Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 84.9%

Time: 23.3s

Alternatives: 11

Speedup: 225.0×

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: 33.5% 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: 84.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := {l\_m}^{2} + t\_2\\ t_4 := t\_3 + t\_3\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-213}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 6.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\mathsf{fma}\left(\sqrt{2}, t\_m, 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t\_m}^{2}, {l\_m}^{2}\right)}{\sqrt{2} \cdot \left(t\_m \cdot x\right)}\right)}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \frac{t\_4 + \frac{\left(t\_4 + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_3}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ (pow l_m 2.0) t_2))
        (t_4 (+ t_3 t_3)))
   (*
    t_s
    (if (<= t_m 4e-213)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 6.6e-189)
        (/
         (* t_m (sqrt 2.0))
         (fma
          (sqrt 2.0)
          t_m
          (*
           0.5
           (/
            (* 2.0 (fma 2.0 (pow t_m 2.0) (pow l_m 2.0)))
            (* (sqrt 2.0) (* t_m x))))))
        (if (<= t_m 6.2e-7)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              t_2
              (/
               (+
                t_4
                (/
                 (+
                  (+ t_4 (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l_m 2.0) x)))
                  (/ t_3 x))
                 x))
               x)))))
          (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 = 2.0 * pow(t_m, 2.0);
	double t_3 = pow(l_m, 2.0) + t_2;
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 4e-213) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 6.6e-189) {
		tmp = (t_m * sqrt(2.0)) / fma(sqrt(2.0), t_m, (0.5 * ((2.0 * fma(2.0, pow(t_m, 2.0), pow(l_m, 2.0))) / (sqrt(2.0) * (t_m * x)))));
	} else if (t_m <= 6.2e-7) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((t_4 + (((t_4 + ((2.0 * (pow(t_m, 2.0) / x)) + (pow(l_m, 2.0) / x))) + (t_3 / x)) / x)) / x))));
	} 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(2.0 * (t_m ^ 2.0))
	t_3 = Float64((l_m ^ 2.0) + t_2)
	t_4 = Float64(t_3 + t_3)
	tmp = 0.0
	if (t_m <= 4e-213)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 6.6e-189)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / fma(sqrt(2.0), t_m, Float64(0.5 * Float64(Float64(2.0 * fma(2.0, (t_m ^ 2.0), (l_m ^ 2.0))) / Float64(sqrt(2.0) * Float64(t_m * x))))));
	elseif (t_m <= 6.2e-7)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(t_4 + Float64(Float64(Float64(t_4 + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x)) / x)) / x)))));
	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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + t$95$3), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-213], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 6.6e-189], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m + N[(0.5 * N[(N[(2.0 * N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e-7], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(t$95$4 + N[(N[(N[(t$95$4 + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 < 3.9999999999999998e-213

   1. Initial program 29.5%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate--l+ 5.8%

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

      3. sub-neg 5.8%

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

      4. metadata-eval 5.8%

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

      5. +-commutative 5.8%

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

      6. sub-neg 5.8%

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

      7. metadata-eval 5.8%

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

      8. +-commutative 5.8%

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

      9. associate-/l* 5.8%

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

   7. Simplified 5.8%

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

   8. Taylor expanded in x around inf 14.7%

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

      1. associate-*l/ 16.5%

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

      2. sqrt-unprod 16.6%

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

      3. metadata-eval 16.6%

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

      4. metadata-eval 16.6%

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

      5. *-rgt-identity 16.6%

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

   10. Applied egg-rr 16.6%

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

   if 3.9999999999999998e-213 < t < 6.6000000000000002e-189

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in x around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. +-commutative 86.5%

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

      3. *-commutative 86.5%

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

      4. fma-define 86.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(\sqrt{2}, t, 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)}} \]

      5. associate-*r/ 86.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2}, t, \color{blue}{\frac{0.5 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}}\right)} \]

   7. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2}, t, \frac{0.5 \cdot \left(\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right) + \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)\right)}{\left(t \cdot x\right) \cdot \sqrt{2}}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 86.5%

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

      2. associate-/l* 86.5%

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

      3. associate-*r* 86.5%

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

      4. count-2 86.5%

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

      5. *-commutative 86.5%

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

      6. *-commutative 86.5%

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

   9. Simplified 86.5%

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

   if 6.6000000000000002e-189 < t < 6.1999999999999999e-7

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in x around -inf 80.9%

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

   if 6.1999999999999999e-7 < t

   1. Initial program 45.0%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in l around 0 95.8%

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

   6. Taylor expanded in t around 0 96.1%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-213}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(\sqrt{2}, t, 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \frac{\left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) + \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) + \frac{\left(\left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) + \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) + \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := {l\_m}^{2} + t\_2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 6.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\mathsf{fma}\left(\sqrt{2}, t\_m, 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t\_m}^{2}, {l\_m}^{2}\right)}{\sqrt{2} \cdot \left(t\_m \cdot x\right)}\right)}\\ \mathbf{elif}\;t\_m \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \frac{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{l\_m}^{2}}{x}\right) + \left(\left(t\_3 + t\_3\right) + \frac{t\_3}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))) (t_3 (+ (pow l_m 2.0) t_2)))
   (*
    t_s
    (if (<= t_m 2.85e-213)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 6.6e-189)
        (/
         (* t_m (sqrt 2.0))
         (fma
          (sqrt 2.0)
          t_m
          (*
           0.5
           (/
            (* 2.0 (fma 2.0 (pow t_m 2.0) (pow l_m 2.0)))
            (* (sqrt 2.0) (* t_m x))))))
        (if (<= t_m 3.75e-6)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              t_2
              (/
               (+
                (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l_m 2.0) x))
                (+ (+ t_3 t_3) (/ t_3 x)))
               x)))))
          (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 = 2.0 * pow(t_m, 2.0);
	double t_3 = pow(l_m, 2.0) + t_2;
	double tmp;
	if (t_m <= 2.85e-213) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 6.6e-189) {
		tmp = (t_m * sqrt(2.0)) / fma(sqrt(2.0), t_m, (0.5 * ((2.0 * fma(2.0, pow(t_m, 2.0), pow(l_m, 2.0))) / (sqrt(2.0) * (t_m * x)))));
	} else if (t_m <= 3.75e-6) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * (pow(t_m, 2.0) / x)) + (pow(l_m, 2.0) / x)) + ((t_3 + t_3) + (t_3 / x))) / x))));
	} 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(2.0 * (t_m ^ 2.0))
	t_3 = Float64((l_m ^ 2.0) + t_2)
	tmp = 0.0
	if (t_m <= 2.85e-213)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 6.6e-189)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / fma(sqrt(2.0), t_m, Float64(0.5 * Float64(Float64(2.0 * fma(2.0, (t_m ^ 2.0), (l_m ^ 2.0))) / Float64(sqrt(2.0) * Float64(t_m * x))))));
	elseif (t_m <= 3.75e-6)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l_m ^ 2.0) / x)) + Float64(Float64(t_3 + t_3) + Float64(t_3 / x))) / x)))));
	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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.85e-213], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 6.6e-189], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m + N[(0.5 * N[(N[(2.0 * N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.75e-6], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 + t$95$3), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 < 2.84999999999999997e-213

   1. Initial program 29.5%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate--l+ 5.8%

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

      3. sub-neg 5.8%

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

      4. metadata-eval 5.8%

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

      5. +-commutative 5.8%

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

      6. sub-neg 5.8%

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

      7. metadata-eval 5.8%

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

      8. +-commutative 5.8%

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

      9. associate-/l* 5.8%

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

   7. Simplified 5.8%

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

   8. Taylor expanded in x around inf 14.7%

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

      1. associate-*l/ 16.5%

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

      2. sqrt-unprod 16.6%

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

      3. metadata-eval 16.6%

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

      4. metadata-eval 16.6%

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

      5. *-rgt-identity 16.6%

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

   10. Applied egg-rr 16.6%

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

   if 2.84999999999999997e-213 < t < 6.6000000000000002e-189

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in x around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. +-commutative 86.5%

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

      3. *-commutative 86.5%

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

      4. fma-define 86.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(\sqrt{2}, t, 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)}} \]

      5. associate-*r/ 86.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2}, t, \color{blue}{\frac{0.5 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}}\right)} \]

   7. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2}, t, \frac{0.5 \cdot \left(\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right) + \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)\right)}{\left(t \cdot x\right) \cdot \sqrt{2}}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 86.5%

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

      2. associate-/l* 86.5%

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

      3. associate-*r* 86.5%

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

      4. count-2 86.5%

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

      5. *-commutative 86.5%

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

      6. *-commutative 86.5%

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

   9. Simplified 86.5%

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

   if 6.6000000000000002e-189 < t < 3.7500000000000001e-6

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in x around -inf 80.3%

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

   if 3.7500000000000001e-6 < t

   1. Initial program 45.0%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in l around 0 95.8%

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

   6. Taylor expanded in t around 0 96.1%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(\sqrt{2}, t, 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) + \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) + \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.1 \cdot 10^{-214}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 6.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\mathsf{fma}\left(\sqrt{2}, t\_m, 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t\_m}^{2}, {l\_m}^{2}\right)}{\sqrt{2} \cdot \left(t\_m \cdot x\right)}\right)}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{{l\_m}^{2} + t\_2}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 7.1e-214)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 6.6e-189)
        (/
         (* t_m (sqrt 2.0))
         (fma
          (sqrt 2.0)
          t_m
          (*
           0.5
           (/
            (* 2.0 (fma 2.0 (pow t_m 2.0) (pow l_m 2.0)))
            (* (sqrt 2.0) (* t_m x))))))
        (if (<= t_m 3.5e-6)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              (/ (+ (pow l_m 2.0) t_2) x)
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))))))
          (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 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 7.1e-214) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 6.6e-189) {
		tmp = (t_m * sqrt(2.0)) / fma(sqrt(2.0), t_m, (0.5 * ((2.0 * fma(2.0, pow(t_m, 2.0), pow(l_m, 2.0))) / (sqrt(2.0) * (t_m * x)))));
	} else if (t_m <= 3.5e-6) {
		tmp = sqrt(2.0) * (t_m / sqrt((((pow(l_m, 2.0) + t_2) / x) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))))));
	} 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(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 7.1e-214)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 6.6e-189)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / fma(sqrt(2.0), t_m, Float64(0.5 * Float64(Float64(2.0 * fma(2.0, (t_m ^ 2.0), (l_m ^ 2.0))) / Float64(sqrt(2.0) * Float64(t_m * x))))));
	elseif (t_m <= 3.5e-6)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64((l_m ^ 2.0) + t_2) / x) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x)))))));
	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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.1e-214], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 6.6e-189], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m + N[(0.5 * N[(N[(2.0 * N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.5e-6], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 < 7.1000000000000001e-214

   1. Initial program 29.5%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate--l+ 5.8%

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

      3. sub-neg 5.8%

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

      4. metadata-eval 5.8%

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

      5. +-commutative 5.8%

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

      6. sub-neg 5.8%

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

      7. metadata-eval 5.8%

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

      8. +-commutative 5.8%

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

      9. associate-/l* 5.8%

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

   7. Simplified 5.8%

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

   8. Taylor expanded in x around inf 14.7%

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

      1. associate-*l/ 16.5%

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

      2. sqrt-unprod 16.6%

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

      3. metadata-eval 16.6%

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

      4. metadata-eval 16.6%

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

      5. *-rgt-identity 16.6%

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

   10. Applied egg-rr 16.6%

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

   if 7.1000000000000001e-214 < t < 6.6000000000000002e-189

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in x around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. +-commutative 86.5%

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

      3. *-commutative 86.5%

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

      4. fma-define 86.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(\sqrt{2}, t, 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)}} \]

      5. associate-*r/ 86.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2}, t, \color{blue}{\frac{0.5 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}}\right)} \]

   7. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{2}, t, \frac{0.5 \cdot \left(\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right) + \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)\right)}{\left(t \cdot x\right) \cdot \sqrt{2}}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 86.5%

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

      2. associate-/l* 86.5%

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

      3. associate-*r* 86.5%

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

      4. count-2 86.5%

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

      5. *-commutative 86.5%

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

      6. *-commutative 86.5%

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

   9. Simplified 86.5%

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

   if 6.6000000000000002e-189 < t < 3.49999999999999995e-6

   1. Initial program 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in x around inf 79.4%

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

   if 3.49999999999999995e-6 < t

   1. Initial program 45.0%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in l around 0 95.8%

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

   6. Taylor expanded in t around 0 96.1%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.1 \cdot 10^{-214}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(\sqrt{2}, t, 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-213}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{2 \cdot {l\_m}^{2}}{\sqrt{2} \cdot \left(t\_m \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{l\_m} \cdot \sqrt{x \cdot {t\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.3e-213)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 1.15e-129)
      (*
       (sqrt 2.0)
       (/
        t_m
        (+
         (* 0.5 (/ (* 2.0 (pow l_m 2.0)) (* (sqrt 2.0) (* t_m x))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 1.05e-106)
        (* (/ 1.0 l_m) (sqrt (* x (pow t_m 2.0))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.3e-213) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 1.15e-129) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((2.0 * pow(l_m, 2.0)) / (sqrt(2.0) * (t_m * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 1.05e-106) {
		tmp = (1.0 / l_m) * sqrt((x * pow(t_m, 2.0)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.3d-213) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 1.15d-129) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * ((2.0d0 * (l_m ** 2.0d0)) / (sqrt(2.0d0) * (t_m * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 1.05d-106) then
        tmp = (1.0d0 / l_m) * sqrt((x * (t_m ** 2.0d0)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.3e-213) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 1.15e-129) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * ((2.0 * Math.pow(l_m, 2.0)) / (Math.sqrt(2.0) * (t_m * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 1.05e-106) {
		tmp = (1.0 / l_m) * Math.sqrt((x * Math.pow(t_m, 2.0)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.3e-213:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 1.15e-129:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * ((2.0 * math.pow(l_m, 2.0)) / (math.sqrt(2.0) * (t_m * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 1.05e-106:
		tmp = (1.0 / l_m) * math.sqrt((x * math.pow(t_m, 2.0)))
	else:
		tmp = math.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)
	tmp = 0.0
	if (t_m <= 1.3e-213)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 1.15e-129)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(2.0 * (l_m ^ 2.0)) / Float64(sqrt(2.0) * Float64(t_m * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 1.05e-106)
		tmp = Float64(Float64(1.0 / l_m) * sqrt(Float64(x * (t_m ^ 2.0))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.3e-213)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 1.15e-129)
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((2.0 * (l_m ^ 2.0)) / (sqrt(2.0) * (t_m * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 1.05e-106)
		tmp = (1.0 / l_m) * sqrt((x * (t_m ^ 2.0)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.3e-213], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.15e-129], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e-106], N[(N[(1.0 / l$95$m), $MachinePrecision] * N[Sqrt[N[(x * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 < 1.3000000000000001e-213

   1. Initial program 29.5%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate--l+ 5.8%

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

      3. sub-neg 5.8%

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

      4. metadata-eval 5.8%

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

      5. +-commutative 5.8%

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

      6. sub-neg 5.8%

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

      7. metadata-eval 5.8%

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

      8. +-commutative 5.8%

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

      9. associate-/l* 5.8%

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

   7. Simplified 5.8%

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

   8. Taylor expanded in x around inf 14.7%

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

      1. associate-*l/ 16.5%

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

      2. sqrt-unprod 16.6%

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

      3. metadata-eval 16.6%

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

      4. metadata-eval 16.6%

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

      5. *-rgt-identity 16.6%

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

   10. Applied egg-rr 16.6%

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

   if 1.3000000000000001e-213 < t < 1.15e-129

   1. Initial program 19.4%

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

   3. Simplified 19.4%

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

   5. Taylor expanded in x around inf 67.0%

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

   6. Taylor expanded in l around inf 67.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)} + t \cdot \sqrt{2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 67.0%

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

      2. associate-*r* 67.0%

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

      3. *-commutative 67.0%

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

      4. *-commutative 67.0%

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

   8. Simplified 67.0%

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

   if 1.15e-129 < t < 1.05000000000000002e-106

   1. Initial program 26.4%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in l around inf 2.1%

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

      1. *-commutative 2.1%

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

      2. associate--l+ 15.6%

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

      3. sub-neg 15.6%

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

      4. metadata-eval 15.6%

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

      5. +-commutative 15.6%

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

      6. sub-neg 15.6%

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

      7. metadata-eval 15.6%

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

      8. +-commutative 15.6%

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

      9. associate-/l* 15.6%

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

   7. Simplified 15.6%

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

   8. Taylor expanded in x around inf 74.3%

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

      1. associate-*l/ 73.7%

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

      2. clear-num 73.9%

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

      3. sqrt-unprod 75.3%

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

      4. metadata-eval 75.3%

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

      5. metadata-eval 75.3%

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

      6. *-rgt-identity 75.3%

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

   10. Applied egg-rr 75.3%

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

       1. associate-/r/ 75.3%

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

   12. Simplified 75.3%

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

       1. add-sqr-sqrt 74.9%

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

       2. sqrt-unprod 75.3%

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

       3. *-commutative 75.3%

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

       4. *-commutative 75.3%

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

       5. swap-sqr 75.3%

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

       6. add-sqr-sqrt 75.3%

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

       7. unpow2 75.3%

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

   14. Applied egg-rr 75.3%

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

   if 1.05000000000000002e-106 < t

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in l around 0 91.1%

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

   6. Taylor expanded in t around 0 91.4%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-213}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{2 \cdot {\ell}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\ell} \cdot \sqrt{x \cdot {t}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-106}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-106)
    (/ (* t_m (sqrt x)) l_m)
    (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 tmp;
	if (t_m <= 1e-106) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1d-106) then
        tmp = (t_m * sqrt(x)) / l_m
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1e-106) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1e-106:
		tmp = (t_m * math.sqrt(x)) / l_m
	else:
		tmp = math.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)
	tmp = 0.0
	if (t_m <= 1e-106)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1e-106)
		tmp = (t_m * sqrt(x)) / l_m;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-106], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999941e-107

   1. Initial program 28.3%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate--l+ 7.1%

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

      3. sub-neg 7.1%

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

      4. metadata-eval 7.1%

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

      5. +-commutative 7.1%

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

      6. sub-neg 7.1%

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

      7. metadata-eval 7.1%

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

      8. +-commutative 7.1%

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

      9. associate-/l* 7.1%

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

   7. Simplified 7.1%

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

   8. Taylor expanded in x around inf 17.3%

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

      1. associate-*l/ 19.6%

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

      2. sqrt-unprod 19.7%

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

      3. metadata-eval 19.7%

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

      4. metadata-eval 19.7%

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

      5. *-rgt-identity 19.7%

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

   10. Applied egg-rr 19.7%

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

   if 9.99999999999999941e-107 < t

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in l around 0 91.1%

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

   6. Taylor expanded in t around 0 91.4%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-106}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-219)
    (* (sqrt x) (/ t_m l_m))
    (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.1e-219) {
		tmp = sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.1d-219) then
        tmp = sqrt(x) * (t_m / l_m)
    else
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.1e-219) {
		tmp = Math.sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.1e-219:
		tmp = math.sqrt(x) * (t_m / l_m)
	else:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.1e-219)
		tmp = Float64(sqrt(x) * Float64(t_m / l_m));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.1e-219)
		tmp = sqrt(x) * (t_m / l_m);
	else
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-219], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e-219

   1. Initial program 29.7%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in l around inf 2.0%

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

      1. *-commutative 2.0%

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

      2. associate--l+ 5.8%

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

      3. sub-neg 5.8%

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

      4. metadata-eval 5.8%

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

      5. +-commutative 5.8%

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

      6. sub-neg 5.8%

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

      7. metadata-eval 5.8%

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

      8. +-commutative 5.8%

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

      9. associate-/l* 5.8%

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

   7. Simplified 5.8%

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

   8. Taylor expanded in x around inf 14.8%

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

      1. associate-*l/ 16.6%

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

      2. clear-num 16.6%

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

      3. sqrt-unprod 16.7%

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

      4. metadata-eval 16.7%

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

      5. metadata-eval 16.7%

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

      6. *-rgt-identity 16.7%

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

   10. Applied egg-rr 16.7%

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

       1. associate-/r/ 16.7%

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

   12. Simplified 16.7%

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

   13. Taylor expanded in l around 0 14.9%

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

   if 1.1e-219 < t

   1. Initial program 40.1%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in l around 0 81.7%

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

   6. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   8. Simplified 81.2%

      \[\leadsto \color{blue}{1 - \frac{\frac{-0.5}{x} + 1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-106}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-106)
    (* t_m (/ (sqrt x) l_m))
    (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1e-106) {
		tmp = t_m * (sqrt(x) / l_m);
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1d-106) then
        tmp = t_m * (sqrt(x) / l_m)
    else
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1e-106) {
		tmp = t_m * (Math.sqrt(x) / l_m);
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1e-106:
		tmp = t_m * (math.sqrt(x) / l_m)
	else:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1e-106)
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1e-106)
		tmp = t_m * (sqrt(x) / l_m);
	else
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-106], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999941e-107

   1. Initial program 28.3%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate--l+ 7.1%

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

      3. sub-neg 7.1%

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

      4. metadata-eval 7.1%

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

      5. +-commutative 7.1%

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

      6. sub-neg 7.1%

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

      7. metadata-eval 7.1%

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

      8. +-commutative 7.1%

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

      9. associate-/l* 7.1%

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

   7. Simplified 7.1%

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

   8. Taylor expanded in x around inf 17.3%

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

      1. *-commutative 17.3%

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

      2. clear-num 17.0%

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

      3. un-div-inv 17.0%

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

      4. sqrt-unprod 17.1%

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

      5. metadata-eval 17.1%

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

      6. metadata-eval 17.1%

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

      7. *-rgt-identity 17.1%

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

   10. Applied egg-rr 17.1%

       \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{\ell}{t}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 19.7%

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

   12. Simplified 19.7%

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

   if 9.99999999999999941e-107 < t

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in l around 0 91.1%

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

   6. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   8. Simplified 90.5%

      \[\leadsto \color{blue}{1 - \frac{\frac{-0.5}{x} + 1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-106}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.22 \cdot 10^{-106}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.22e-106)
    (/ (* t_m (sqrt x)) l_m)
    (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.22e-106) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.22d-106) then
        tmp = (t_m * sqrt(x)) / l_m
    else
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.22e-106) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.22e-106:
		tmp = (t_m * math.sqrt(x)) / l_m
	else:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.22e-106)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.22e-106)
		tmp = (t_m * sqrt(x)) / l_m;
	else
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.22e-106], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2200000000000001e-106

   1. Initial program 28.3%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate--l+ 7.1%

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

      3. sub-neg 7.1%

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

      4. metadata-eval 7.1%

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

      5. +-commutative 7.1%

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

      6. sub-neg 7.1%

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

      7. metadata-eval 7.1%

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

      8. +-commutative 7.1%

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

      9. associate-/l* 7.1%

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

   7. Simplified 7.1%

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

   8. Taylor expanded in x around inf 17.3%

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

      1. associate-*l/ 19.6%

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

      2. sqrt-unprod 19.7%

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

      3. metadata-eval 19.7%

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

      4. metadata-eval 19.7%

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

      5. *-rgt-identity 19.7%

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

   10. Applied egg-rr 19.7%

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

   if 1.2200000000000001e-106 < t

   1. Initial program 46.1%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in l around 0 91.1%

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

   6. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   8. Simplified 90.5%

      \[\leadsto \color{blue}{1 - \frac{\frac{-0.5}{x} + 1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.22 \cdot 10^{-106}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.6% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

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 + (((-1.0d0) - ((-0.5d0) / x)) / x))
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 + ((-1.0 - (-0.5 / x)) / x));
}

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 + ((-1.0 - (-0.5 / x)) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x)))
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 + ((-1.0 - (-0.5 / x)) / x));
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[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

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

3. Simplified 34.1%

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

5. Taylor expanded in l around 0 36.6%

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

6. Taylor expanded in x around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unsub-neg 0.0%

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

8. Simplified 36.4%

   \[\leadsto \color{blue}{1 - \frac{\frac{-0.5}{x} + 1}{x}} \]

9. Final simplification 36.4%

   \[\leadsto 1 + \frac{-1 - \frac{-0.5}{x}}{x} \]
10. Add Preprocessing

Alternative 10: 76.4% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

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 + ((-1.0d0) / x))
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 + (-1.0 / x));
}

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 + (-1.0 / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
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 + (-1.0 / x));
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[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

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

3. Simplified 34.1%

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

5. Taylor expanded in l around 0 36.6%

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

6. Taylor expanded in x around inf 36.0%

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

7. Final simplification 36.0%

   \[\leadsto 1 + \frac{-1}{x} \]
8. Add Preprocessing

Alternative 11: 75.7% accurate, 225.0× 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 34.2%

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

3. Simplified 34.1%

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

5. Taylor expanded in l around 0 36.6%

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

6. Taylor expanded in x around inf 35.3%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 35.3%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(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)))))