2.0))
        (if (<= t_m 1.05e+68)
          (*
           (sqrt (/ 2.0 (fma t_2 (* t_m t_m) (* (/ (* l_m l_m) x) 2.0))))
           t_m)
          (/ t_3 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_3))))))))

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 = (4.0 / x) + 2.0;
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 4.2e-204) {
		tmp = (sqrt((x - 1.0)) * t_m) / l_m;
	} else if (t_m <= 3.5e-177) {
		tmp = (t_m / (sqrt(t_2) * t_m)) * sqrt(2.0);
	} else if (t_m <= 1.05e+68) {
		tmp = sqrt((2.0 / fma(t_2, (t_m * t_m), (((l_m * l_m) / x) * 2.0)))) * t_m;
	} else {
		tmp = t_3 / (sqrt(((x - -1.0) / (x - 1.0))) * t_3);
	}
	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(Float64(4.0 / x) + 2.0)
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 4.2e-204)
		tmp = Float64(Float64(sqrt(Float64(x - 1.0)) * t_m) / l_m);
	elseif (t_m <= 3.5e-177)
		tmp = Float64(Float64(t_m / Float64(sqrt(t_2) * t_m)) * sqrt(2.0));
	elseif (t_m <= 1.05e+68)
		tmp = Float64(sqrt(Float64(2.0 / fma(t_2, Float64(t_m * t_m), Float64(Float64(Float64(l_m * l_m) / x) * 2.0)))) * t_m);
	else
		tmp = Float64(t_3 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_3));
	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[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-204], N[(N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 3.5e-177], N[(N[(t$95$m / N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+68], N[(N[Sqrt[N[(2.0 / N[(t$95$2 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$3 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.20000000000000018e-204

   1. Initial program 27.5%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 16.9%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 18.1%

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

   if 4.20000000000000018e-204 < t < 3.5000000000000002e-177

   1. Initial program 1.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{t}{\sqrt{2 + 4 \cdot \frac{1}{x}} \cdot t} \cdot \sqrt{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.6%

       \[\leadsto \frac{t}{\sqrt{\frac{4}{x} + 2} \cdot t} \cdot \sqrt{2} \]

   if 3.5000000000000002e-177 < t < 1.05e68

   1. Initial program 48.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 48.4

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   7. Applied rewrites 66.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 82.8%

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

   if 1.05e68 < t

   1. Initial program 26.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t}{\ell}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{4}{x} + 2} \cdot t} \cdot \sqrt{2}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(\frac{4}{x} + 2, t \cdot t, \frac{\ell \cdot \ell}{x} \cdot 2\right)}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{4}{x} + 2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t\_m}{l\_m}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{t\_m}{\sqrt{t\_2} \cdot t\_m} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(t\_2, t\_m \cdot t\_m, \frac{l\_m \cdot l\_m}{x} \cdot 2\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t\_m} \cdot \sqrt{2}\\ \end{array} \end{array} \end{array} \]

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 (+ (/ 4.0 x) 2.0)))
   (*
    t_s
    (if (<= t_m 4.2e-204)
      (/ (* (sqrt (- x 1.0)) t_m) l_m)
      (if (<= t_m 3.5e-177)
        (* (/ t_m (* (sqrt t_2) t_m)) (sqrt 2.0))
        (if (<= t_m 5.5e+79)
          (*
           (sqrt (/ 2.0 (fma t_2 (* t_m t_m) (* (/ (* l_m l_m) x) 2.0))))
           t_m)
          (*
           (/ t_m (* (sqrt (* (/ (- -1.0 x) (- 1.0 x)) 2.0)) t_m))
           (sqrt 2.0))))))))

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) {
	double t_2 = (4.0 / x) + 2.0;
	double tmp;
	if (t_m <= 4.2e-204) {
		tmp = (sqrt((x - 1.0)) * t_m) / l_m;
	} else if (t_m <= 3.5e-177) {
		tmp = (t_m / (sqrt(t_2) * t_m)) * sqrt(2.0);
	} else if (t_m <= 5.5e+79) {
		tmp = sqrt((2.0 / fma(t_2, (t_m * t_m), (((l_m * l_m) / x) * 2.0)))) * t_m;
	} else {
		tmp = (t_m / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * sqrt(2.0);
	}
	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(Float64(4.0 / x) + 2.0)
	tmp = 0.0
	if (t_m <= 4.2e-204)
		tmp = Float64(Float64(sqrt(Float64(x - 1.0)) * t_m) / l_m);
	elseif (t_m <= 3.5e-177)
		tmp = Float64(Float64(t_m / Float64(sqrt(t_2) * t_m)) * sqrt(2.0));
	elseif (t_m <= 5.5e+79)
		tmp = Float64(sqrt(Float64(2.0 / fma(t_2, Float64(t_m * t_m), Float64(Float64(Float64(l_m * l_m) / x) * 2.0)))) * t_m);
	else
		tmp = Float64(Float64(t_m / Float64(sqrt(Float64(Float64(Float64(-1.0 - x) / Float64(1.0 - x)) * 2.0)) * t_m)) * sqrt(2.0));
	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[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-204], N[(N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 3.5e-177], N[(N[(t$95$m / N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+79], N[(N[Sqrt[N[(2.0 / N[(t$95$2 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m / 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] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.20000000000000018e-204

   1. Initial program 27.5%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 16.9%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 18.1%

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

   if 4.20000000000000018e-204 < t < 3.5000000000000002e-177

   1. Initial program 1.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{t}{\sqrt{2 + 4 \cdot \frac{1}{x}} \cdot t} \cdot \sqrt{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.6%

       \[\leadsto \frac{t}{\sqrt{\frac{4}{x} + 2} \cdot t} \cdot \sqrt{2} \]

   if 3.5000000000000002e-177 < t < 5.50000000000000007e79

   1. Initial program 47.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 47.8

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   7. Applied rewrites 64.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 82.6%

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

   if 5.50000000000000007e79 < t

   1. Initial program 24.9%

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

   3. Taylor expanded in 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 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t}{\ell}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{4}{x} + 2} \cdot t} \cdot \sqrt{2}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(\frac{4}{x} + 2, t \cdot t, \frac{\ell \cdot \ell}{x} \cdot 2\right)}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t\_m} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{l\_m} \cdot \left(\sqrt{x - 1} \cdot t\_m\right)\\ \end{array} \end{array} \]

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 l_m) 2e+307)
    (* (/ t_m (* (sqrt (* (/ (- -1.0 x) (- 1.0 x)) 2.0)) t_m)) (sqrt 2.0))
    (* (/ 1.0 l_m) (* (sqrt (- x 1.0)) t_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 * l_m) <= 2e+307) {
		tmp = (t_m / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * sqrt(2.0);
	} else {
		tmp = (1.0 / l_m) * (sqrt((x - 1.0)) * t_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 * l_m) <= 2d+307) then
        tmp = (t_m / (sqrt(((((-1.0d0) - x) / (1.0d0 - x)) * 2.0d0)) * t_m)) * sqrt(2.0d0)
    else
        tmp = (1.0d0 / l_m) * (sqrt((x - 1.0d0)) * t_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 * l_m) <= 2e+307) {
		tmp = (t_m / (Math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * Math.sqrt(2.0);
	} else {
		tmp = (1.0 / l_m) * (Math.sqrt((x - 1.0)) * t_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 * l_m) <= 2e+307:
		tmp = (t_m / (math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * math.sqrt(2.0)
	else:
		tmp = (1.0 / l_m) * (math.sqrt((x - 1.0)) * t_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 (Float64(l_m * l_m) <= 2e+307)
		tmp = Float64(Float64(t_m / Float64(sqrt(Float64(Float64(Float64(-1.0 - x) / Float64(1.0 - x)) * 2.0)) * t_m)) * sqrt(2.0));
	else
		tmp = Float64(Float64(1.0 / l_m) * Float64(sqrt(Float64(x - 1.0)) * t_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 * l_m) <= 2e+307)
		tmp = (t_m / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * sqrt(2.0);
	else
		tmp = (1.0 / l_m) * (sqrt((x - 1.0)) * t_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[N[(l$95$m * l$95$m), $MachinePrecision], 2e+307], N[(N[(t$95$m / 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] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999997e307

   1. Initial program 37.9%

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

   3. Taylor expanded in 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 44.1

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

   5. Applied rewrites 44.1%

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

   7. Applied rewrites 44.0%

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

   if 1.99999999999999997e307 < (*.f64 l l)

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.5%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.0%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\ell} \cdot \left(\sqrt{x - 1} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+307}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{l\_m} \cdot \left(\sqrt{x - 1} \cdot t\_m\right)\\ \end{array} \end{array} \]

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 l_m) 2e+307) 1.0 (* (/ 1.0 l_m) (* (sqrt (- x 1.0)) t_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 * l_m) <= 2e+307) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / l_m) * (sqrt((x - 1.0)) * t_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 * l_m) <= 2d+307) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 / l_m) * (sqrt((x - 1.0d0)) * t_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 * l_m) <= 2e+307) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / l_m) * (Math.sqrt((x - 1.0)) * t_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 * l_m) <= 2e+307:
		tmp = 1.0
	else:
		tmp = (1.0 / l_m) * (math.sqrt((x - 1.0)) * t_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 (Float64(l_m * l_m) <= 2e+307)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 / l_m) * Float64(sqrt(Float64(x - 1.0)) * t_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 * l_m) <= 2e+307)
		tmp = 1.0;
	else
		tmp = (1.0 / l_m) * (sqrt((x - 1.0)) * t_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[N[(l$95$m * l$95$m), $MachinePrecision], 2e+307], 1.0, N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999997e307

   1. Initial program 37.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   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 42.7

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 43.3%

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

   if 1.99999999999999997e307 < (*.f64 l l)

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.5%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.0%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+307}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\ell} \cdot \left(\sqrt{x - 1} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+307}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x - 1} \cdot t\_m}{l\_m}\\ \end{array} \end{array} \]

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 l_m) 2e+307) 1.0 (/ (* (sqrt (- x 1.0)) t_m) 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 * l_m) <= 2e+307) {
		tmp = 1.0;
	} else {
		tmp = (sqrt((x - 1.0)) * t_m) / 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 * l_m) <= 2d+307) then
        tmp = 1.0d0
    else
        tmp = (sqrt((x - 1.0d0)) * t_m) / 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 * l_m) <= 2e+307) {
		tmp = 1.0;
	} else {
		tmp = (Math.sqrt((x - 1.0)) * t_m) / 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 * l_m) <= 2e+307:
		tmp = 1.0
	else:
		tmp = (math.sqrt((x - 1.0)) * t_m) / 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 (Float64(l_m * l_m) <= 2e+307)
		tmp = 1.0;
	else
		tmp = Float64(Float64(sqrt(Float64(x - 1.0)) * t_m) / 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 * l_m) <= 2e+307)
		tmp = 1.0;
	else
		tmp = (sqrt((x - 1.0)) * t_m) / 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[N[(l$95$m * l$95$m), $MachinePrecision], 2e+307], 1.0, N[(N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999997e307

   1. Initial program 37.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   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 42.7

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 43.3%

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

   if 1.99999999999999997e307 < (*.f64 l l)

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.5%

       \[\leadsto \sqrt{x - 1} \cdot \frac{t}{\ell} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.0%

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

Alternative 11: 78.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+307}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{l\_m}\\ \end{array} \end{array} \]

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 l_m) 2e+307) 1.0 (/ (* (sqrt x) t_m) 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 * l_m) <= 2e+307) {
		tmp = 1.0;
	} else {
		tmp = (sqrt(x) * t_m) / 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 * l_m) <= 2d+307) then
        tmp = 1.0d0
    else
        tmp = (sqrt(x) * t_m) / 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 * l_m) <= 2e+307) {
		tmp = 1.0;
	} else {
		tmp = (Math.sqrt(x) * t_m) / 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 * l_m) <= 2e+307:
		tmp = 1.0
	else:
		tmp = (math.sqrt(x) * t_m) / 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 (Float64(l_m * l_m) <= 2e+307)
		tmp = 1.0;
	else
		tmp = Float64(Float64(sqrt(x) * t_m) / 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 * l_m) <= 2e+307)
		tmp = 1.0;
	else
		tmp = (sqrt(x) * t_m) / 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[N[(l$95$m * l$95$m), $MachinePrecision], 2e+307], 1.0, N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999997e307

   1. Initial program 37.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   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 42.7

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 43.3%

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

   if 1.99999999999999997e307 < (*.f64 l l)

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 52.0%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+307}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+307}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{x}\\ \end{array} \end{array} \]

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 l_m) 2e+307) 1.0 (* (/ t_m l_m) (sqrt x)))))

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 * l_m) <= 2e+307) {
		tmp = 1.0;
	} else {
		tmp = (t_m / l_m) * sqrt(x);
	}
	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 * l_m) <= 2d+307) then
        tmp = 1.0d0
    else
        tmp = (t_m / l_m) * sqrt(x)
    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 * l_m) <= 2e+307) {
		tmp = 1.0;
	} else {
		tmp = (t_m / l_m) * Math.sqrt(x);
	}
	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 * l_m) <= 2e+307:
		tmp = 1.0
	else:
		tmp = (t_m / l_m) * math.sqrt(x)
	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 (Float64(l_m * l_m) <= 2e+307)
		tmp = 1.0;
	else
		tmp = Float64(Float64(t_m / l_m) * sqrt(x));
	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 * l_m) <= 2e+307)
		tmp = 1.0;
	else
		tmp = (t_m / l_m) * sqrt(x);
	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[N[(l$95$m * l$95$m), $MachinePrecision], 2e+307], 1.0, N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999997e307

   1. Initial program 37.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   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 42.7

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 43.3%

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

   if 1.99999999999999997e307 < (*.f64 l l)

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.9%

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

   10. Applied rewrites 43.5%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+307}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.7% 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 31.2%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. 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 36.9

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

5. Applied rewrites 36.9%

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

7. Applied rewrites 37.5%

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

Reproduce

herbie shell --seed 2024236 
(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)))))