Toniolo and Linder, Equation (7)

Percentage Accurate: 33.0% → 86.2%

Time: 11.8s

Alternatives: 14

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + \left(\frac{2}{x \cdot x} + 2\right)}{x}, -1, -2\right)}{-x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(l\_m, \frac{l\_m}{\left(x \cdot \sqrt{2}\right) \cdot t\_m}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{t\_3}{x}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2.25 \cdot 10^{+70}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, \frac{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{l\_m \cdot l\_m}{x}\right) + \mathsf{fma}\left(2, t\_2, \frac{t\_2}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

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 (fma (* t_m t_m) 2.0 (* l_m l_m))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5.8e-219)
      (/
       t_3
       (*
        (sqrt
         (/ (fma (/ (+ (/ 2.0 x) (+ (/ 2.0 (* x x)) 2.0)) x) -1.0 -2.0) (- x)))
        l_m))
      (if (<= t_m 6e-156)
        (/
         t_3
         (fma
          l_m
          (/ l_m (* (* x (sqrt 2.0)) t_m))
          (fma t_m (sqrt 2.0) (/ t_3 x))))
        (if (<= t_m 2.25e+70)
          (/
           t_3
           (sqrt
            (fma
             (* 2.0 t_m)
             t_m
             (/
              (+
               (fma (/ (* t_m t_m) x) 2.0 (/ (* l_m l_m) x))
               (fma 2.0 t_2 (/ t_2 x)))
              x))))
          (/ 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 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 5.8e-219) {
		tmp = t_3 / (sqrt((fma((((2.0 / x) + ((2.0 / (x * x)) + 2.0)) / x), -1.0, -2.0) / -x)) * l_m);
	} else if (t_m <= 6e-156) {
		tmp = t_3 / fma(l_m, (l_m / ((x * sqrt(2.0)) * t_m)), fma(t_m, sqrt(2.0), (t_3 / x)));
	} else if (t_m <= 2.25e+70) {
		tmp = t_3 / sqrt(fma((2.0 * t_m), t_m, ((fma(((t_m * t_m) / x), 2.0, ((l_m * l_m) / x)) + fma(2.0, t_2, (t_2 / x))) / x)));
	} 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 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 5.8e-219)
		tmp = Float64(t_3 / Float64(sqrt(Float64(fma(Float64(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / Float64(x * x)) + 2.0)) / x), -1.0, -2.0) / Float64(-x))) * l_m));
	elseif (t_m <= 6e-156)
		tmp = Float64(t_3 / fma(l_m, Float64(l_m / Float64(Float64(x * sqrt(2.0)) * t_m)), fma(t_m, sqrt(2.0), Float64(t_3 / x))));
	elseif (t_m <= 2.25e+70)
		tmp = Float64(t_3 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(Float64(fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(l_m * l_m) / x)) + fma(2.0, t_2, Float64(t_2 / x))) / x))));
	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[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e-219], N[(t$95$3 / N[(N[Sqrt[N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + -2.0), $MachinePrecision] / (-x)), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e-156], N[(t$95$3 / N[(l$95$m * N[(l$95$m / N[(N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.25e+70], N[(t$95$3 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(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] + N[(2.0 * t$95$2 + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 < 5.79999999999999968e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 16.1%

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

   if 5.79999999999999968e-219 < t < 6e-156

   1. Initial program 8.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.8%

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

   10. Applied rewrites 72.8%

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

   if 6e-156 < t < 2.25e70

   1. Initial program 69.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{-1 \cdot \frac{\left(-1 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right) - \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)}{x} + 2 \cdot {t}^{2}}}} \]
   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 98.3%

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

   if 2.25e70 < t

   1. Initial program 35.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + \left(\frac{2}{x \cdot x} + 2\right)}{x}, -1, -2\right)}{-x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\ell, \frac{\ell}{\left(x \cdot \sqrt{2}\right) \cdot t}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\sqrt{2} \cdot t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2 \cdot t, t, \frac{\mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\ell \cdot \ell}{x}\right) + \mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + \left(\frac{2}{x \cdot x} + 2\right)}{x}, -1, -2\right)}{-x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(l\_m, \frac{l\_m}{\left(x \cdot \sqrt{2}\right) \cdot t\_m}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{t\_2}{x}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2.25 \cdot 10^{+70}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x} + t\_m \cdot t\_m, \frac{l\_m \cdot l\_m}{x} + \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= t_m 5.8e-219)
      (/
       t_2
       (*
        (sqrt
         (/ (fma (/ (+ (/ 2.0 x) (+ (/ 2.0 (* x x)) 2.0)) x) -1.0 -2.0) (- x)))
        l_m))
      (if (<= t_m 6e-156)
        (/
         t_2
         (fma
          l_m
          (/ l_m (* (* x (sqrt 2.0)) t_m))
          (fma t_m (sqrt 2.0) (/ t_2 x))))
        (if (<= t_m 2.25e+70)
          (/
           t_2
           (sqrt
            (fma
             2.0
             (+ (/ (* t_m t_m) x) (* t_m t_m))
             (+ (/ (* l_m l_m) x) (/ (fma (* t_m t_m) 2.0 (* l_m l_m)) x)))))
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 5.8e-219) {
		tmp = t_2 / (sqrt((fma((((2.0 / x) + ((2.0 / (x * x)) + 2.0)) / x), -1.0, -2.0) / -x)) * l_m);
	} else if (t_m <= 6e-156) {
		tmp = t_2 / fma(l_m, (l_m / ((x * sqrt(2.0)) * t_m)), fma(t_m, sqrt(2.0), (t_2 / x)));
	} else if (t_m <= 2.25e+70) {
		tmp = t_2 / sqrt(fma(2.0, (((t_m * t_m) / x) + (t_m * t_m)), (((l_m * l_m) / x) + (fma((t_m * t_m), 2.0, (l_m * l_m)) / x))));
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 5.8e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(Float64(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / Float64(x * x)) + 2.0)) / x), -1.0, -2.0) / Float64(-x))) * l_m));
	elseif (t_m <= 6e-156)
		tmp = Float64(t_2 / fma(l_m, Float64(l_m / Float64(Float64(x * sqrt(2.0)) * t_m)), fma(t_m, sqrt(2.0), Float64(t_2 / x))));
	elseif (t_m <= 2.25e+70)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(Float64(Float64(t_m * t_m) / x) + Float64(t_m * t_m)), Float64(Float64(Float64(l_m * l_m) / x) + Float64(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)) / x)))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e-219], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + -2.0), $MachinePrecision] / (-x)), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e-156], N[(t$95$2 / N[(l$95$m * N[(l$95$m / N[(N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.25e+70], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $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 < 5.79999999999999968e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 16.1%

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

   if 5.79999999999999968e-219 < t < 6e-156

   1. Initial program 8.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.8%

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

   10. Applied rewrites 72.8%

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

   if 6e-156 < t < 2.25e70

   1. Initial program 69.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]
   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 97.8%

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

   if 2.25e70 < t

   1. Initial program 35.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + \left(\frac{2}{x \cdot x} + 2\right)}{x}, -1, -2\right)}{-x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\ell, \frac{\ell}{\left(x \cdot \sqrt{2}\right) \cdot t}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\sqrt{2} \cdot t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x} + t \cdot t, \frac{\ell \cdot \ell}{x} + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + \left(\frac{2}{x \cdot x} + 2\right)}{x}, -1, -2\right)}{-x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(l\_m, \frac{l\_m}{\left(x \cdot \sqrt{2}\right) \cdot t\_m}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{t\_2}{x}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5.8e-219)
      (/
       t_2
       (*
        (sqrt
         (/ (fma (/ (+ (/ 2.0 x) (+ (/ 2.0 (* x x)) 2.0)) x) -1.0 -2.0) (- x)))
        l_m))
      (/
       t_2
       (fma
        l_m
        (/ l_m (* (* x (sqrt 2.0)) t_m))
        (fma t_m (sqrt 2.0) (/ t_2 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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 5.8e-219) {
		tmp = t_2 / (sqrt((fma((((2.0 / x) + ((2.0 / (x * x)) + 2.0)) / x), -1.0, -2.0) / -x)) * l_m);
	} else {
		tmp = t_2 / fma(l_m, (l_m / ((x * sqrt(2.0)) * t_m)), fma(t_m, sqrt(2.0), (t_2 / 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 5.8e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(Float64(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / Float64(x * x)) + 2.0)) / x), -1.0, -2.0) / Float64(-x))) * l_m));
	else
		tmp = Float64(t_2 / fma(l_m, Float64(l_m / Float64(Float64(x * sqrt(2.0)) * t_m)), fma(t_m, sqrt(2.0), Float64(t_2 / x))));
	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, 5.8e-219], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + -2.0), $MachinePrecision] / (-x)), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(l$95$m * N[(l$95$m / N[(N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.79999999999999968e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 16.1%

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

   if 5.79999999999999968e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.4%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + \left(\frac{2}{x \cdot x} + 2\right)}{x}, -1, -2\right)}{-x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\ell, \frac{\ell}{\left(x \cdot \sqrt{2}\right) \cdot t}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\sqrt{2} \cdot t}{x}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + \left(\frac{2}{x \cdot x} + 2\right)}{x}, -1, -2\right)}{-x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{l\_m}{x \cdot \sqrt{2}}, \frac{l\_m}{t\_m}, t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5.8e-219)
      (/
       t_2
       (*
        (sqrt
         (/ (fma (/ (+ (/ 2.0 x) (+ (/ 2.0 (* x x)) 2.0)) x) -1.0 -2.0) (- x)))
        l_m))
      (/ t_2 (fma (/ l_m (* x (sqrt 2.0))) (/ l_m t_m) 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 <= 5.8e-219) {
		tmp = t_2 / (sqrt((fma((((2.0 / x) + ((2.0 / (x * x)) + 2.0)) / x), -1.0, -2.0) / -x)) * l_m);
	} else {
		tmp = t_2 / fma((l_m / (x * sqrt(2.0))), (l_m / t_m), 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 <= 5.8e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(Float64(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / Float64(x * x)) + 2.0)) / x), -1.0, -2.0) / Float64(-x))) * l_m));
	else
		tmp = Float64(t_2 / fma(Float64(l_m / Float64(x * sqrt(2.0))), Float64(l_m / t_m), 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, 5.8e-219], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + -2.0), $MachinePrecision] / (-x)), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(l$95$m / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.79999999999999968e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 16.1%

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

   if 5.79999999999999968e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 87.4%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 86.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + \left(\frac{2}{x \cdot x} + 2\right)}{x}, -1, -2\right)}{-x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\ell}{x \cdot \sqrt{2}}, \frac{\ell}{t}, \sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + 2}{x}, -1, -2\right)}{-x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{l\_m}{x \cdot \sqrt{2}}, \frac{l\_m}{t\_m}, t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5.8e-219)
      (/ t_2 (* (sqrt (/ (fma (/ (+ (/ 2.0 x) 2.0) x) -1.0 -2.0) (- x))) l_m))
      (/ t_2 (fma (/ l_m (* x (sqrt 2.0))) (/ l_m t_m) 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 <= 5.8e-219) {
		tmp = t_2 / (sqrt((fma((((2.0 / x) + 2.0) / x), -1.0, -2.0) / -x)) * l_m);
	} else {
		tmp = t_2 / fma((l_m / (x * sqrt(2.0))), (l_m / t_m), 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 <= 5.8e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(Float64(Float64(Float64(2.0 / x) + 2.0) / x), -1.0, -2.0) / Float64(-x))) * l_m));
	else
		tmp = Float64(t_2 / fma(Float64(l_m / Float64(x * sqrt(2.0))), Float64(l_m / t_m), 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, 5.8e-219], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision] * -1.0 + -2.0), $MachinePrecision] / (-x)), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(l$95$m / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.79999999999999968e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 16.1%

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

   if 5.79999999999999968e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 87.4%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 86.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(\frac{\frac{2}{x} + 2}{x}, -1, -2\right)}{-x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\ell}{x \cdot \sqrt{2}}, \frac{\ell}{t}, \sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\frac{2}{x} + 2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{l\_m}{x \cdot \sqrt{2}}, \frac{l\_m}{t\_m}, t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5.8e-219)
      (/ t_2 (* (sqrt (/ (+ (/ 2.0 x) 2.0) x)) l_m))
      (/ t_2 (fma (/ l_m (* x (sqrt 2.0))) (/ l_m t_m) 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 <= 5.8e-219) {
		tmp = t_2 / (sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	} else {
		tmp = t_2 / fma((l_m / (x * sqrt(2.0))), (l_m / t_m), 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 <= 5.8e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 / x) + 2.0) / x)) * l_m));
	else
		tmp = Float64(t_2 / fma(Float64(l_m / Float64(x * sqrt(2.0))), Float64(l_m / t_m), 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, 5.8e-219], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(l$95$m / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.79999999999999968e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.1%

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

   if 5.79999999999999968e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 87.4%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 86.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\frac{2}{x} + 2}{x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\ell}{x \cdot \sqrt{2}}, \frac{\ell}{t}, \sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\frac{2}{x} + 2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(l\_m, \frac{l\_m}{\left(x \cdot \sqrt{2}\right) \cdot t\_m}, t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5.8e-219)
      (/ t_2 (* (sqrt (/ (+ (/ 2.0 x) 2.0) x)) l_m))
      (/ t_2 (fma l_m (/ l_m (* (* x (sqrt 2.0)) t_m)) 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 <= 5.8e-219) {
		tmp = t_2 / (sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	} else {
		tmp = t_2 / fma(l_m, (l_m / ((x * sqrt(2.0)) * t_m)), 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 <= 5.8e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 / x) + 2.0) / x)) * l_m));
	else
		tmp = Float64(t_2 / fma(l_m, Float64(l_m / Float64(Float64(x * sqrt(2.0)) * t_m)), 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, 5.8e-219], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(l$95$m * N[(l$95$m / N[(N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.79999999999999968e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.1%

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

   if 5.79999999999999968e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.4%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 86.1%

       \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\ell, \frac{\ell}{\left(x \cdot \sqrt{2}\right) \cdot t}, \sqrt{2} \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\frac{2}{x} + 2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 6e-219)
      (/ t_2 (* (sqrt (/ (+ (/ 2.0 x) 2.0) x)) l_m))
      (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 6e-219) {
		tmp = t_2 / (sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * t_m
    if (t_m <= 6d-219) then
        tmp = t_2 / (sqrt((((2.0d0 / x) + 2.0d0) / x)) * l_m)
    else
        tmp = t_2 / (sqrt(((x - (-1.0d0)) / (x - 1.0d0))) * t_2)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 6e-219) {
		tmp = t_2 / (Math.sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	} else {
		tmp = t_2 / (Math.sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 6e-219:
		tmp = t_2 / (math.sqrt((((2.0 / x) + 2.0) / x)) * l_m)
	else:
		tmp = t_2 / (math.sqrt(((x - -1.0) / (x - 1.0))) * t_2)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 6e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 / x) + 2.0) / x)) * l_m));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 6e-219)
		tmp = t_2 / (sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	else
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6e-219], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000002e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.1%

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

   if 6.0000000000000002e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.3

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

   5. Applied rewrites 85.3%

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

Alternative 9: 80.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\frac{2}{x} + 2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1} \cdot 2} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 6e-219)
      (/ t_2 (* (sqrt (/ (+ (/ 2.0 x) 2.0) x)) l_m))
      (/ t_2 (* (sqrt (* (/ (- x -1.0) (- x 1.0)) 2.0)) 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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 6e-219) {
		tmp = t_2 / (sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	} else {
		tmp = t_2 / (sqrt((((x - -1.0) / (x - 1.0)) * 2.0)) * 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) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * t_m
    if (t_m <= 6d-219) then
        tmp = t_2 / (sqrt((((2.0d0 / x) + 2.0d0) / x)) * l_m)
    else
        tmp = t_2 / (sqrt((((x - (-1.0d0)) / (x - 1.0d0)) * 2.0d0)) * 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 t_2 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 6e-219) {
		tmp = t_2 / (Math.sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	} else {
		tmp = t_2 / (Math.sqrt((((x - -1.0) / (x - 1.0)) * 2.0)) * 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):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 6e-219:
		tmp = t_2 / (math.sqrt((((2.0 / x) + 2.0) / x)) * l_m)
	else:
		tmp = t_2 / (math.sqrt((((x - -1.0) / (x - 1.0)) * 2.0)) * 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 6e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 / x) + 2.0) / x)) * l_m));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(x - -1.0) / Float64(x - 1.0)) * 2.0)) * 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)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 6e-219)
		tmp = t_2 / (sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	else
		tmp = t_2 / (sqrt((((x - -1.0) / (x - 1.0)) * 2.0)) * 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6e-219], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000002e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.1%

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

   if 6.0000000000000002e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.3%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\frac{2}{x} + 2}{x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1} \cdot 2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1} \cdot 2} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 6e-219)
      (/ t_2 (* (sqrt (/ 2.0 x)) l_m))
      (/ t_2 (* (sqrt (* (/ (- x -1.0) (- x 1.0)) 2.0)) 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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 6e-219) {
		tmp = t_2 / (sqrt((2.0 / x)) * l_m);
	} else {
		tmp = t_2 / (sqrt((((x - -1.0) / (x - 1.0)) * 2.0)) * 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) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * t_m
    if (t_m <= 6d-219) then
        tmp = t_2 / (sqrt((2.0d0 / x)) * l_m)
    else
        tmp = t_2 / (sqrt((((x - (-1.0d0)) / (x - 1.0d0)) * 2.0d0)) * 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 t_2 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 6e-219) {
		tmp = t_2 / (Math.sqrt((2.0 / x)) * l_m);
	} else {
		tmp = t_2 / (Math.sqrt((((x - -1.0) / (x - 1.0)) * 2.0)) * 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):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 6e-219:
		tmp = t_2 / (math.sqrt((2.0 / x)) * l_m)
	else:
		tmp = t_2 / (math.sqrt((((x - -1.0) / (x - 1.0)) * 2.0)) * 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 6e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(2.0 / x)) * l_m));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(x - -1.0) / Float64(x - 1.0)) * 2.0)) * 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)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 6e-219)
		tmp = t_2 / (sqrt((2.0 / x)) * l_m);
	else
		tmp = t_2 / (sqrt((((x - -1.0) / (x - 1.0)) * 2.0)) * 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6e-219], N[(t$95$2 / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000002e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.1%

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

   if 6.0000000000000002e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.3%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1} \cdot 2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\left(\frac{1}{x} + 1\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 6e-219)
      (/ t_2 (* (sqrt (/ 2.0 x)) l_m))
      (/ t_2 (* (+ (/ 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 <= 6e-219) {
		tmp = t_2 / (sqrt((2.0 / x)) * l_m);
	} else {
		tmp = t_2 / (((1.0 / x) + 1.0) * t_2);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * t_m
    if (t_m <= 6d-219) then
        tmp = t_2 / (sqrt((2.0d0 / x)) * l_m)
    else
        tmp = t_2 / (((1.0d0 / x) + 1.0d0) * t_2)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 6e-219) {
		tmp = t_2 / (Math.sqrt((2.0 / x)) * l_m);
	} else {
		tmp = t_2 / (((1.0 / x) + 1.0) * t_2);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 6e-219:
		tmp = t_2 / (math.sqrt((2.0 / x)) * l_m)
	else:
		tmp = t_2 / (((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 <= 6e-219)
		tmp = Float64(t_2 / Float64(sqrt(Float64(2.0 / x)) * l_m));
	else
		tmp = Float64(t_2 / Float64(Float64(Float64(1.0 / x) + 1.0) * t_2));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 6e-219)
		tmp = t_2 / (sqrt((2.0 / x)) * l_m);
	else
		tmp = t_2 / (((1.0 / x) + 1.0) * t_2);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6e-219], N[(t$95$2 / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(N[(1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000002e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.1%

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

   if 6.0000000000000002e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.7%

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

Alternative 12: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{4}{x} + 2} \cdot t\_m} \cdot t\_m\\ \end{array} \end{array} \]

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 (<= t_m 6e-219)
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 2.0 x)) l_m))
    (* (/ (sqrt 2.0) (* (sqrt (+ (/ 4.0 x) 2.0)) t_m)) 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 (t_m <= 6e-219) {
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	} else {
		tmp = (sqrt(2.0) / (sqrt(((4.0 / x) + 2.0)) * t_m)) * 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 (t_m <= 6d-219) then
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((2.0d0 / x)) * l_m)
    else
        tmp = (sqrt(2.0d0) / (sqrt(((4.0d0 / x) + 2.0d0)) * t_m)) * 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 (t_m <= 6e-219) {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((2.0 / x)) * l_m);
	} else {
		tmp = (Math.sqrt(2.0) / (Math.sqrt(((4.0 / x) + 2.0)) * t_m)) * 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 t_m <= 6e-219:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((2.0 / x)) * l_m)
	else:
		tmp = (math.sqrt(2.0) / (math.sqrt(((4.0 / x) + 2.0)) * t_m)) * 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 (t_m <= 6e-219)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(2.0 / x)) * l_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(4.0 / x) + 2.0)) * t_m)) * 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 (t_m <= 6e-219)
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	else
		tmp = (sqrt(2.0) / (sqrt(((4.0 / x) + 2.0)) * t_m)) * 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[t$95$m, 6e-219], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000002e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.1%

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

   if 6.0000000000000002e-219 < t

   1. Initial program 41.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.0

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

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 84.4%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{4}{x} + 2} \cdot t} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 (<= t_m 6e-219) (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 2.0 x)) l_m)) 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) {
	double tmp;
	if (t_m <= 6e-219) {
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	} else {
		tmp = 1.0;
	}
	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 (t_m <= 6d-219) then
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((2.0d0 / x)) * l_m)
    else
        tmp = 1.0d0
    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 (t_m <= 6e-219) {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((2.0 / x)) * l_m);
	} else {
		tmp = 1.0;
	}
	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 t_m <= 6e-219:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((2.0 / x)) * l_m)
	else:
		tmp = 1.0
	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)
	tmp = 0.0
	if (t_m <= 6e-219)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(2.0 / x)) * l_m));
	else
		tmp = 1.0;
	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 (t_m <= 6e-219)
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	else
		tmp = 1.0;
	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[t$95$m, 6e-219], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000002e-219

   1. Initial program 25.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 1.7

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.1%

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

   if 6.0000000000000002e-219 < t

   1. Initial program 41.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 83.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 75.4% 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 33.1%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 38.7

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

5. Applied rewrites 38.7%

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

7. Applied rewrites 39.3%

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

Reproduce

herbie shell --seed 2024283 
(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)))))