Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 84.9%

Time: 24.1s

Alternatives: 10

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: 33.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2} + {l\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.1 \cdot 10^{-275}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_2 + t\_2}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, {t\_m}^{2}, \frac{{l\_m}^{2}}{x}\right) + \mathsf{fma}\left({t\_m}^{2}, \frac{2}{x}, \frac{\mathsf{fma}\left(2, {t\_m}^{2}, {l\_m}^{2}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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)) (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 5.1e-275)
      (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (/ x (+ 2.0 (/ 2.0 x))))))
      (if (<= t_m 1.8e-165)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (/ (+ t_2 t_2) (* t_m (* (sqrt 2.0) x))))
           (* t_m (sqrt 2.0)))))
        (if (<= t_m 2.1e+15)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (fma 2.0 (pow t_m 2.0) (/ (pow l_m 2.0) x))
              (fma
               (pow t_m 2.0)
               (/ 2.0 x)
               (/ (fma 2.0 (pow t_m 2.0) (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)) + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 5.1e-275) {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / x)))));
	} else if (t_m <= 1.8e-165) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_2 + t_2) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 2.1e+15) {
		tmp = t_m * (sqrt(2.0) / sqrt((fma(2.0, pow(t_m, 2.0), (pow(l_m, 2.0) / x)) + fma(pow(t_m, 2.0), (2.0 / x), (fma(2.0, pow(t_m, 2.0), 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(Float64(2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 5.1e-275)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(Float64(x / Float64(2.0 + Float64(2.0 / x))))));
	elseif (t_m <= 1.8e-165)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_2 + t_2) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 2.1e+15)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(fma(2.0, (t_m ^ 2.0), Float64((l_m ^ 2.0) / x)) + fma((t_m ^ 2.0), Float64(2.0 / x), Float64(fma(2.0, (t_m ^ 2.0), (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[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.1e-275], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x / N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e-165], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$2 + t$95$2), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e+15], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $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 < 5.09999999999999984e-275

   1. Initial program 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in l around inf 2.9%

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

   6. Taylor expanded in x around inf 9.4%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in l around 0 8.6%

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

       1. associate-/l* 8.6%

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

       2. associate-*l* 10.0%

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

       3. associate-*r/ 10.0%

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

       4. metadata-eval 10.0%

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

   11. Simplified 10.0%

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

   if 5.09999999999999984e-275 < t < 1.79999999999999992e-165

   1. Initial program 2.3%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in x around inf 59.4%

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

   if 1.79999999999999992e-165 < t < 2.1e15

   1. Initial program 56.3%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in x around inf 80.0%

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

      1. clear-num 80.2%

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

      2. un-div-inv 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-/r/ 80.5%

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

   9. Simplified 80.5%

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

   if 2.1e15 < t

   1. Initial program 32.0%

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

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
   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 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{-275}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right) + \mathsf{fma}\left({t}^{2}, \frac{2}{x}, \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {l\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-275}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_3 + t\_3}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (+ t_2 (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 7.5e-275)
      (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (/ x (+ 2.0 (/ 2.0 x))))))
      (if (<= t_m 1.7e-165)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x))))
           (* t_m (sqrt 2.0)))))
        (if (<= t_m 8.5e+14)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ t_3 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 = t_2 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 7.5e-275) {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / x)))));
	} else if (t_m <= 1.7e-165) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 8.5e+14) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + (t_3 / x))));
	} 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    if (t_m <= 7.5d-275) then
        tmp = t_m * ((sqrt(2.0d0) / l_m) * sqrt((x / (2.0d0 + (2.0d0 / x)))))
    else if (t_m <= 1.7d-165) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * ((t_3 + t_3) / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 8.5d+14) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + (t_3 / x))))
    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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 7.5e-275) {
		tmp = t_m * ((Math.sqrt(2.0) / l_m) * Math.sqrt((x / (2.0 + (2.0 / x)))));
	} else if (t_m <= 1.7e-165) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 8.5e+14) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + (t_3 / x))));
	} 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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 7.5e-275:
		tmp = t_m * ((math.sqrt(2.0) / l_m) * math.sqrt((x / (2.0 + (2.0 / x)))))
	elif t_m <= 1.7e-165:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 8.5e+14:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + (t_3 / x))))
	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)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 7.5e-275)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(Float64(x / Float64(2.0 + Float64(2.0 / x))))));
	elseif (t_m <= 1.7e-165)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 8.5e+14)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x)))));
	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)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 7.5e-275)
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / x)))));
	elseif (t_m <= 1.7e-165)
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 8.5e+14)
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + (t_3 / x))));
	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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.5e-275], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x / N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e-165], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+14], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(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] + N[(t$95$3 / 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 < 7.49999999999999943e-275

   1. Initial program 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in l around inf 2.9%

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

   6. Taylor expanded in x around inf 9.4%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in l around 0 8.6%

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

       1. associate-/l* 8.6%

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

       2. associate-*l* 10.0%

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

       3. associate-*r/ 10.0%

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

       4. metadata-eval 10.0%

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

   11. Simplified 10.0%

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

   if 7.49999999999999943e-275 < t < 1.7e-165

   1. Initial program 2.3%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in x around inf 59.4%

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

   if 1.7e-165 < t < 8.5e14

   1. Initial program 56.3%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in x around inf 80.0%

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

   if 8.5e14 < t

   1. Initial program 32.0%

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

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
   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 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-275}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-274}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\mathsf{fma}\left(t\_m, \sqrt{2}, \frac{0.5}{t\_m} \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t\_m}^{2}, {l\_m}^{2}\right)}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_2 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= t_m 2.5e-274)
      (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (/ x (+ 2.0 (/ 2.0 x))))))
      (if (<= t_m 1.8e-165)
        (*
         (sqrt 2.0)
         (/
          t_m
          (fma
           t_m
           (sqrt 2.0)
           (*
            (/ 0.5 t_m)
            (/
             (* 2.0 (fma 2.0 (pow t_m 2.0) (pow l_m 2.0)))
             (* (sqrt 2.0) x))))))
        (if (<= t_m 1.7e+14)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_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 <= 2.5e-274) {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / x)))));
	} else if (t_m <= 1.8e-165) {
		tmp = sqrt(2.0) * (t_m / fma(t_m, sqrt(2.0), ((0.5 / t_m) * ((2.0 * fma(2.0, pow(t_m, 2.0), pow(l_m, 2.0))) / (sqrt(2.0) * x)))));
	} else if (t_m <= 1.7e+14) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_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 <= 2.5e-274)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(Float64(x / Float64(2.0 + Float64(2.0 / x))))));
	elseif (t_m <= 1.8e-165)
		tmp = Float64(sqrt(2.0) * Float64(t_m / fma(t_m, sqrt(2.0), Float64(Float64(0.5 / t_m) * Float64(Float64(2.0 * fma(2.0, (t_m ^ 2.0), (l_m ^ 2.0))) / Float64(sqrt(2.0) * x))))));
	elseif (t_m <= 1.7e+14)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (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, 2.5e-274], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x / N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e-165], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(0.5 / t$95$m), $MachinePrecision] * 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] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e+14], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(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] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $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 < 2.5e-274

   1. Initial program 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in l around inf 2.9%

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

   6. Taylor expanded in x around inf 9.4%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in l around 0 8.6%

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

       1. associate-/l* 8.6%

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

       2. associate-*l* 10.0%

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

       3. associate-*r/ 10.0%

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

       4. metadata-eval 10.0%

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

   11. Simplified 10.0%

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

   if 2.5e-274 < t < 1.79999999999999992e-165

   1. Initial program 2.3%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in x around inf 59.4%

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

      1. +-commutative 59.4%

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

      2. fma-define 59.4%

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

      3. associate-*r/ 59.4%

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

      4. times-frac 58.4%

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

      5. cancel-sign-sub-inv 58.4%

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

      6. metadata-eval 58.4%

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

      7. distribute-rgt1-in 58.4%

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

      8. metadata-eval 58.4%

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

      9. fma-define 58.4%

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

   7. Simplified 58.4%

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

   if 1.79999999999999992e-165 < t < 1.7e14

   1. Initial program 56.3%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in x around inf 80.0%

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

   if 1.7e14 < t

   1. Initial program 32.0%

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

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
   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 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-274}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{t} \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{l\_m}^{2}}{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.1 \cdot 10^{-274}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\mathsf{fma}\left(t\_m, \sqrt{2}, \frac{0.5}{t\_m} \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t\_m}^{2}, {l\_m}^{2}\right)}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \left(t\_2 + {t\_m}^{2} \cdot \left(2 + 4 \cdot \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (/ (pow l_m 2.0) x)))
   (*
    t_s
    (if (<= t_m 4.1e-274)
      (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (/ x (+ 2.0 (/ 2.0 x))))))
      (if (<= t_m 1.8e-165)
        (*
         (sqrt 2.0)
         (/
          t_m
          (fma
           t_m
           (sqrt 2.0)
           (*
            (/ 0.5 t_m)
            (/
             (* 2.0 (fma 2.0 (pow t_m 2.0) (pow l_m 2.0)))
             (* (sqrt 2.0) x))))))
        (if (<= t_m 8e+14)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+ t_2 (+ t_2 (* (pow t_m 2.0) (+ 2.0 (* 4.0 (/ 1.0 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 = pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 4.1e-274) {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / x)))));
	} else if (t_m <= 1.8e-165) {
		tmp = sqrt(2.0) * (t_m / fma(t_m, sqrt(2.0), ((0.5 / t_m) * ((2.0 * fma(2.0, pow(t_m, 2.0), pow(l_m, 2.0))) / (sqrt(2.0) * x)))));
	} else if (t_m <= 8e+14) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + (t_2 + (pow(t_m, 2.0) * (2.0 + (4.0 * (1.0 / 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((l_m ^ 2.0) / x)
	tmp = 0.0
	if (t_m <= 4.1e-274)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(Float64(x / Float64(2.0 + Float64(2.0 / x))))));
	elseif (t_m <= 1.8e-165)
		tmp = Float64(sqrt(2.0) * Float64(t_m / fma(t_m, sqrt(2.0), Float64(Float64(0.5 / t_m) * Float64(Float64(2.0 * fma(2.0, (t_m ^ 2.0), (l_m ^ 2.0))) / Float64(sqrt(2.0) * x))))));
	elseif (t_m <= 8e+14)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(t_2 + Float64((t_m ^ 2.0) * Float64(2.0 + Float64(4.0 * Float64(1.0 / 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[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.1e-274], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x / N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e-165], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(0.5 / t$95$m), $MachinePrecision] * 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] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+14], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(2.0 + N[(4.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $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 < 4.09999999999999987e-274

   1. Initial program 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in l around inf 2.9%

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

   6. Taylor expanded in x around inf 9.4%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in l around 0 8.6%

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

       1. associate-/l* 8.6%

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

       2. associate-*l* 10.0%

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

       3. associate-*r/ 10.0%

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

       4. metadata-eval 10.0%

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

   11. Simplified 10.0%

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

   if 4.09999999999999987e-274 < t < 1.79999999999999992e-165

   1. Initial program 2.3%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in x around inf 59.4%

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

      1. +-commutative 59.4%

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

      2. fma-define 59.4%

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

      3. associate-*r/ 59.4%

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

      4. times-frac 58.4%

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

      5. cancel-sign-sub-inv 58.4%

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

      6. metadata-eval 58.4%

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

      7. distribute-rgt1-in 58.4%

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

      8. metadata-eval 58.4%

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

      9. fma-define 58.4%

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

   7. Simplified 58.4%

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

   if 1.79999999999999992e-165 < t < 8e14

   1. Initial program 56.3%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in x around inf 80.0%

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

   6. Taylor expanded in t around 0 79.9%

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

   if 8e14 < t

   1. Initial program 32.0%

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

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
   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 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{-274}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{t} \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + 4 \cdot \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{l\_m}^{2}}{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.1 \cdot 10^{-274}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{-229}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \left(t\_2 + {t\_m}^{2} \cdot \left(2 + 4 \cdot \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (/ (pow l_m 2.0) x)))
   (*
    t_s
    (if (<= t_m 4.1e-274)
      (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (/ x (+ 2.0 (/ 2.0 x))))))
      (if (<= t_m 5.8e-229)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 1.1e+16)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+ t_2 (+ t_2 (* (pow t_m 2.0) (+ 2.0 (* 4.0 (/ 1.0 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 = pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 4.1e-274) {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / x)))));
	} else if (t_m <= 5.8e-229) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.1e+16) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + (t_2 + (pow(t_m, 2.0) * (2.0 + (4.0 * (1.0 / x))))))));
	} 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) :: t_2
    real(8) :: tmp
    t_2 = (l_m ** 2.0d0) / x
    if (t_m <= 4.1d-274) then
        tmp = t_m * ((sqrt(2.0d0) / l_m) * sqrt((x / (2.0d0 + (2.0d0 / x)))))
    else if (t_m <= 5.8d-229) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 1.1d+16) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((t_2 + (t_2 + ((t_m ** 2.0d0) * (2.0d0 + (4.0d0 * (1.0d0 / x))))))))
    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 t_2 = Math.pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 4.1e-274) {
		tmp = t_m * ((Math.sqrt(2.0) / l_m) * Math.sqrt((x / (2.0 + (2.0 / x)))));
	} else if (t_m <= 5.8e-229) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.1e+16) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + (t_2 + (Math.pow(t_m, 2.0) * (2.0 + (4.0 * (1.0 / x))))))));
	} 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):
	t_2 = math.pow(l_m, 2.0) / x
	tmp = 0
	if t_m <= 4.1e-274:
		tmp = t_m * ((math.sqrt(2.0) / l_m) * math.sqrt((x / (2.0 + (2.0 / x)))))
	elif t_m <= 5.8e-229:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 1.1e+16:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + (t_2 + (math.pow(t_m, 2.0) * (2.0 + (4.0 * (1.0 / x))))))))
	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)
	t_2 = Float64((l_m ^ 2.0) / x)
	tmp = 0.0
	if (t_m <= 4.1e-274)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(Float64(x / Float64(2.0 + Float64(2.0 / x))))));
	elseif (t_m <= 5.8e-229)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 1.1e+16)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(t_2 + Float64((t_m ^ 2.0) * Float64(2.0 + Float64(4.0 * Float64(1.0 / x)))))))));
	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)
	t_2 = (l_m ^ 2.0) / x;
	tmp = 0.0;
	if (t_m <= 4.1e-274)
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / x)))));
	elseif (t_m <= 5.8e-229)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 1.1e+16)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + (t_2 + ((t_m ^ 2.0) * (2.0 + (4.0 * (1.0 / x))))))));
	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_] := Block[{t$95$2 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.1e-274], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x / N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.8e-229], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.1e+16], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(2.0 + N[(4.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $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 < 4.09999999999999987e-274

   1. Initial program 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in l around inf 2.9%

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

   6. Taylor expanded in x around inf 9.4%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in l around 0 8.6%

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

       1. associate-/l* 8.6%

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

       2. associate-*l* 10.0%

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

       3. associate-*r/ 10.0%

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

       4. metadata-eval 10.0%

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

   11. Simplified 10.0%

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

   if 4.09999999999999987e-274 < t < 5.7999999999999999e-229

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in l around 0 52.5%

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

   6. Taylor expanded in x around inf 52.5%

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

   if 5.7999999999999999e-229 < t < 1.1e16

   1. Initial program 48.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in x around inf 77.0%

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

   6. Taylor expanded in t around 0 77.0%

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

   if 1.1e16 < t

   1. Initial program 32.0%

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

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
   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 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{-274}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-229}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + 4 \cdot \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.45 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \end{array} \end{array} \]

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 (<= l_m 1.45e+187)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* (sqrt 2.0) (/ t_m (* l_m (sqrt (/ (+ 2.0 (/ 2.0 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 (l_m <= 1.45e+187) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((2.0 + (2.0 / 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 (l_m <= 1.45d+187) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt(((2.0d0 + (2.0d0 / 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 (l_m <= 1.45e+187) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((2.0 + (2.0 / 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 l_m <= 1.45e+187:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((2.0 + (2.0 / 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 (l_m <= 1.45e+187)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(2.0 + Float64(2.0 / 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 (l_m <= 1.45e+187)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((2.0 + (2.0 / 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[l$95$m, 1.45e+187], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.45e187

   1. Initial program 40.6%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in l around 0 44.2%

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

   6. Taylor expanded in t around 0 44.3%

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

   if 1.45e187 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 0.0%

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

   6. Taylor expanded in x around inf 17.9%

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

   8. Simplified 18.1%

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

   9. Taylor expanded in l around 0 72.1%

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

       1. associate-*r/ 72.1%

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

       2. metadata-eval 72.1%

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

   11. Simplified 72.1%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.1 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.1e+187)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (/ x (+ 2.0 (/ 2.0 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) {
	double tmp;
	if (l_m <= 1.1e+187) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / 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 (l_m <= 1.1d+187) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * ((sqrt(2.0d0) / l_m) * sqrt((x / (2.0d0 + (2.0d0 / 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 (l_m <= 1.1e+187) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * ((Math.sqrt(2.0) / l_m) * Math.sqrt((x / (2.0 + (2.0 / 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 l_m <= 1.1e+187:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * ((math.sqrt(2.0) / l_m) * math.sqrt((x / (2.0 + (2.0 / 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 (l_m <= 1.1e+187)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(Float64(x / Float64(2.0 + Float64(2.0 / 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 (l_m <= 1.1e+187)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt((x / (2.0 + (2.0 / 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[l$95$m, 1.1e+187], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x / N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.0999999999999999e187

   1. Initial program 40.6%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in l around 0 44.2%

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

   6. Taylor expanded in t around 0 44.3%

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

   if 1.0999999999999999e187 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 0.0%

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

   6. Taylor expanded in x around inf 17.9%

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

   8. Simplified 18.1%

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

   9. Taylor expanded in l around 0 65.5%

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

       1. associate-/l* 65.3%

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

       2. associate-*l* 71.8%

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

       3. associate-*r/ 71.8%

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

       4. metadata-eval 71.8%

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

   11. Simplified 71.8%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{x}{2 + \frac{2}{x}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

3. Simplified 38.3%

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

5. Taylor expanded in l around 0 42.7%

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

6. Taylor expanded in t around 0 42.8%

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

7. Final simplification 42.8%

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

Alternative 9: 75.9% 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 38.4%

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

3. Simplified 38.3%

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

5. Taylor expanded in l around 0 42.7%

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

6. Taylor expanded in x around inf 41.7%

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

7. Final simplification 41.7%

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

Alternative 10: 75.2% 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 38.4%

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

3. Simplified 38.3%

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

5. Taylor expanded in l around 0 42.7%

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

6. Taylor expanded in x around inf 41.2%

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

Reproduce

herbie shell --seed 2024111 
(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)))))