Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 85.5%

Time: 10.2s

Alternatives: 13

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{l\_m \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, \frac{\mathsf{fma}\left(t\_3, -2, \frac{\frac{t\_3}{x} + \mathsf{fma}\left(2, t\_3, \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{l\_m \cdot l\_m}{x}\right)\right)}{-x}\right)}{-x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)) (t_3 (fma (* t_m t_m) 2.0 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 1.5e-241)
      (* (* (/ (- -1.0) (* l_m (sqrt (/ 2.0 x)))) (sqrt 2.0)) t_m)
      (if (<= t_m 8e-165)
        (/ t_2 (fma (/ 0.5 (* x (sqrt 2.0))) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (if (<= t_m 6e+56)
          (/
           t_2
           (sqrt
            (fma
             (* 2.0 t_m)
             t_m
             (/
              (fma
               t_3
               -2.0
               (/
                (+
                 (/ t_3 x)
                 (fma 2.0 t_3 (fma (/ (* t_m t_m) x) 2.0 (/ (* l_m l_m) x))))
                (- x)))
              (- x)))))
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double t_3 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double tmp;
	if (t_m <= 1.5e-241) {
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_m;
	} else if (t_m <= 8e-165) {
		tmp = t_2 / fma((0.5 / (x * sqrt(2.0))), (((l_m * l_m) / t_m) * 2.0), t_2);
	} else if (t_m <= 6e+56) {
		tmp = t_2 / sqrt(fma((2.0 * t_m), t_m, (fma(t_3, -2.0, (((t_3 / x) + fma(2.0, t_3, fma(((t_m * t_m) / x), 2.0, ((l_m * l_m) / x)))) / -x)) / -x)));
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	t_3 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 1.5e-241)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / Float64(l_m * sqrt(Float64(2.0 / x)))) * sqrt(2.0)) * t_m);
	elseif (t_m <= 8e-165)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	elseif (t_m <= 6e+56)
		tmp = Float64(t_2 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(fma(t_3, -2.0, Float64(Float64(Float64(t_3 / x) + fma(2.0, t_3, fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(l_m * l_m) / x)))) / Float64(-x))) / Float64(-x)))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e-241], N[(N[(N[((--1.0) / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 8e-165], N[(t$95$2 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+56], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(t$95$3 * -2.0 + N[(N[(N[(t$95$3 / x), $MachinePrecision] + N[(2.0 * t$95$3 + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.5e-241

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.7%

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

   if 1.5e-241 < t < 8.0000000000000001e-165

   1. Initial program 8.7%

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

   3. Taylor expanded in 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 69.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.1%

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

   if 8.0000000000000001e-165 < t < 6.00000000000000012e56

   1. Initial program 62.4%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 89.6%

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

   if 6.00000000000000012e56 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2 \cdot t, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), -2, \frac{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\ell \cdot \ell}{x}\right)\right)}{-x}\right)}{-x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{l\_m \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_3, \frac{t\_3}{x}\right) + \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{l\_m \cdot l\_m}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)) (t_3 (fma (* t_m t_m) 2.0 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 1.5e-241)
      (* (* (/ (- -1.0) (* l_m (sqrt (/ 2.0 x)))) (sqrt 2.0)) t_m)
      (if (<= t_m 8e-165)
        (/ t_2 (fma (/ 0.5 (* x (sqrt 2.0))) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (if (<= t_m 6e+56)
          (/
           t_2
           (sqrt
            (fma
             (* 2.0 t_m)
             t_m
             (/
              (+
               (fma 2.0 t_3 (/ t_3 x))
               (fma (/ (* t_m t_m) x) 2.0 (/ (* l_m l_m) x)))
              x))))
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double t_3 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double tmp;
	if (t_m <= 1.5e-241) {
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_m;
	} else if (t_m <= 8e-165) {
		tmp = t_2 / fma((0.5 / (x * sqrt(2.0))), (((l_m * l_m) / t_m) * 2.0), t_2);
	} else if (t_m <= 6e+56) {
		tmp = t_2 / sqrt(fma((2.0 * t_m), t_m, ((fma(2.0, t_3, (t_3 / x)) + fma(((t_m * t_m) / x), 2.0, ((l_m * l_m) / x))) / x)));
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	t_3 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 1.5e-241)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / Float64(l_m * sqrt(Float64(2.0 / x)))) * sqrt(2.0)) * t_m);
	elseif (t_m <= 8e-165)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	elseif (t_m <= 6e+56)
		tmp = Float64(t_2 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(Float64(fma(2.0, t_3, Float64(t_3 / x)) + fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(l_m * l_m) / x))) / x))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e-241], N[(N[(N[((--1.0) / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 8e-165], N[(t$95$2 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+56], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(N[(2.0 * t$95$3 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.5e-241

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.7%

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

   if 1.5e-241 < t < 8.0000000000000001e-165

   1. Initial program 8.7%

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

   3. Taylor expanded in 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 69.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.1%

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

   if 8.0000000000000001e-165 < t < 6.00000000000000012e56

   1. Initial program 62.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 89.3%

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

   if 6.00000000000000012e56 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2 \cdot t, t, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}\right) + \mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\ell \cdot \ell}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{l\_m \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x} + t\_m \cdot t\_m, \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x} + \frac{l\_m \cdot l\_m}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.5e-241)
      (* (* (/ (- -1.0) (* l_m (sqrt (/ 2.0 x)))) (sqrt 2.0)) t_m)
      (if (<= t_m 8e-165)
        (/ t_2 (fma (/ 0.5 (* x (sqrt 2.0))) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (if (<= t_m 6e+56)
          (/
           t_2
           (sqrt
            (fma
             2.0
             (+ (/ (* t_m t_m) x) (* t_m t_m))
             (+ (/ (fma (* t_m t_m) 2.0 (* l_m l_m)) x) (/ (* l_m l_m) x)))))
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.5e-241) {
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_m;
	} else if (t_m <= 8e-165) {
		tmp = t_2 / fma((0.5 / (x * sqrt(2.0))), (((l_m * l_m) / t_m) * 2.0), t_2);
	} else if (t_m <= 6e+56) {
		tmp = t_2 / sqrt(fma(2.0, (((t_m * t_m) / x) + (t_m * t_m)), ((fma((t_m * t_m), 2.0, (l_m * l_m)) / x) + ((l_m * l_m) / x))));
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.5e-241)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / Float64(l_m * sqrt(Float64(2.0 / x)))) * sqrt(2.0)) * t_m);
	elseif (t_m <= 8e-165)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	elseif (t_m <= 6e+56)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(Float64(Float64(t_m * t_m) / x) + Float64(t_m * t_m)), Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)) / x) + Float64(Float64(l_m * l_m) / x)))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e-241], N[(N[(N[((--1.0) / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 8e-165], N[(t$95$2 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+56], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.5e-241

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.7%

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

   if 1.5e-241 < t < 8.0000000000000001e-165

   1. Initial program 8.7%

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

   3. Taylor expanded in 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 69.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.1%

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

   if 8.0000000000000001e-165 < t < 6.00000000000000012e56

   1. Initial program 62.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 89.0%

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

   if 6.00000000000000012e56 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x} + t \cdot t, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \mathsf{fma}\left(l\_m, l\_m, \left(t\_m \cdot t\_m\right) \cdot 2\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{l\_m \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\frac{2}{\frac{t\_3 + t\_3}{x} - \left(l\_m \cdot l\_m - t\_3\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)) (t_3 (fma l_m l_m (* (* t_m t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 1.5e-241)
      (* (* (/ (- -1.0) (* l_m (sqrt (/ 2.0 x)))) (sqrt 2.0)) t_m)
      (if (<= t_m 8e-165)
        (/ t_2 (fma (/ 0.5 (* x (sqrt 2.0))) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (if (<= t_m 2.2e-83)
          (* (sqrt (/ 2.0 (- (/ (+ t_3 t_3) x) (- (* l_m l_m) t_3)))) t_m)
          (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double t_3 = fma(l_m, l_m, ((t_m * t_m) * 2.0));
	double tmp;
	if (t_m <= 1.5e-241) {
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_m;
	} else if (t_m <= 8e-165) {
		tmp = t_2 / fma((0.5 / (x * sqrt(2.0))), (((l_m * l_m) / t_m) * 2.0), t_2);
	} else if (t_m <= 2.2e-83) {
		tmp = sqrt((2.0 / (((t_3 + t_3) / x) - ((l_m * l_m) - t_3)))) * t_m;
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	t_3 = fma(l_m, l_m, Float64(Float64(t_m * t_m) * 2.0))
	tmp = 0.0
	if (t_m <= 1.5e-241)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / Float64(l_m * sqrt(Float64(2.0 / x)))) * sqrt(2.0)) * t_m);
	elseif (t_m <= 8e-165)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	elseif (t_m <= 2.2e-83)
		tmp = Float64(sqrt(Float64(2.0 / Float64(Float64(Float64(t_3 + t_3) / x) - Float64(Float64(l_m * l_m) - t_3)))) * t_m);
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(l$95$m * l$95$m + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e-241], N[(N[(N[((--1.0) / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 8e-165], N[(t$95$2 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e-83], N[(N[Sqrt[N[(2.0 / N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] / x), $MachinePrecision] - N[(N[(l$95$m * l$95$m), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.5e-241

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.7%

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

   if 1.5e-241 < t < 8.0000000000000001e-165

   1. Initial program 8.7%

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

   3. Taylor expanded in 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 69.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.1%

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

   if 8.0000000000000001e-165 < t < 2.20000000000000008e-83

   1. Initial program 43.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 8.9

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

   5. Applied rewrites 8.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 9.0

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

   7. Applied rewrites 8.8%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   10. Applied rewrites 73.6%

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

   if 2.20000000000000008e-83 < t

   1. Initial program 43.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\frac{2}{\frac{\mathsf{fma}\left(\ell, \ell, \left(t \cdot t\right) \cdot 2\right) + \mathsf{fma}\left(\ell, \ell, \left(t \cdot t\right) \cdot 2\right)}{x} - \left(\ell \cdot \ell - \mathsf{fma}\left(\ell, \ell, \left(t \cdot t\right) \cdot 2\right)\right)}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{l\_m \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{l\_m \cdot l\_m}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 1.5e-241)
      (* (* (/ (- -1.0) (* l_m (sqrt (/ 2.0 x)))) (sqrt 2.0)) t_m)
      (if (<= t_m 3.2e-15)
        (/ t_2 (fma (/ 0.5 (* x (sqrt 2.0))) (* (/ (* l_m l_m) t_m) 2.0) t_2))
        (/ 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 <= 1.5e-241) {
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_m;
	} else if (t_m <= 3.2e-15) {
		tmp = t_2 / fma((0.5 / (x * sqrt(2.0))), (((l_m * l_m) / t_m) * 2.0), t_2);
	} 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 <= 1.5e-241)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / Float64(l_m * sqrt(Float64(2.0 / x)))) * sqrt(2.0)) * t_m);
	elseif (t_m <= 3.2e-15)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(Float64(l_m * l_m) / t_m) * 2.0), t_2));
	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, 1.5e-241], N[(N[(N[((--1.0) / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 3.2e-15], N[(t$95$2 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $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 3 regimes

2. if t < 1.5e-241

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.7%

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

   if 1.5e-241 < t < 3.1999999999999999e-15

   1. Initial program 42.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{1}{2} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}} \]
   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 69.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.2%

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

   if 3.1999999999999999e-15 < t

   1. Initial program 38.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{l\_m \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.4e-241)
      (* (* (/ (- -1.0) (* l_m (sqrt (/ 2.0 x)))) (sqrt 2.0)) t_m)
      (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.4e-241) {
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_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 <= 2.4d-241) then
        tmp = ((-(-1.0d0) / (l_m * sqrt((2.0d0 / x)))) * sqrt(2.0d0)) * t_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 <= 2.4e-241) {
		tmp = ((-(-1.0) / (l_m * Math.sqrt((2.0 / x)))) * Math.sqrt(2.0)) * t_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 <= 2.4e-241:
		tmp = ((-(-1.0) / (l_m * math.sqrt((2.0 / x)))) * math.sqrt(2.0)) * t_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 <= 2.4e-241)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / Float64(l_m * sqrt(Float64(2.0 / x)))) * sqrt(2.0)) * t_m);
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

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 <= 2.4e-241)
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_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, 2.4e-241], N[(N[(N[((--1.0) / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.4e-241

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.7%

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

   if 2.4e-241 < t

   1. Initial program 39.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{l\_m \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \]

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 2.4e-241)
    (* (* (/ (- -1.0) (* l_m (sqrt (/ 2.0 x)))) (sqrt 2.0)) t_m)
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ (fma 2.0 x 2.0) (- x 1.0))) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-241) {
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_m;
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt((fma(2.0, x, 2.0) / (x - 1.0))) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.4e-241)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / Float64(l_m * sqrt(Float64(2.0 / x)))) * sqrt(2.0)) * t_m);
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(fma(2.0, x, 2.0) / Float64(x - 1.0))) * t_m));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-241], N[(N[(N[((--1.0) / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.4e-241

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.7%

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

   if 2.4e-241 < t

   1. Initial program 39.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 85.7%

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

   9. Applied rewrites 85.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-/.f64 85.7

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

   11. Applied rewrites 85.7%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{l\_m \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x - 1}{x - -1}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

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 2.4e-241)
    (* (* (/ (- -1.0) (* l_m (sqrt (/ 2.0 x)))) (sqrt 2.0)) t_m)
    (* (* (sqrt (/ (- x 1.0) (- x -1.0))) (sqrt 0.5)) (sqrt 2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-241) {
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_m;
	} else {
		tmp = (sqrt(((x - 1.0) / (x - -1.0))) * sqrt(0.5)) * sqrt(2.0);
	}
	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 <= 2.4d-241) then
        tmp = ((-(-1.0d0) / (l_m * sqrt((2.0d0 / x)))) * sqrt(2.0d0)) * t_m
    else
        tmp = (sqrt(((x - 1.0d0) / (x - (-1.0d0)))) * sqrt(0.5d0)) * sqrt(2.0d0)
    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 <= 2.4e-241) {
		tmp = ((-(-1.0) / (l_m * Math.sqrt((2.0 / x)))) * Math.sqrt(2.0)) * t_m;
	} else {
		tmp = (Math.sqrt(((x - 1.0) / (x - -1.0))) * Math.sqrt(0.5)) * Math.sqrt(2.0);
	}
	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 <= 2.4e-241:
		tmp = ((-(-1.0) / (l_m * math.sqrt((2.0 / x)))) * math.sqrt(2.0)) * t_m
	else:
		tmp = (math.sqrt(((x - 1.0) / (x - -1.0))) * math.sqrt(0.5)) * math.sqrt(2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.4e-241)
		tmp = Float64(Float64(Float64(Float64(-(-1.0)) / Float64(l_m * sqrt(Float64(2.0 / x)))) * sqrt(2.0)) * t_m);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0))) * sqrt(0.5)) * sqrt(2.0));
	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 <= 2.4e-241)
		tmp = ((-(-1.0) / (l_m * sqrt((2.0 / x)))) * sqrt(2.0)) * t_m;
	else
		tmp = (sqrt(((x - 1.0) / (x - -1.0))) * sqrt(0.5)) * sqrt(2.0);
	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, 2.4e-241], N[(N[(N[((--1.0) / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.4e-241

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.7%

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

   if 2.4e-241 < t

   1. Initial program 39.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 85.7%

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

   9. Applied rewrites 85.5%

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

   10. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 N/A

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 84.4

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

   12. Applied rewrites 84.4%

       \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{x - -1}}\right)} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{--1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x - 1}{x - -1}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-286}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{\left(l\_m \cdot l\_m\right) \cdot 2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x - 1}{x - -1}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

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 2.3e-286)
    (/ (* (sqrt 2.0) t_m) (sqrt (/ (* (* l_m l_m) 2.0) x)))
    (* (* (sqrt (/ (- x 1.0) (- x -1.0))) (sqrt 0.5)) (sqrt 2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.3e-286) {
		tmp = (sqrt(2.0) * t_m) / sqrt((((l_m * l_m) * 2.0) / x));
	} else {
		tmp = (sqrt(((x - 1.0) / (x - -1.0))) * sqrt(0.5)) * sqrt(2.0);
	}
	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 <= 2.3d-286) then
        tmp = (sqrt(2.0d0) * t_m) / sqrt((((l_m * l_m) * 2.0d0) / x))
    else
        tmp = (sqrt(((x - 1.0d0) / (x - (-1.0d0)))) * sqrt(0.5d0)) * sqrt(2.0d0)
    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 <= 2.3e-286) {
		tmp = (Math.sqrt(2.0) * t_m) / Math.sqrt((((l_m * l_m) * 2.0) / x));
	} else {
		tmp = (Math.sqrt(((x - 1.0) / (x - -1.0))) * Math.sqrt(0.5)) * Math.sqrt(2.0);
	}
	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 <= 2.3e-286:
		tmp = (math.sqrt(2.0) * t_m) / math.sqrt((((l_m * l_m) * 2.0) / x))
	else:
		tmp = (math.sqrt(((x - 1.0) / (x - -1.0))) * math.sqrt(0.5)) * math.sqrt(2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.3e-286)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(Float64(Float64(l_m * l_m) * 2.0) / x)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0))) * sqrt(0.5)) * sqrt(2.0));
	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 <= 2.3e-286)
		tmp = (sqrt(2.0) * t_m) / sqrt((((l_m * l_m) * 2.0) / x));
	else
		tmp = (sqrt(((x - 1.0) / (x - -1.0))) * sqrt(0.5)) * sqrt(2.0);
	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, 2.3e-286], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.3000000000000002e-286

   1. Initial program 26.4%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 2.2

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 18.0%

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

   if 2.3000000000000002e-286 < t

   1. Initial program 37.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 82.8%

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

   9. Applied rewrites 82.6%

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

   10. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 N/A

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

       8. sub-neg N/A

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

       9. lower--.f64 81.6

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

   12. Applied rewrites 81.6%

       \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{x - -1}}\right)} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-286}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(\ell \cdot \ell\right) \cdot 2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x - 1}{x - -1}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\sqrt{\frac{x - 1}{x - -1}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \end{array} \]

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 (* (* (sqrt (/ (- x 1.0) (- x -1.0))) (sqrt 0.5)) (sqrt 2.0))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * ((sqrt(((x - 1.0) / (x - -1.0))) * sqrt(0.5)) * sqrt(2.0));
}

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 * ((sqrt(((x - 1.0d0) / (x - (-1.0d0)))) * sqrt(0.5d0)) * sqrt(2.0d0))
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 * ((Math.sqrt(((x - 1.0) / (x - -1.0))) * Math.sqrt(0.5)) * Math.sqrt(2.0));
}

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 * ((math.sqrt(((x - 1.0) / (x - -1.0))) * math.sqrt(0.5)) * math.sqrt(2.0))

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 * Float64(Float64(sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0))) * sqrt(0.5)) * sqrt(2.0)))
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 * ((sqrt(((x - 1.0) / (x - -1.0))) * sqrt(0.5)) * sqrt(2.0));
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 * N[(N[(N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 43.6

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

5. Applied rewrites 43.6%

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

7. Applied rewrites 43.6%

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

9. Applied rewrites 43.5%

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

10. Taylor expanded in t around inf

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

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-sqrt.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower--.f64 N/A

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

    6. +-commutative N/A

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

    7. metadata-eval N/A

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

    8. sub-neg N/A

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

    9. lower--.f64 43.0

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

12. Applied rewrites 43.0%

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

13. Final simplification 43.0%

    \[\leadsto \left(\sqrt{\frac{x - 1}{x - -1}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2} \]
14. Add Preprocessing

Alternative 11: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}\right) \end{array} \]

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 (* (* (sqrt 0.5) (sqrt 2.0)) (sqrt (/ (- x 1.0) (- x -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 * ((sqrt(0.5) * sqrt(2.0)) * sqrt(((x - 1.0) / (x - -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 * ((sqrt(0.5d0) * sqrt(2.0d0)) * sqrt(((x - 1.0d0) / (x - (-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 * ((Math.sqrt(0.5) * Math.sqrt(2.0)) * Math.sqrt(((x - 1.0) / (x - -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 * ((math.sqrt(0.5) * math.sqrt(2.0)) * math.sqrt(((x - 1.0) / (x - -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 * Float64(Float64(sqrt(0.5) * sqrt(2.0)) * sqrt(Float64(Float64(x - 1.0) / Float64(x - -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 * ((sqrt(0.5) * sqrt(2.0)) * sqrt(((x - 1.0) / (x - -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 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 43.0

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

5. Applied rewrites 43.0%

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

6. Final simplification 43.0%

   \[\leadsto \left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}} \]
7. Add Preprocessing

Alternative 12: 76.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{4}{x} + 2} \cdot t\_m} \end{array} \]

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 (/ (* (sqrt 2.0) t_m) (* (sqrt (+ (/ 4.0 x) 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) {
	return t_s * ((sqrt(2.0) * t_m) / (sqrt(((4.0 / x) + 2.0)) * t_m));
}

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 * ((sqrt(2.0d0) * t_m) / (sqrt(((4.0d0 / x) + 2.0d0)) * t_m))
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 * ((Math.sqrt(2.0) * t_m) / (Math.sqrt(((4.0 / x) + 2.0)) * t_m));
}

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 * ((math.sqrt(2.0) * t_m) / (math.sqrt(((4.0 / x) + 2.0)) * t_m))

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 * Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(Float64(4.0 / x) + 2.0)) * t_m)))
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 * ((sqrt(2.0) * t_m) / (sqrt(((4.0 / x) + 2.0)) * t_m));
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 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 43.6

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

5. Applied rewrites 43.6%

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

7. Applied rewrites 43.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 42.9%

    \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{x} + 2} \cdot t} \]
11. Add Preprocessing

Alternative 13: 75.7% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 42.0

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

5. Applied rewrites 42.0%

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

7. Applied rewrites 42.6%

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

Reproduce

herbie shell --seed 2024288 
(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)))))