Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 84.3%

Time: 12.8s

Alternatives: 10

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-251}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\_m\right)\right)}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.18 \cdot 10^{-181}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{\left(--2\right) \cdot t\_3}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{--1}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, 0\right) - \frac{-2 \cdot t\_3}{x}}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)) (t_3 (fma (* t_m t_m) 2.0 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 1.25e-251)
      (/ (* (sqrt x) (* (sqrt 2.0) (* (sqrt 0.5) t_m))) l_m)
      (if (<= t_m 1.18e-181)
        (/ t_2 (fma (/ (* (- -2.0) t_3) (* (* (sqrt 2.0) x) t_m)) 0.5 t_2))
        (if (<= t_m 6e+86)
          (*
           (/ (- -1.0) (sqrt (- (fma (* t_m t_m) 2.0 0.0) (/ (* -2.0 t_3) x))))
           t_2)
          (sqrt (/ (- x 1.0) (- x -1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double t_3 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double tmp;
	if (t_m <= 1.25e-251) {
		tmp = (sqrt(x) * (sqrt(2.0) * (sqrt(0.5) * t_m))) / l_m;
	} else if (t_m <= 1.18e-181) {
		tmp = t_2 / fma(((-(-2.0) * t_3) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else if (t_m <= 6e+86) {
		tmp = (-(-1.0) / sqrt((fma((t_m * t_m), 2.0, 0.0) - ((-2.0 * t_3) / x)))) * t_2;
	} else {
		tmp = sqrt(((x - 1.0) / (x - -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)
	t_2 = Float64(sqrt(2.0) * t_m)
	t_3 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 1.25e-251)
		tmp = Float64(Float64(sqrt(x) * Float64(sqrt(2.0) * Float64(sqrt(0.5) * t_m))) / l_m);
	elseif (t_m <= 1.18e-181)
		tmp = Float64(t_2 / fma(Float64(Float64(Float64(-(-2.0)) * t_3) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	elseif (t_m <= 6e+86)
		tmp = Float64(Float64(Float64(-(-1.0)) / sqrt(Float64(fma(Float64(t_m * t_m), 2.0, 0.0) - Float64(Float64(-2.0 * t_3) / x)))) * t_2);
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.25e-251], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.18e-181], N[(t$95$2 / N[(N[(N[((--2.0) * t$95$3), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+86], N[(N[((--1.0) / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + 0.0), $MachinePrecision] - N[(N[(-2.0 * t$95$3), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.2500000000000001e-251

   1. Initial program 37.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 4.0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 8.2

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

   8. Applied rewrites 8.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 13.9%

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

   if 1.2500000000000001e-251 < t < 1.17999999999999994e-181

   1. Initial program 3.0%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.0%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 68.9%

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

   if 1.17999999999999994e-181 < t < 5.99999999999999954e86

   1. Initial program 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\left(\left(-t\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -2}{x}}}} \]

   if 5.99999999999999954e86 < t

   1. Initial program 31.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 95.2

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

   8. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.7%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-251}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\right)\right)}{\ell}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\left(--2\right) \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{--1}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - \frac{-2 \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}} \cdot \left(\sqrt{2} \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < 2

   1. Initial program 52.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 40.5

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 39.9

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

   8. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.5%

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

   if 2 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 1.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 17.1

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

   8. Applied rewrites 17.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 30.4%

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

4. Final simplification 37.5%

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

Alternative 3: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{x - 1}{x - -1}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\_m\right)\right)}{l\_m}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{-182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{--1}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, 0\right) - \frac{-2 \cdot \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x}}} \cdot \left(\sqrt{2} \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (sqrt (/ (- x 1.0) (- x -1.0)))))
   (*
    t_s
    (if (<= t_m 2.6e-239)
      (/ (* (sqrt x) (* (sqrt 2.0) (* (sqrt 0.5) t_m))) l_m)
      (if (<= t_m 9e-182)
        t_2
        (if (<= t_m 6e+86)
          (*
           (/
            (- -1.0)
            (sqrt
             (-
              (fma (* t_m t_m) 2.0 0.0)
              (/ (* -2.0 (fma (* t_m t_m) 2.0 (* l_m l_m))) x))))
           (* (sqrt 2.0) t_m))
          t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(((x - 1.0) / (x - -1.0)));
	double tmp;
	if (t_m <= 2.6e-239) {
		tmp = (sqrt(x) * (sqrt(2.0) * (sqrt(0.5) * t_m))) / l_m;
	} else if (t_m <= 9e-182) {
		tmp = t_2;
	} else if (t_m <= 6e+86) {
		tmp = (-(-1.0) / sqrt((fma((t_m * t_m), 2.0, 0.0) - ((-2.0 * fma((t_m * t_m), 2.0, (l_m * l_m))) / x)))) * (sqrt(2.0) * t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)))
	tmp = 0.0
	if (t_m <= 2.6e-239)
		tmp = Float64(Float64(sqrt(x) * Float64(sqrt(2.0) * Float64(sqrt(0.5) * t_m))) / l_m);
	elseif (t_m <= 9e-182)
		tmp = t_2;
	elseif (t_m <= 6e+86)
		tmp = Float64(Float64(Float64(-(-1.0)) / sqrt(Float64(fma(Float64(t_m * t_m), 2.0, 0.0) - Float64(Float64(-2.0 * fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))) / x)))) * Float64(sqrt(2.0) * t_m));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-239], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 9e-182], t$95$2, If[LessEqual[t$95$m, 6e+86], N[(N[((--1.0) / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + 0.0), $MachinePrecision] - N[(N[(-2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.60000000000000003e-239

   1. Initial program 36.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 8.1

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

   8. Applied rewrites 8.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 13.8%

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

   if 2.60000000000000003e-239 < t < 8.9999999999999998e-182 or 5.99999999999999954e86 < t

   1. Initial program 28.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 92.1

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

   8. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.5%

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

   if 8.9999999999999998e-182 < t < 5.99999999999999954e86

   1. Initial program 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\left(\left(-t\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -2}{x}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\right)\right)}{\ell}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{--1}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - \frac{-2 \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}} \cdot \left(\sqrt{2} \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{x - 1}{x - -1}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\_m\right)\right)}{l\_m}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{-182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right) \cdot 2}{x}\right)}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (sqrt (/ (- x 1.0) (- x -1.0)))))
   (*
    t_s
    (if (<= t_m 2.6e-239)
      (/ (* (sqrt x) (* (sqrt 2.0) (* (sqrt 0.5) t_m))) l_m)
      (if (<= t_m 9e-182)
        t_2
        (if (<= t_m 6e+86)
          (/
           (sqrt 2.0)
           (/
            (sqrt
             (fma
              (* 2.0 t_m)
              t_m
              (/ (* (fma (* t_m t_m) 2.0 (* l_m l_m)) 2.0) x)))
            t_m))
          t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(((x - 1.0) / (x - -1.0)));
	double tmp;
	if (t_m <= 2.6e-239) {
		tmp = (sqrt(x) * (sqrt(2.0) * (sqrt(0.5) * t_m))) / l_m;
	} else if (t_m <= 9e-182) {
		tmp = t_2;
	} else if (t_m <= 6e+86) {
		tmp = sqrt(2.0) / (sqrt(fma((2.0 * t_m), t_m, ((fma((t_m * t_m), 2.0, (l_m * l_m)) * 2.0) / x))) / t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)))
	tmp = 0.0
	if (t_m <= 2.6e-239)
		tmp = Float64(Float64(sqrt(x) * Float64(sqrt(2.0) * Float64(sqrt(0.5) * t_m))) / l_m);
	elseif (t_m <= 9e-182)
		tmp = t_2;
	elseif (t_m <= 6e+86)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(fma(Float64(2.0 * t_m), t_m, Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)) * 2.0) / x))) / t_m));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-239], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 9e-182], t$95$2, If[LessEqual[t$95$m, 6e+86], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.60000000000000003e-239

   1. Initial program 36.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 8.1

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

   8. Applied rewrites 8.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 13.8%

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

   if 2.60000000000000003e-239 < t < 8.9999999999999998e-182 or 5.99999999999999954e86 < t

   1. Initial program 28.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 92.1

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

   8. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.5%

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

   if 8.9999999999999998e-182 < t < 5.99999999999999954e86

   1. Initial program 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 83.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. clear-num N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -2}{x}}}{t}}} \]

       5. un-div-inv N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 83.7%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\right)\right)}{\ell}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2 \cdot t, t, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{x}\right)}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{x - 1}{x - -1}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\_m\right)\right)}{l\_m}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{-182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, 0\right) - \frac{-2 \cdot \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x}}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (sqrt (/ (- x 1.0) (- x -1.0)))))
   (*
    t_s
    (if (<= t_m 2.6e-239)
      (/ (* (sqrt x) (* (sqrt 2.0) (* (sqrt 0.5) t_m))) l_m)
      (if (<= t_m 9e-182)
        t_2
        (if (<= t_m 6e+86)
          (*
           (/
            t_m
            (sqrt
             (-
              (fma (* t_m t_m) 2.0 0.0)
              (/ (* -2.0 (fma (* t_m t_m) 2.0 (* l_m l_m))) x))))
           (sqrt 2.0))
          t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(((x - 1.0) / (x - -1.0)));
	double tmp;
	if (t_m <= 2.6e-239) {
		tmp = (sqrt(x) * (sqrt(2.0) * (sqrt(0.5) * t_m))) / l_m;
	} else if (t_m <= 9e-182) {
		tmp = t_2;
	} else if (t_m <= 6e+86) {
		tmp = (t_m / sqrt((fma((t_m * t_m), 2.0, 0.0) - ((-2.0 * fma((t_m * t_m), 2.0, (l_m * l_m))) / x)))) * sqrt(2.0);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)))
	tmp = 0.0
	if (t_m <= 2.6e-239)
		tmp = Float64(Float64(sqrt(x) * Float64(sqrt(2.0) * Float64(sqrt(0.5) * t_m))) / l_m);
	elseif (t_m <= 9e-182)
		tmp = t_2;
	elseif (t_m <= 6e+86)
		tmp = Float64(Float64(t_m / sqrt(Float64(fma(Float64(t_m * t_m), 2.0, 0.0) - Float64(Float64(-2.0 * fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))) / x)))) * sqrt(2.0));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-239], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 9e-182], t$95$2, If[LessEqual[t$95$m, 6e+86], N[(N[(t$95$m / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + 0.0), $MachinePrecision] - N[(N[(-2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.60000000000000003e-239

   1. Initial program 36.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 8.1

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

   8. Applied rewrites 8.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 13.8%

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

   if 2.60000000000000003e-239 < t < 8.9999999999999998e-182 or 5.99999999999999954e86 < t

   1. Initial program 28.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 92.1

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

   8. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.5%

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

   if 8.9999999999999998e-182 < t < 5.99999999999999954e86

   1. Initial program 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -2}{x}}} \cdot \sqrt{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\right)\right)}{\ell}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - \frac{-2 \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{x - 1}{x - -1}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\_m\right)\right)}{l\_m}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{-182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right), \frac{--2}{x}, \left(t\_m \cdot t\_m\right) \cdot 2\right)}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (sqrt (/ (- x 1.0) (- x -1.0)))))
   (*
    t_s
    (if (<= t_m 2.6e-239)
      (/ (* (sqrt x) (* (sqrt 2.0) (* (sqrt 0.5) t_m))) l_m)
      (if (<= t_m 9e-182)
        t_2
        (if (<= t_m 6e+86)
          (*
           (/
            t_m
            (sqrt
             (fma
              (fma (* t_m t_m) 2.0 (* l_m l_m))
              (/ (- -2.0) x)
              (* (* t_m t_m) 2.0))))
           (sqrt 2.0))
          t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(((x - 1.0) / (x - -1.0)));
	double tmp;
	if (t_m <= 2.6e-239) {
		tmp = (sqrt(x) * (sqrt(2.0) * (sqrt(0.5) * t_m))) / l_m;
	} else if (t_m <= 9e-182) {
		tmp = t_2;
	} else if (t_m <= 6e+86) {
		tmp = (t_m / sqrt(fma(fma((t_m * t_m), 2.0, (l_m * l_m)), (-(-2.0) / x), ((t_m * t_m) * 2.0)))) * sqrt(2.0);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)))
	tmp = 0.0
	if (t_m <= 2.6e-239)
		tmp = Float64(Float64(sqrt(x) * Float64(sqrt(2.0) * Float64(sqrt(0.5) * t_m))) / l_m);
	elseif (t_m <= 9e-182)
		tmp = t_2;
	elseif (t_m <= 6e+86)
		tmp = Float64(Float64(t_m / sqrt(fma(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)), Float64(Float64(-(-2.0)) / x), Float64(Float64(t_m * t_m) * 2.0)))) * sqrt(2.0));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-239], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 9e-182], t$95$2, If[LessEqual[t$95$m, 6e+86], N[(N[(t$95$m / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[((--2.0) / x), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.60000000000000003e-239

   1. Initial program 36.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 8.1

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

   8. Applied rewrites 8.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 13.8%

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

   if 2.60000000000000003e-239 < t < 8.9999999999999998e-182 or 5.99999999999999954e86 < t

   1. Initial program 28.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 92.1

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

   8. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.5%

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

   if 8.9999999999999998e-182 < t < 5.99999999999999954e86

   1. Initial program 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 83.7%

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

   11. Applied rewrites 83.7%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\right)\right)}{\ell}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \frac{--2}{x}, \left(t \cdot t\right) \cdot 2\right)}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{x - 1}{x - -1}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\_m\right)\right)}{l\_m}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{-182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, 0\right) - \frac{l\_m \cdot l\_m}{x} \cdot -2}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (sqrt (/ (- x 1.0) (- x -1.0)))))
   (*
    t_s
    (if (<= t_m 2.6e-239)
      (/ (* (sqrt x) (* (sqrt 2.0) (* (sqrt 0.5) t_m))) l_m)
      (if (<= t_m 9e-182)
        t_2
        (if (<= t_m 6e+86)
          (*
           (/
            t_m
            (sqrt (- (fma (* t_m t_m) 2.0 0.0) (* (/ (* l_m l_m) x) -2.0))))
           (sqrt 2.0))
          t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(((x - 1.0) / (x - -1.0)));
	double tmp;
	if (t_m <= 2.6e-239) {
		tmp = (sqrt(x) * (sqrt(2.0) * (sqrt(0.5) * t_m))) / l_m;
	} else if (t_m <= 9e-182) {
		tmp = t_2;
	} else if (t_m <= 6e+86) {
		tmp = (t_m / sqrt((fma((t_m * t_m), 2.0, 0.0) - (((l_m * l_m) / x) * -2.0)))) * sqrt(2.0);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)))
	tmp = 0.0
	if (t_m <= 2.6e-239)
		tmp = Float64(Float64(sqrt(x) * Float64(sqrt(2.0) * Float64(sqrt(0.5) * t_m))) / l_m);
	elseif (t_m <= 9e-182)
		tmp = t_2;
	elseif (t_m <= 6e+86)
		tmp = Float64(Float64(t_m / sqrt(Float64(fma(Float64(t_m * t_m), 2.0, 0.0) - Float64(Float64(Float64(l_m * l_m) / x) * -2.0)))) * sqrt(2.0));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-239], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 9e-182], t$95$2, If[LessEqual[t$95$m, 6e+86], N[(N[(t$95$m / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + 0.0), $MachinePrecision] - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.60000000000000003e-239

   1. Initial program 36.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 8.1

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

   8. Applied rewrites 8.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 13.8%

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

   if 2.60000000000000003e-239 < t < 8.9999999999999998e-182 or 5.99999999999999954e86 < t

   1. Initial program 28.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 92.1

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

   8. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.5%

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

   if 8.9999999999999998e-182 < t < 5.99999999999999954e86

   1. Initial program 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 83.7%

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

   10. Taylor expanded in t around 0

       \[\leadsto \frac{t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - -2 \cdot \color{blue}{\frac{{\ell}^{2}}{x}}}} \cdot \sqrt{2} \]
   11. Step-by-step derivation

   12. Applied rewrites 83.7%

       \[\leadsto \frac{t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - -2 \cdot \color{blue}{\frac{\ell \cdot \ell}{x}}}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot t\right)\right)}{\ell}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+86}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, 0\right) - \frac{\ell \cdot \ell}{x} \cdot -2}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{x - 1}{x - -1}}\\ t_3 := \frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{\left(l\_m + l\_m\right) \cdot l\_m}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-260}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{-182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{-156}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (sqrt (/ (- x 1.0) (- x -1.0))))
        (t_3 (/ (* (sqrt 2.0) t_m) (sqrt (/ (* (+ l_m l_m) l_m) x)))))
   (*
    t_s
    (if (<= t_m 6.5e-260)
      t_3
      (if (<= t_m 9e-182) t_2 (if (<= t_m 6.2e-156) t_3 t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(((x - 1.0) / (x - -1.0)));
	double t_3 = (sqrt(2.0) * t_m) / sqrt((((l_m + l_m) * l_m) / x));
	double tmp;
	if (t_m <= 6.5e-260) {
		tmp = t_3;
	} else if (t_m <= 9e-182) {
		tmp = t_2;
	} else if (t_m <= 6.2e-156) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = sqrt(((x - 1.0d0) / (x - (-1.0d0))))
    t_3 = (sqrt(2.0d0) * t_m) / sqrt((((l_m + l_m) * l_m) / x))
    if (t_m <= 6.5d-260) then
        tmp = t_3
    else if (t_m <= 9d-182) then
        tmp = t_2
    else if (t_m <= 6.2d-156) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(((x - 1.0) / (x - -1.0)));
	double t_3 = (Math.sqrt(2.0) * t_m) / Math.sqrt((((l_m + l_m) * l_m) / x));
	double tmp;
	if (t_m <= 6.5e-260) {
		tmp = t_3;
	} else if (t_m <= 9e-182) {
		tmp = t_2;
	} else if (t_m <= 6.2e-156) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(((x - 1.0) / (x - -1.0)))
	t_3 = (math.sqrt(2.0) * t_m) / math.sqrt((((l_m + l_m) * l_m) / x))
	tmp = 0
	if t_m <= 6.5e-260:
		tmp = t_3
	elif t_m <= 9e-182:
		tmp = t_2
	elif t_m <= 6.2e-156:
		tmp = t_3
	else:
		tmp = t_2
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)))
	t_3 = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(Float64(Float64(l_m + l_m) * l_m) / x)))
	tmp = 0.0
	if (t_m <= 6.5e-260)
		tmp = t_3;
	elseif (t_m <= 9e-182)
		tmp = t_2;
	elseif (t_m <= 6.2e-156)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(((x - 1.0) / (x - -1.0)));
	t_3 = (sqrt(2.0) * t_m) / sqrt((((l_m + l_m) * l_m) / x));
	tmp = 0.0;
	if (t_m <= 6.5e-260)
		tmp = t_3;
	elseif (t_m <= 9e-182)
		tmp = t_2;
	elseif (t_m <= 6.2e-156)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(N[(l$95$m + l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.5e-260], t$95$3, If[LessEqual[t$95$m, 9e-182], t$95$2, If[LessEqual[t$95$m, 6.2e-156], t$95$3, t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 6.50000000000000002e-260 or 8.9999999999999998e-182 < t < 6.1999999999999996e-156

   1. Initial program 36.4%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 19.1%

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

   10. Applied rewrites 19.1%

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

   if 6.50000000000000002e-260 < t < 8.9999999999999998e-182 or 6.1999999999999996e-156 < t

   1. Initial program 38.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 79.8

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

   8. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.0%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(\ell + \ell\right) \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(\ell + \ell\right) \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.7% accurate, 3.0× speedup

Math

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

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 (sqrt (/ (- x 1.0) (- x -1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 38.5

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

5. Applied rewrites 38.5%

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

6. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. lower--.f64 38.0

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

8. Applied rewrites 38.0%

   \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{x - -1}}} \]
9. Step-by-step derivation

10. Applied rewrites 38.5%

    \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{x - -1}}} \]
11. Add Preprocessing

Alternative 10: 75.5% 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 37.3%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 38.0

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

5. Applied rewrites 38.0%

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

7. Applied rewrites 38.5%

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

Reproduce

herbie shell --seed 2024263 
(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)))))