Toniolo and Linder, Equation (7)

Percentage Accurate: 32.5% → 84.2%

Time: 16.3s

Alternatives: 12

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 32.5% 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: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t_4 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \mathsf{fma}\left(\ell, \ell, t\_2\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2 \cdot \mathsf{fma}\left(t\_m, t\_m, \frac{t\_2}{x}\right)}} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{x \cdot t\_3}\right)}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_3}{\sqrt{2 \cdot \left(t\_m \cdot t\_m\right) + \frac{t\_2 + \left(t\_2 + \frac{t\_4 + \frac{t\_4}{x}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{t\_m \cdot \sqrt{2 \cdot \frac{x + 1}{-1 + x}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma 2.0 (* t_m t_m) (* l l)))
        (t_3 (* t_m (sqrt 2.0)))
        (t_4 (fma 2.0 (* t_m t_m) (fma l l t_2))))
   (*
    t_s
    (if (<= t_m 4.2e-281)
      (* (/ t_m (sqrt (* 2.0 (fma t_m t_m (/ t_2 x))))) (sqrt 2.0))
      (if (<= t_m 1.7e-145)
        (/ t_3 (fma t_m (sqrt 2.0) (/ (* l l) (* x t_3))))
        (if (<= t_m 3e+48)
          (/
           t_3
           (sqrt
            (+
             (* 2.0 (* t_m t_m))
             (/ (+ t_2 (+ t_2 (/ (+ t_4 (/ t_4 x)) x))) x))))
          (/ t_3 (* t_m (sqrt (* 2.0 (/ (+ x 1.0) (+ -1.0 x))))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma(2.0, (t_m * t_m), (l * l));
	double t_3 = t_m * sqrt(2.0);
	double t_4 = fma(2.0, (t_m * t_m), fma(l, l, t_2));
	double tmp;
	if (t_m <= 4.2e-281) {
		tmp = (t_m / sqrt((2.0 * fma(t_m, t_m, (t_2 / x))))) * sqrt(2.0);
	} else if (t_m <= 1.7e-145) {
		tmp = t_3 / fma(t_m, sqrt(2.0), ((l * l) / (x * t_3)));
	} else if (t_m <= 3e+48) {
		tmp = t_3 / sqrt(((2.0 * (t_m * t_m)) + ((t_2 + (t_2 + ((t_4 + (t_4 / x)) / x))) / x)));
	} else {
		tmp = t_3 / (t_m * sqrt((2.0 * ((x + 1.0) / (-1.0 + x)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(2.0, Float64(t_m * t_m), Float64(l * l))
	t_3 = Float64(t_m * sqrt(2.0))
	t_4 = fma(2.0, Float64(t_m * t_m), fma(l, l, t_2))
	tmp = 0.0
	if (t_m <= 4.2e-281)
		tmp = Float64(Float64(t_m / sqrt(Float64(2.0 * fma(t_m, t_m, Float64(t_2 / x))))) * sqrt(2.0));
	elseif (t_m <= 1.7e-145)
		tmp = Float64(t_3 / fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(x * t_3))));
	elseif (t_m <= 3e+48)
		tmp = Float64(t_3 / sqrt(Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(Float64(t_2 + Float64(t_2 + Float64(Float64(t_4 + Float64(t_4 / x)) / x))) / x))));
	else
		tmp = Float64(t_3 / Float64(t_m * sqrt(Float64(2.0 * Float64(Float64(x + 1.0) / Float64(-1.0 + x))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l + t$95$2), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-281], N[(N[(t$95$m / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e-145], N[(t$95$3 / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(x * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+48], N[(t$95$3 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[(t$95$2 + N[(N[(t$95$4 + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t$95$m * N[Sqrt[N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.1999999999999998e-281

   1. Initial program 2.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

   7. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 100.0%

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

   if 4.1999999999999998e-281 < t < 1.6999999999999999e-145

   1. Initial program 6.2%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 70.5%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 70.5%

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

   if 1.6999999999999999e-145 < t < 3e48

   1. Initial program 48.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 39.0%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 82.5%

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

   if 3e48 < t

   1. Initial program 35.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{t}{\sqrt{2 \cdot \mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \cdot \sqrt{2}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \frac{\ell \cdot \ell}{x \cdot \left(t \cdot \sqrt{2}\right)}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+48}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \left(\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\ell, \ell, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(\ell, \ell, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\right)\right)}{x}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{x + 1}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_2 \cdot -2\\ t_4 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2 \cdot \mathsf{fma}\left(t\_m, t\_m, \frac{t\_2}{x}\right)}} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{x \cdot t\_4}\right)}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_4}{\sqrt{2 \cdot \left(t\_m \cdot t\_m\right) - \frac{t\_3 + \frac{t\_3 + \frac{t\_3}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{t\_m \cdot \sqrt{2 \cdot \frac{x + 1}{-1 + x}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma 2.0 (* t_m t_m) (* l l)))
        (t_3 (* t_2 -2.0))
        (t_4 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 4.2e-281)
      (* (/ t_m (sqrt (* 2.0 (fma t_m t_m (/ t_2 x))))) (sqrt 2.0))
      (if (<= t_m 1.7e-145)
        (/ t_4 (fma t_m (sqrt 2.0) (/ (* l l) (* x t_4))))
        (if (<= t_m 3e+48)
          (/
           t_4
           (sqrt
            (- (* 2.0 (* t_m t_m)) (/ (+ t_3 (/ (+ t_3 (/ t_3 x)) x)) x))))
          (/ t_4 (* t_m (sqrt (* 2.0 (/ (+ x 1.0) (+ -1.0 x))))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma(2.0, (t_m * t_m), (l * l));
	double t_3 = t_2 * -2.0;
	double t_4 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 4.2e-281) {
		tmp = (t_m / sqrt((2.0 * fma(t_m, t_m, (t_2 / x))))) * sqrt(2.0);
	} else if (t_m <= 1.7e-145) {
		tmp = t_4 / fma(t_m, sqrt(2.0), ((l * l) / (x * t_4)));
	} else if (t_m <= 3e+48) {
		tmp = t_4 / sqrt(((2.0 * (t_m * t_m)) - ((t_3 + ((t_3 + (t_3 / x)) / x)) / x)));
	} else {
		tmp = t_4 / (t_m * sqrt((2.0 * ((x + 1.0) / (-1.0 + x)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(2.0, Float64(t_m * t_m), Float64(l * l))
	t_3 = Float64(t_2 * -2.0)
	t_4 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 4.2e-281)
		tmp = Float64(Float64(t_m / sqrt(Float64(2.0 * fma(t_m, t_m, Float64(t_2 / x))))) * sqrt(2.0));
	elseif (t_m <= 1.7e-145)
		tmp = Float64(t_4 / fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(x * t_4))));
	elseif (t_m <= 3e+48)
		tmp = Float64(t_4 / sqrt(Float64(Float64(2.0 * Float64(t_m * t_m)) - Float64(Float64(t_3 + Float64(Float64(t_3 + Float64(t_3 / x)) / x)) / x))));
	else
		tmp = Float64(t_4 / Float64(t_m * sqrt(Float64(2.0 * Float64(Float64(x + 1.0) / Float64(-1.0 + x))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * -2.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-281], N[(N[(t$95$m / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e-145], N[(t$95$4 / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(x * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+48], N[(t$95$4 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$3 + N[(N[(t$95$3 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$4 / N[(t$95$m * N[Sqrt[N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.1999999999999998e-281

   1. Initial program 4.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 9.1%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

   7. Applied rewrites 56.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 55.9%

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

   if 4.1999999999999998e-281 < t < 1.6999999999999999e-145

   1. Initial program 7.8%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 46.8

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 62.1%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 62.0%

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

   if 1.6999999999999999e-145 < t < 3e48

   1. Initial program 55.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 41.6%

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

   5. Taylor expanded in x around -inf

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

   7. Applied rewrites 84.8%

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

   if 3e48 < t

   1. Initial program 29.6%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 93.7

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

   5. Applied rewrites 93.7%

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

   7. Applied rewrites 93.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{t}{\sqrt{2 \cdot \mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \cdot \sqrt{2}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \frac{\ell \cdot \ell}{x \cdot \left(t \cdot \sqrt{2}\right)}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+48}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) - \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot -2 + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot -2 + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot -2}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{x + 1}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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)))))