Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 83.8%

Time: 11.3s

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: 33.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\left(2 + \frac{2}{x \cdot x}\right) + \frac{2}{x}}{x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 5.9 \cdot 10^{+44}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{\mathsf{fma}\left(t\_m \cdot \left(\frac{t\_m}{x} + t\_m\right), 2, {x}^{-1} \cdot \mathsf{fma}\left(l\_m, l\_m, \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\right)\right)}{2}}}{t\_m}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.65e-144)
      (/ t_2 (* (sqrt (/ (+ (+ 2.0 (/ 2.0 (* x x))) (/ 2.0 x)) x)) l_m))
      (if (<= t_m 5.9e+44)
        (pow
         (/
          (sqrt
           (/
            (fma
             (* t_m (+ (/ t_m x) t_m))
             2.0
             (* (pow x -1.0) (fma l_m l_m (fma (* t_m t_m) 2.0 (* l_m l_m)))))
            2.0))
          t_m)
         -1.0)
        (/ t_2 (* (sqrt (/ (fma x 2.0 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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.65e-144) {
		tmp = t_2 / (sqrt((((2.0 + (2.0 / (x * x))) + (2.0 / x)) / x)) * l_m);
	} else if (t_m <= 5.9e+44) {
		tmp = pow((sqrt((fma((t_m * ((t_m / x) + t_m)), 2.0, (pow(x, -1.0) * fma(l_m, l_m, fma((t_m * t_m), 2.0, (l_m * l_m))))) / 2.0)) / t_m), -1.0);
	} else {
		tmp = t_2 / (sqrt((fma(x, 2.0, 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.65e-144)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 + Float64(2.0 / Float64(x * x))) + Float64(2.0 / x)) / x)) * l_m));
	elseif (t_m <= 5.9e+44)
		tmp = Float64(sqrt(Float64(fma(Float64(t_m * Float64(Float64(t_m / x) + t_m)), 2.0, Float64((x ^ -1.0) * fma(l_m, l_m, fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))))) / 2.0)) / t_m) ^ -1.0;
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(x, 2.0, 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-144], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.9e+44], N[Power[N[(N[Sqrt[N[(N[(N[(t$95$m * N[(N[(t$95$m / x), $MachinePrecision] + t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[Power[x, -1.0], $MachinePrecision] * N[(l$95$m * l$95$m + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision], -1.0], $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999998e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 24.7%

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

   if 1.64999999999999998e-144 < t < 5.89999999999999965e44

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

   5. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   7. Applied rewrites 79.0%

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

   if 5.89999999999999965e44 < t

   1. Initial program 30.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.2%

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

   9. Applied rewrites 93.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(2 + \frac{2}{x \cdot x}\right) + \frac{2}{x}}{x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+44}:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{\mathsf{fma}\left(t \cdot \left(\frac{t}{x} + t\right), 2, {x}^{-1} \cdot \mathsf{fma}\left(\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)\right)\right)}{2}}}{t}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\left(2 + \frac{2}{x \cdot x}\right) + \frac{2}{x}}{x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 5.9 \cdot 10^{+44}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\mathsf{fma}\left(t\_m \cdot \left(\frac{t\_m}{x} + t\_m\right), 2, {x}^{-1} \cdot \mathsf{fma}\left(l\_m, l\_m, \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.65e-144)
      (/ t_2 (* (sqrt (/ (+ (+ 2.0 (/ 2.0 (* x x))) (/ 2.0 x)) x)) l_m))
      (if (<= t_m 5.9e+44)
        (*
         t_m
         (sqrt
          (/
           2.0
           (fma
            (* t_m (+ (/ t_m x) t_m))
            2.0
            (*
             (pow x -1.0)
             (fma l_m l_m (fma (* t_m t_m) 2.0 (* l_m l_m))))))))
        (/ t_2 (* (sqrt (/ (fma x 2.0 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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.65e-144) {
		tmp = t_2 / (sqrt((((2.0 + (2.0 / (x * x))) + (2.0 / x)) / x)) * l_m);
	} else if (t_m <= 5.9e+44) {
		tmp = t_m * sqrt((2.0 / fma((t_m * ((t_m / x) + t_m)), 2.0, (pow(x, -1.0) * fma(l_m, l_m, fma((t_m * t_m), 2.0, (l_m * l_m)))))));
	} else {
		tmp = t_2 / (sqrt((fma(x, 2.0, 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.65e-144)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 + Float64(2.0 / Float64(x * x))) + Float64(2.0 / x)) / x)) * l_m));
	elseif (t_m <= 5.9e+44)
		tmp = Float64(t_m * sqrt(Float64(2.0 / fma(Float64(t_m * Float64(Float64(t_m / x) + t_m)), 2.0, Float64((x ^ -1.0) * fma(l_m, l_m, fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))))))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(x, 2.0, 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-144], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.9e+44], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m / x), $MachinePrecision] + t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[Power[x, -1.0], $MachinePrecision] * N[(l$95$m * l$95$m + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999998e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 24.7%

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

   if 1.64999999999999998e-144 < t < 5.89999999999999965e44

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

   5. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

   7. Applied rewrites 78.6%

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

   if 5.89999999999999965e44 < t

   1. Initial program 30.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.2%

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

   9. Applied rewrites 93.2%

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

Alternative 3: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{\left(2 + \frac{2}{x \cdot x}\right) + \frac{2}{x}}{x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_2, \frac{t\_2}{x}\right) + \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{l\_m \cdot l\_m}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l_m l_m))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.65e-144)
      (/ t_3 (* (sqrt (/ (+ (+ 2.0 (/ 2.0 (* x x))) (/ 2.0 x)) x)) l_m))
      (if (<= t_m 5.8e+44)
        (/
         t_3
         (sqrt
          (fma
           (* 2.0 t_m)
           t_m
           (/
            (+
             (fma 2.0 t_2 (/ t_2 x))
             (fma (/ (* t_m t_m) x) 2.0 (/ (* l_m l_m) x)))
            x))))
        (/ t_3 (* (sqrt (/ (fma x 2.0 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 t_2 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.65e-144) {
		tmp = t_3 / (sqrt((((2.0 + (2.0 / (x * x))) + (2.0 / x)) / x)) * l_m);
	} else if (t_m <= 5.8e+44) {
		tmp = t_3 / sqrt(fma((2.0 * t_m), t_m, ((fma(2.0, t_2, (t_2 / x)) + fma(((t_m * t_m) / x), 2.0, ((l_m * l_m) / x))) / x)));
	} else {
		tmp = t_3 / (sqrt((fma(x, 2.0, 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)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.65e-144)
		tmp = Float64(t_3 / Float64(sqrt(Float64(Float64(Float64(2.0 + Float64(2.0 / Float64(x * x))) + Float64(2.0 / x)) / x)) * l_m));
	elseif (t_m <= 5.8e+44)
		tmp = Float64(t_3 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(Float64(fma(2.0, t_2, Float64(t_2 / x)) + fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(l_m * l_m) / x))) / x))));
	else
		tmp = Float64(t_3 / Float64(sqrt(Float64(fma(x, 2.0, 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_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-144], N[(t$95$3 / N[(N[Sqrt[N[(N[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.8e+44], N[(t$95$3 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(N[(2.0 * t$95$2 + N[(t$95$2 / 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$3 / N[(N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999998e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 24.7%

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

   if 1.64999999999999998e-144 < t < 5.8000000000000004e44

   1. Initial program 44.1%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 79.0%

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

   if 5.8000000000000004e44 < t

   1. Initial program 30.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.2%

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

   9. Applied rewrites 93.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(2 + \frac{2}{x \cdot x}\right) + \frac{2}{x}}{x}} \cdot \ell}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2 \cdot t, t, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}\right) + \mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\ell \cdot \ell}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\left(2 + \frac{2}{x \cdot x}\right) + \frac{2}{x}}{x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x} + t\_m \cdot t\_m, \frac{l\_m \cdot l\_m}{x} + \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.65e-144)
      (/ t_2 (* (sqrt (/ (+ (+ 2.0 (/ 2.0 (* x x))) (/ 2.0 x)) x)) l_m))
      (if (<= t_m 5.8e+44)
        (/
         t_2
         (sqrt
          (fma
           2.0
           (+ (/ (* t_m t_m) x) (* t_m t_m))
           (+ (/ (* l_m l_m) x) (/ (fma (* t_m t_m) 2.0 (* l_m l_m)) x)))))
        (/ t_2 (* (sqrt (/ (fma x 2.0 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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.65e-144) {
		tmp = t_2 / (sqrt((((2.0 + (2.0 / (x * x))) + (2.0 / x)) / x)) * l_m);
	} else if (t_m <= 5.8e+44) {
		tmp = t_2 / sqrt(fma(2.0, (((t_m * t_m) / x) + (t_m * t_m)), (((l_m * l_m) / x) + (fma((t_m * t_m), 2.0, (l_m * l_m)) / x))));
	} else {
		tmp = t_2 / (sqrt((fma(x, 2.0, 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.65e-144)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 + Float64(2.0 / Float64(x * x))) + Float64(2.0 / x)) / x)) * l_m));
	elseif (t_m <= 5.8e+44)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(Float64(Float64(t_m * t_m) / x) + Float64(t_m * t_m)), Float64(Float64(Float64(l_m * l_m) / x) + Float64(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)) / x)))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(x, 2.0, 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-144], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.8e+44], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999998e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 24.7%

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

   if 1.64999999999999998e-144 < t < 5.8000000000000004e44

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

   5. Applied rewrites 78.6%

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

   if 5.8000000000000004e44 < t

   1. Initial program 30.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.2%

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

   9. Applied rewrites 93.2%

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

Alternative 5: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\left(2 + \frac{2}{x \cdot x}\right) + \frac{2}{x}}{x}} \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x} + t\_m \cdot t\_m, \frac{l\_m \cdot l\_m}{x} \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.65e-144)
      (/ t_2 (* (sqrt (/ (+ (+ 2.0 (/ 2.0 (* x x))) (/ 2.0 x)) x)) l_m))
      (if (<= t_m 5.8e+44)
        (/
         t_2
         (sqrt
          (fma
           2.0
           (+ (/ (* t_m t_m) x) (* t_m t_m))
           (* (/ (* l_m l_m) x) 2.0))))
        (/ t_2 (* (sqrt (/ (fma x 2.0 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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.65e-144) {
		tmp = t_2 / (sqrt((((2.0 + (2.0 / (x * x))) + (2.0 / x)) / x)) * l_m);
	} else if (t_m <= 5.8e+44) {
		tmp = t_2 / sqrt(fma(2.0, (((t_m * t_m) / x) + (t_m * t_m)), (((l_m * l_m) / x) * 2.0)));
	} else {
		tmp = t_2 / (sqrt((fma(x, 2.0, 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.65e-144)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 + Float64(2.0 / Float64(x * x))) + Float64(2.0 / x)) / x)) * l_m));
	elseif (t_m <= 5.8e+44)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(Float64(Float64(t_m * t_m) / x) + Float64(t_m * t_m)), Float64(Float64(Float64(l_m * l_m) / x) * 2.0))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(x, 2.0, 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-144], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.8e+44], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999998e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 24.7%

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

   if 1.64999999999999998e-144 < t < 5.8000000000000004e44

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 77.4%

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

   if 5.8000000000000004e44 < t

   1. Initial program 30.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.2%

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

   9. Applied rewrites 93.2%

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

Alternative 6: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\left(2 + \frac{2}{x \cdot x}\right) + \frac{2}{x}}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.95e-144)
      (/ t_2 (* (sqrt (/ (+ (+ 2.0 (/ 2.0 (* x x))) (/ 2.0 x)) x)) l_m))
      (/ t_2 (* (sqrt (/ (fma x 2.0 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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.95e-144) {
		tmp = t_2 / (sqrt((((2.0 + (2.0 / (x * x))) + (2.0 / x)) / x)) * l_m);
	} else {
		tmp = t_2 / (sqrt((fma(x, 2.0, 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.95e-144)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(Float64(2.0 + Float64(2.0 / Float64(x * x))) + Float64(2.0 / x)) / x)) * l_m));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(fma(x, 2.0, 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.95e-144], N[(t$95$2 / N[(N[Sqrt[N[(N[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x * 2.0 + 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 < 1.95000000000000007e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 24.7%

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

   if 1.95000000000000007e-144 < t

   1. Initial program 33.8%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 87.2

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 87.3%

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

   9. Applied rewrites 87.3%

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

Alternative 7: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\left(\sqrt{0.5 \cdot x} \cdot t\_m\right) \cdot \frac{\sqrt{2}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \]

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 1.95e-144)
    (* (* (sqrt (* 0.5 x)) t_m) (/ (sqrt 2.0) l_m))
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ (fma x 2.0 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 <= 1.95e-144) {
		tmp = (sqrt((0.5 * x)) * t_m) * (sqrt(2.0) / l_m);
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt((fma(x, 2.0, 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 <= 1.95e-144)
		tmp = Float64(Float64(sqrt(Float64(0.5 * x)) * t_m) * Float64(sqrt(2.0) / l_m));
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(fma(x, 2.0, 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, 1.95e-144], N[(N[(N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(x * 2.0 + 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 < 1.95000000000000007e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 21.5%

       \[\leadsto \sqrt{0.5 \cdot x} \cdot \frac{\color{blue}{\sqrt{2}} \cdot t}{\ell} \]
   12. Step-by-step derivation

   13. Applied rewrites 24.3%

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

   if 1.95000000000000007e-144 < t

   1. Initial program 33.8%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 87.2

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 87.3%

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

   9. Applied rewrites 87.3%

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

Alternative 8: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\left(\sqrt{0.5 \cdot x} \cdot t\_m\right) \cdot \frac{\sqrt{2}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\\ \end{array} \end{array} \]

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 1.95e-144)
    (* (* (sqrt (* 0.5 x)) t_m) (/ (sqrt 2.0) l_m))
    (* t_m (/ (sqrt 2.0) (* (sqrt (/ (fma x 2.0 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 <= 1.95e-144) {
		tmp = (sqrt((0.5 * x)) * t_m) * (sqrt(2.0) / l_m);
	} else {
		tmp = t_m * (sqrt(2.0) / (sqrt((fma(x, 2.0, 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 <= 1.95e-144)
		tmp = Float64(Float64(sqrt(Float64(0.5 * x)) * t_m) * Float64(sqrt(2.0) / l_m));
	else
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(sqrt(Float64(fma(x, 2.0, 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, 1.95e-144], N[(N[(N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.95000000000000007e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 21.5%

       \[\leadsto \sqrt{0.5 \cdot x} \cdot \frac{\color{blue}{\sqrt{2}} \cdot t}{\ell} \]
   12. Step-by-step derivation

   13. Applied rewrites 24.3%

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

   if 1.95000000000000007e-144 < t

   1. Initial program 33.8%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 87.2

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 87.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.0

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

   9. Applied rewrites 87.0%

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

Alternative 9: 78.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\left(\sqrt{0.5 \cdot x} \cdot t\_m\right) \cdot \frac{\sqrt{2}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}} \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-144)
    (* (* (sqrt (* 0.5 x)) t_m) (/ (sqrt 2.0) l_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 <= 1.95e-144) {
		tmp = (sqrt((0.5 * x)) * t_m) * (sqrt(2.0) / l_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 <= 1.95d-144) then
        tmp = (sqrt((0.5d0 * x)) * t_m) * (sqrt(2.0d0) / l_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 <= 1.95e-144) {
		tmp = (Math.sqrt((0.5 * x)) * t_m) * (Math.sqrt(2.0) / l_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 <= 1.95e-144:
		tmp = (math.sqrt((0.5 * x)) * t_m) * (math.sqrt(2.0) / l_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 <= 1.95e-144)
		tmp = Float64(Float64(sqrt(Float64(0.5 * x)) * t_m) * Float64(sqrt(2.0) / l_m));
	else
		tmp = Float64(sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0))) * Float64(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 <= 1.95e-144)
		tmp = (sqrt((0.5 * x)) * t_m) * (sqrt(2.0) / l_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, 1.95e-144], N[(N[(N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.95000000000000007e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 21.5%

       \[\leadsto \sqrt{0.5 \cdot x} \cdot \frac{\color{blue}{\sqrt{2}} \cdot t}{\ell} \]
   12. Step-by-step derivation

   13. Applied rewrites 24.3%

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

   if 1.95000000000000007e-144 < t

   1. Initial program 33.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   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 85.9

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

   5. Applied rewrites 85.9%

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

Alternative 10: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\left(\sqrt{0.5 \cdot x} \cdot t\_m\right) \cdot \frac{\sqrt{2}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{4}{x} + 2} \cdot t\_m}\\ \end{array} \end{array} \]

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 1.95e-144)
    (* (* (sqrt (* 0.5 x)) t_m) (/ (sqrt 2.0) l_m))
    (/ (* (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) {
	double tmp;
	if (t_m <= 1.95e-144) {
		tmp = (sqrt((0.5 * x)) * t_m) * (sqrt(2.0) / l_m);
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt(((4.0 / x) + 2.0)) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.95d-144) then
        tmp = (sqrt((0.5d0 * x)) * t_m) * (sqrt(2.0d0) / l_m)
    else
        tmp = (sqrt(2.0d0) * t_m) / (sqrt(((4.0d0 / x) + 2.0d0)) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.95e-144) {
		tmp = (Math.sqrt((0.5 * x)) * t_m) * (Math.sqrt(2.0) / l_m);
	} else {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt(((4.0 / x) + 2.0)) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.95e-144:
		tmp = (math.sqrt((0.5 * x)) * t_m) * (math.sqrt(2.0) / l_m)
	else:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt(((4.0 / x) + 2.0)) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.95e-144)
		tmp = Float64(Float64(sqrt(Float64(0.5 * x)) * t_m) * Float64(sqrt(2.0) / l_m));
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(Float64(4.0 / x) + 2.0)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.95e-144)
		tmp = (sqrt((0.5 * x)) * t_m) * (sqrt(2.0) / l_m);
	else
		tmp = (sqrt(2.0) * t_m) / (sqrt(((4.0 / x) + 2.0)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.95e-144], N[(N[(N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if t < 1.95000000000000007e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 21.5%

       \[\leadsto \sqrt{0.5 \cdot x} \cdot \frac{\color{blue}{\sqrt{2}} \cdot t}{\ell} \]
   12. Step-by-step derivation

   13. Applied rewrites 24.3%

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

   if 1.95000000000000007e-144 < t

   1. Initial program 33.8%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 87.2

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 87.3%

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

   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 85.3%

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

Alternative 11: 78.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\left(\sqrt{0.5 \cdot x} \cdot t\_m\right) \cdot \frac{\sqrt{2}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-144) (* (* (sqrt (* 0.5 x)) t_m) (/ (sqrt 2.0) l_m)) 1.0)))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.95e-144) {
		tmp = (sqrt((0.5 * x)) * t_m) * (sqrt(2.0) / l_m);
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.95d-144) then
        tmp = (sqrt((0.5d0 * x)) * t_m) * (sqrt(2.0d0) / l_m)
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.95e-144) {
		tmp = (Math.sqrt((0.5 * x)) * t_m) * (Math.sqrt(2.0) / l_m);
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.95e-144:
		tmp = (math.sqrt((0.5 * x)) * t_m) * (math.sqrt(2.0) / l_m)
	else:
		tmp = 1.0
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.95e-144)
		tmp = Float64(Float64(sqrt(Float64(0.5 * x)) * t_m) * Float64(sqrt(2.0) / l_m));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.95e-144)
		tmp = (sqrt((0.5 * x)) * t_m) * (sqrt(2.0) / l_m);
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.95e-144], N[(N[(N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.95000000000000007e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 21.5%

       \[\leadsto \sqrt{0.5 \cdot x} \cdot \frac{\color{blue}{\sqrt{2}} \cdot t}{\ell} \]
   12. Step-by-step derivation

   13. Applied rewrites 24.3%

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

   if 1.95000000000000007e-144 < t

   1. Initial program 33.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 82.3

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

   5. Applied rewrites 82.3%

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

   7. Applied rewrites 83.6%

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

Alternative 12: 78.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{0.5 \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-144) (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (* 0.5 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) {
	double tmp;
	if (t_m <= 1.95e-144) {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((0.5 * x)));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.95d-144) then
        tmp = t_m * ((sqrt(2.0d0) / l_m) * sqrt((0.5d0 * x)))
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.95e-144) {
		tmp = t_m * ((Math.sqrt(2.0) / l_m) * Math.sqrt((0.5 * x)));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.95e-144:
		tmp = t_m * ((math.sqrt(2.0) / l_m) * math.sqrt((0.5 * x)))
	else:
		tmp = 1.0
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.95e-144)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(Float64(0.5 * x))));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.95e-144)
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((0.5 * x)));
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.95e-144], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.95000000000000007e-144

   1. Initial program 27.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 21.5%

       \[\leadsto \sqrt{0.5 \cdot x} \cdot \frac{\color{blue}{\sqrt{2}} \cdot t}{\ell} \]
   12. Step-by-step derivation

   13. Applied rewrites 24.3%

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

   if 1.95000000000000007e-144 < t

   1. Initial program 33.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 82.3

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

   5. Applied rewrites 82.3%

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

   7. Applied rewrites 83.6%

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

Alternative 13: 75.1% 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 30.4%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 38.5

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

5. Applied rewrites 38.5%

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

7. Applied rewrites 39.0%

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

Reproduce

herbie shell --seed 2024321 
(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)))))