Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 84.5%

Time: 22.3s

Alternatives: 12

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: 33.3% 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: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t_4 := t\_2 + l\_m \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-198}:\\ \;\;\;\;\frac{t\_3}{\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t\_m \leq 2.05 \cdot 10^{-160}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+38}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2 + \frac{\left(\frac{t\_2}{x} + \frac{l\_m \cdot l\_m}{x}\right) + \left(\left(t\_4 + t\_4\right) + \frac{t\_4}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m)))
        (t_3 (* t_m (sqrt 2.0)))
        (t_4 (+ t_2 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 1.35e-198)
      (/ t_3 (* (* (sqrt 2.0) l_m) (sqrt (/ 1.0 x))))
      (if (<= t_m 2.05e-160)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 3e+38)
          (/
           t_3
           (sqrt
            (+
             t_2
             (/
              (+ (+ (/ t_2 x) (/ (* l_m l_m) x)) (+ (+ t_4 t_4) (/ t_4 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 * (t_m * t_m);
	double t_3 = t_m * sqrt(2.0);
	double t_4 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 1.35e-198) {
		tmp = t_3 / ((sqrt(2.0) * l_m) * sqrt((1.0 / x)));
	} else if (t_m <= 2.05e-160) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 3e+38) {
		tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = t_m * sqrt(2.0d0)
    t_4 = t_2 + (l_m * l_m)
    if (t_m <= 1.35d-198) then
        tmp = t_3 / ((sqrt(2.0d0) * l_m) * sqrt((1.0d0 / x)))
    else if (t_m <= 2.05d-160) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 3d+38) then
        tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * (t_m * t_m);
	double t_3 = t_m * Math.sqrt(2.0);
	double t_4 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 1.35e-198) {
		tmp = t_3 / ((Math.sqrt(2.0) * l_m) * Math.sqrt((1.0 / x)));
	} else if (t_m <= 2.05e-160) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 3e+38) {
		tmp = t_3 / Math.sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * (t_m * t_m)
	t_3 = t_m * math.sqrt(2.0)
	t_4 = t_2 + (l_m * l_m)
	tmp = 0
	if t_m <= 1.35e-198:
		tmp = t_3 / ((math.sqrt(2.0) * l_m) * math.sqrt((1.0 / x)))
	elif t_m <= 2.05e-160:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 3e+38:
		tmp = t_3 / math.sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * Float64(t_m * t_m))
	t_3 = Float64(t_m * sqrt(2.0))
	t_4 = Float64(t_2 + Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 1.35e-198)
		tmp = Float64(t_3 / Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(1.0 / x))));
	elseif (t_m <= 2.05e-160)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 3e+38)
		tmp = Float64(t_3 / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(t_2 / x) + Float64(Float64(l_m * l_m) / x)) + Float64(Float64(t_4 + t_4) + Float64(t_4 / x))) / 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 * t_m);
	t_3 = t_m * sqrt(2.0);
	t_4 = t_2 + (l_m * l_m);
	tmp = 0.0;
	if (t_m <= 1.35e-198)
		tmp = t_3 / ((sqrt(2.0) * l_m) * sqrt((1.0 / x)));
	elseif (t_m <= 2.05e-160)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 3e+38)
		tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.35e-198], N[(t$95$3 / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.05e-160], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+38], N[(t$95$3 / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 + t$95$4), $MachinePrecision] + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 1.3500000000000001e-198

   1. Initial program 28.8%

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

   3. Taylor expanded in x around inf 45.3%

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

      1. associate--l+ 45.3%

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

      2. associate-*r/ 45.3%

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

      3. unpow2 45.3%

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

      4. sub-neg 45.3%

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

      5. unpow2 45.3%

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

      6. unpow2 45.3%

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

      7. mul-1-neg 45.3%

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

      8. remove-double-neg 45.3%

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

      9. unpow2 45.3%

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

      10. unpow2 45.3%

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

   5. Simplified 45.3%

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

   6. Taylor expanded in t around 0 17.8%

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

   if 1.3500000000000001e-198 < t < 2.05000000000000001e-160

   1. Initial program 4.3%

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

   3. Taylor expanded in l around 0 63.8%

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

   4. Taylor expanded in x around inf 63.8%

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

   if 2.05000000000000001e-160 < t < 3.0000000000000001e38

   1. Initial program 69.6%

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

   3. Taylor expanded in x around -inf 90.0%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. unpow2 90.0%

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

   5. Simplified 90.0%

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

   if 3.0000000000000001e38 < t

   1. Initial program 29.1%

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

   3. Taylor expanded in l around 0 95.4%

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

   4. Taylor expanded in t around 0 95.4%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-198}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-160}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+38}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \left(\left(\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right) + \left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)\right) + \frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t_4 := t\_2 + l\_m \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_3}{t\_m \cdot \left(\sqrt{2 + \frac{4}{x}} + \left(\frac{l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}\right) \cdot \frac{\sqrt{0.5}}{x}\right)}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2 + \frac{\left(\frac{t\_2}{x} + \frac{l\_m \cdot l\_m}{x}\right) + \left(\left(t\_4 + t\_4\right) + \frac{t\_4}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m)))
        (t_3 (* t_m (sqrt 2.0)))
        (t_4 (+ t_2 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 4.2e-161)
      (/
       t_3
       (*
        t_m
        (+
         (sqrt (+ 2.0 (/ 4.0 x)))
         (* (* (/ l_m t_m) (/ l_m t_m)) (/ (sqrt 0.5) x)))))
      (if (<= t_m 2e+32)
        (/
         t_3
         (sqrt
          (+
           t_2
           (/
            (+ (+ (/ t_2 x) (/ (* l_m l_m) x)) (+ (+ t_4 t_4) (/ t_4 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 * (t_m * t_m);
	double t_3 = t_m * sqrt(2.0);
	double t_4 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 4.2e-161) {
		tmp = t_3 / (t_m * (sqrt((2.0 + (4.0 / x))) + (((l_m / t_m) * (l_m / t_m)) * (sqrt(0.5) / x))));
	} else if (t_m <= 2e+32) {
		tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = t_m * sqrt(2.0d0)
    t_4 = t_2 + (l_m * l_m)
    if (t_m <= 4.2d-161) then
        tmp = t_3 / (t_m * (sqrt((2.0d0 + (4.0d0 / x))) + (((l_m / t_m) * (l_m / t_m)) * (sqrt(0.5d0) / x))))
    else if (t_m <= 2d+32) then
        tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * (t_m * t_m);
	double t_3 = t_m * Math.sqrt(2.0);
	double t_4 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 4.2e-161) {
		tmp = t_3 / (t_m * (Math.sqrt((2.0 + (4.0 / x))) + (((l_m / t_m) * (l_m / t_m)) * (Math.sqrt(0.5) / x))));
	} else if (t_m <= 2e+32) {
		tmp = t_3 / Math.sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * (t_m * t_m)
	t_3 = t_m * math.sqrt(2.0)
	t_4 = t_2 + (l_m * l_m)
	tmp = 0
	if t_m <= 4.2e-161:
		tmp = t_3 / (t_m * (math.sqrt((2.0 + (4.0 / x))) + (((l_m / t_m) * (l_m / t_m)) * (math.sqrt(0.5) / x))))
	elif t_m <= 2e+32:
		tmp = t_3 / math.sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * Float64(t_m * t_m))
	t_3 = Float64(t_m * sqrt(2.0))
	t_4 = Float64(t_2 + Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 4.2e-161)
		tmp = Float64(t_3 / Float64(t_m * Float64(sqrt(Float64(2.0 + Float64(4.0 / x))) + Float64(Float64(Float64(l_m / t_m) * Float64(l_m / t_m)) * Float64(sqrt(0.5) / x)))));
	elseif (t_m <= 2e+32)
		tmp = Float64(t_3 / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(t_2 / x) + Float64(Float64(l_m * l_m) / x)) + Float64(Float64(t_4 + t_4) + Float64(t_4 / x))) / 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 * t_m);
	t_3 = t_m * sqrt(2.0);
	t_4 = t_2 + (l_m * l_m);
	tmp = 0.0;
	if (t_m <= 4.2e-161)
		tmp = t_3 / (t_m * (sqrt((2.0 + (4.0 / x))) + (((l_m / t_m) * (l_m / t_m)) * (sqrt(0.5) / x))));
	elseif (t_m <= 2e+32)
		tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-161], N[(t$95$3 / N[(t$95$m * N[(N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+32], N[(t$95$3 / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 + t$95$4), $MachinePrecision] + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 3 regimes

2. if t < 4.2000000000000001e-161

   1. Initial program 27.5%

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

   3. Taylor expanded in x around inf 43.8%

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

      1. associate--l+ 43.8%

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

      2. associate-*r/ 43.8%

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

      3. unpow2 43.8%

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

      4. sub-neg 43.8%

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

      5. unpow2 43.8%

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

      6. unpow2 43.8%

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

      7. mul-1-neg 43.8%

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

      8. remove-double-neg 43.8%

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

      9. unpow2 43.8%

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

      10. unpow2 43.8%

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

   5. Simplified 43.8%

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

   6. Taylor expanded in t around inf 5.8%

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

      1. associate-*r/ 5.8%

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

      2. metadata-eval 5.8%

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

      3. associate-*l/ 5.8%

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

      4. times-frac 5.8%

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

      5. unpow2 5.8%

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

      6. unpow2 5.8%

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

      7. times-frac 10.8%

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

      8. associate-*r/ 10.8%

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

      9. metadata-eval 10.8%

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

   8. Simplified 10.8%

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

   9. Taylor expanded in x around inf 10.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{t \cdot \left(\sqrt{2 + \frac{4}{x}} + \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\frac{\sqrt{0.5}}{x}}\right)} \]

   if 4.2000000000000001e-161 < t < 2.00000000000000011e32

   1. Initial program 69.6%

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

   3. Taylor expanded in x around -inf 90.0%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. unpow2 90.0%

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

   5. Simplified 90.0%

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

   if 2.00000000000000011e32 < t

   1. Initial program 29.1%

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

   3. Taylor expanded in l around 0 95.4%

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

   4. Taylor expanded in t around 0 95.4%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2 + \frac{4}{x}} + \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt{0.5}}{x}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \left(\left(\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right) + \left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)\right) + \frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t_4 := t\_2 + l\_m \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-198}:\\ \;\;\;\;\frac{t\_3}{l\_m \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{+36}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2 + \frac{\left(\frac{t\_2}{x} + \frac{l\_m \cdot l\_m}{x}\right) + \left(\left(t\_4 + t\_4\right) + \frac{t\_4}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m)))
        (t_3 (* t_m (sqrt 2.0)))
        (t_4 (+ t_2 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 1.55e-198)
      (/ t_3 (* l_m (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x))))))
      (if (<= t_m 1.8e-160)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 1.15e+36)
          (/
           t_3
           (sqrt
            (+
             t_2
             (/
              (+ (+ (/ t_2 x) (/ (* l_m l_m) x)) (+ (+ t_4 t_4) (/ t_4 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 * (t_m * t_m);
	double t_3 = t_m * sqrt(2.0);
	double t_4 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 1.55e-198) {
		tmp = t_3 / (l_m * sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else if (t_m <= 1.8e-160) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.15e+36) {
		tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = t_m * sqrt(2.0d0)
    t_4 = t_2 + (l_m * l_m)
    if (t_m <= 1.55d-198) then
        tmp = t_3 / (l_m * sqrt(((2.0d0 / x) + (2.0d0 / (x * x)))))
    else if (t_m <= 1.8d-160) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 1.15d+36) then
        tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * (t_m * t_m);
	double t_3 = t_m * Math.sqrt(2.0);
	double t_4 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 1.55e-198) {
		tmp = t_3 / (l_m * Math.sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else if (t_m <= 1.8e-160) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.15e+36) {
		tmp = t_3 / Math.sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * (t_m * t_m)
	t_3 = t_m * math.sqrt(2.0)
	t_4 = t_2 + (l_m * l_m)
	tmp = 0
	if t_m <= 1.55e-198:
		tmp = t_3 / (l_m * math.sqrt(((2.0 / x) + (2.0 / (x * x)))))
	elif t_m <= 1.8e-160:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 1.15e+36:
		tmp = t_3 / math.sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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 * Float64(t_m * t_m))
	t_3 = Float64(t_m * sqrt(2.0))
	t_4 = Float64(t_2 + Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 1.55e-198)
		tmp = Float64(t_3 / Float64(l_m * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))))));
	elseif (t_m <= 1.8e-160)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 1.15e+36)
		tmp = Float64(t_3 / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(t_2 / x) + Float64(Float64(l_m * l_m) / x)) + Float64(Float64(t_4 + t_4) + Float64(t_4 / x))) / 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 * t_m);
	t_3 = t_m * sqrt(2.0);
	t_4 = t_2 + (l_m * l_m);
	tmp = 0.0;
	if (t_m <= 1.55e-198)
		tmp = t_3 / (l_m * sqrt(((2.0 / x) + (2.0 / (x * x)))));
	elseif (t_m <= 1.8e-160)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 1.15e+36)
		tmp = t_3 / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_4 + t_4) + (t_4 / x))) / 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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-198], N[(t$95$3 / N[(l$95$m * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e-160], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.15e+36], N[(t$95$3 / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 + t$95$4), $MachinePrecision] + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 1.5499999999999999e-198

   1. Initial program 28.8%

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

   3. Taylor expanded in x around -inf 45.7%

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

      1. +-commutative 45.7%

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

      2. mul-1-neg 45.7%

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

      3. unsub-neg 45.7%

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

      4. unpow2 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in l around inf 18.0%

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

      1. associate-*r/ 18.0%

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

      2. metadata-eval 18.0%

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

      3. associate-*r/ 18.0%

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

      4. metadata-eval 18.0%

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

      5. unpow2 18.0%

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

   8. Simplified 18.0%

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

   if 1.5499999999999999e-198 < t < 1.7999999999999999e-160

   1. Initial program 4.3%

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

   3. Taylor expanded in l around 0 63.8%

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

   4. Taylor expanded in x around inf 63.8%

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

   if 1.7999999999999999e-160 < t < 1.14999999999999998e36

   1. Initial program 69.6%

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

   3. Taylor expanded in x around -inf 90.0%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. unpow2 90.0%

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

   5. Simplified 90.0%

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

   if 1.14999999999999998e36 < t

   1. Initial program 29.1%

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

   3. Taylor expanded in l around 0 95.4%

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

   4. Taylor expanded in t around 0 95.4%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-198}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+36}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \left(\left(\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right) + \left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)\right) + \frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-198}:\\ \;\;\;\;\frac{t\_3}{l\_m \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 1.62 \cdot 10^{+38}:\\ \;\;\;\;\frac{t\_3}{\sqrt{2 \cdot \frac{t\_m \cdot t\_m}{x} + \left(t\_2 + \left(l\_m \cdot \frac{l\_m}{x} + \frac{t\_2 + l\_m \cdot l\_m}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m))) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 3.4e-198)
      (/ t_3 (* l_m (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x))))))
      (if (<= t_m 1.6e-161)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 1.62e+38)
          (/
           t_3
           (sqrt
            (+
             (* 2.0 (/ (* t_m t_m) x))
             (+ t_2 (+ (* l_m (/ l_m x)) (/ (+ t_2 (* l_m l_m)) 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 * (t_m * t_m);
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 3.4e-198) {
		tmp = t_3 / (l_m * sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else if (t_m <= 1.6e-161) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.62e+38) {
		tmp = t_3 / sqrt(((2.0 * ((t_m * t_m) / x)) + (t_2 + ((l_m * (l_m / x)) + ((t_2 + (l_m * l_m)) / 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 * t_m)
    t_3 = t_m * sqrt(2.0d0)
    if (t_m <= 3.4d-198) then
        tmp = t_3 / (l_m * sqrt(((2.0d0 / x) + (2.0d0 / (x * x)))))
    else if (t_m <= 1.6d-161) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 1.62d+38) then
        tmp = t_3 / sqrt(((2.0d0 * ((t_m * t_m) / x)) + (t_2 + ((l_m * (l_m / x)) + ((t_2 + (l_m * l_m)) / 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 * (t_m * t_m);
	double t_3 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 3.4e-198) {
		tmp = t_3 / (l_m * Math.sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else if (t_m <= 1.6e-161) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.62e+38) {
		tmp = t_3 / Math.sqrt(((2.0 * ((t_m * t_m) / x)) + (t_2 + ((l_m * (l_m / x)) + ((t_2 + (l_m * l_m)) / 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 * (t_m * t_m)
	t_3 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 3.4e-198:
		tmp = t_3 / (l_m * math.sqrt(((2.0 / x) + (2.0 / (x * x)))))
	elif t_m <= 1.6e-161:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 1.62e+38:
		tmp = t_3 / math.sqrt(((2.0 * ((t_m * t_m) / x)) + (t_2 + ((l_m * (l_m / x)) + ((t_2 + (l_m * l_m)) / 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 * Float64(t_m * t_m))
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 3.4e-198)
		tmp = Float64(t_3 / Float64(l_m * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))))));
	elseif (t_m <= 1.6e-161)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 1.62e+38)
		tmp = Float64(t_3 / sqrt(Float64(Float64(2.0 * Float64(Float64(t_m * t_m) / x)) + Float64(t_2 + Float64(Float64(l_m * Float64(l_m / x)) + Float64(Float64(t_2 + Float64(l_m * l_m)) / 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 * t_m);
	t_3 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 3.4e-198)
		tmp = t_3 / (l_m * sqrt(((2.0 / x) + (2.0 / (x * x)))));
	elseif (t_m <= 1.6e-161)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 1.62e+38)
		tmp = t_3 / sqrt(((2.0 * ((t_m * t_m) / x)) + (t_2 + ((l_m * (l_m / x)) + ((t_2 + (l_m * l_m)) / 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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.4e-198], N[(t$95$3 / N[(l$95$m * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e-161], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.62e+38], N[(t$95$3 / N[Sqrt[N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(l$95$m * N[(l$95$m / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[(l$95$m * l$95$m), $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 < 3.3999999999999998e-198

   1. Initial program 28.8%

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

   3. Taylor expanded in x around -inf 45.7%

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

      1. +-commutative 45.7%

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

      2. mul-1-neg 45.7%

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

      3. unsub-neg 45.7%

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

      4. unpow2 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in l around inf 18.0%

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

      1. associate-*r/ 18.0%

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

      2. metadata-eval 18.0%

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

      3. associate-*r/ 18.0%

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

      4. metadata-eval 18.0%

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

      5. unpow2 18.0%

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

   8. Simplified 18.0%

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

   if 3.3999999999999998e-198 < t < 1.59999999999999993e-161

   1. Initial program 4.3%

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

   3. Taylor expanded in l around 0 63.8%

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

   4. Taylor expanded in x around inf 63.8%

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

   if 1.59999999999999993e-161 < t < 1.62000000000000001e38

   1. Initial program 69.6%

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

   3. Taylor expanded in x around inf 88.6%

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

      1. associate--l+ 88.6%

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

      2. associate-*r/ 88.6%

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

      3. unpow2 88.6%

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

      4. sub-neg 88.6%

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

      5. unpow2 88.6%

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

      6. unpow2 88.6%

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

      7. mul-1-neg 88.6%

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

      8. remove-double-neg 88.6%

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

      9. unpow2 88.6%

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

      10. unpow2 88.6%

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

   5. Simplified 88.6%

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

      1. associate-/l* 88.6%

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

      2. associate-+l+ 88.6%

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

      3. associate-/l* 88.6%

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

      4. +-commutative 88.6%

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

   7. Applied egg-rr 88.6%

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

   if 1.62000000000000001e38 < t

   1. Initial program 29.1%

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

   3. Taylor expanded in l around 0 95.4%

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

   4. Taylor expanded in t around 0 95.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-198}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{+38}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \frac{\ell}{x} + \frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-198}:\\ \;\;\;\;\frac{t\_2}{l\_m \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{-160}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{t\_2}{\sqrt{2 \cdot \left(t\_m \cdot t\_m\right) + \left(l\_m \cdot l\_m\right) \cdot \frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 4e-198)
      (/ t_2 (* l_m (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x))))))
      (if (<= t_m 1.3e-160)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 9.5e+37)
          (/
           t_2
           (sqrt
            (+ (* 2.0 (* t_m t_m)) (* (* l_m l_m) (/ (+ 2.0 (/ 2.0 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 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 4e-198) {
		tmp = t_2 / (l_m * sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else if (t_m <= 1.3e-160) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 9.5e+37) {
		tmp = t_2 / sqrt(((2.0 * (t_m * t_m)) + ((l_m * l_m) * ((2.0 + (2.0 / x)) / 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 = t_m * sqrt(2.0d0)
    if (t_m <= 4d-198) then
        tmp = t_2 / (l_m * sqrt(((2.0d0 / x) + (2.0d0 / (x * x)))))
    else if (t_m <= 1.3d-160) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 9.5d+37) then
        tmp = t_2 / sqrt(((2.0d0 * (t_m * t_m)) + ((l_m * l_m) * ((2.0d0 + (2.0d0 / x)) / 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 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 4e-198) {
		tmp = t_2 / (l_m * Math.sqrt(((2.0 / x) + (2.0 / (x * x)))));
	} else if (t_m <= 1.3e-160) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 9.5e+37) {
		tmp = t_2 / Math.sqrt(((2.0 * (t_m * t_m)) + ((l_m * l_m) * ((2.0 + (2.0 / x)) / 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 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 4e-198:
		tmp = t_2 / (l_m * math.sqrt(((2.0 / x) + (2.0 / (x * x)))))
	elif t_m <= 1.3e-160:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 9.5e+37:
		tmp = t_2 / math.sqrt(((2.0 * (t_m * t_m)) + ((l_m * l_m) * ((2.0 + (2.0 / x)) / 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(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 4e-198)
		tmp = Float64(t_2 / Float64(l_m * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))))));
	elseif (t_m <= 1.3e-160)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 9.5e+37)
		tmp = Float64(t_2 / sqrt(Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(Float64(l_m * l_m) * Float64(Float64(2.0 + Float64(2.0 / x)) / 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 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 4e-198)
		tmp = t_2 / (l_m * sqrt(((2.0 / x) + (2.0 / (x * x)))));
	elseif (t_m <= 1.3e-160)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 9.5e+37)
		tmp = t_2 / sqrt(((2.0 * (t_m * t_m)) + ((l_m * l_m) * ((2.0 + (2.0 / x)) / 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[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-198], N[(t$95$2 / N[(l$95$m * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e-160], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+37], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(2.0 + N[(2.0 / 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 < 3.9999999999999996e-198

   1. Initial program 28.8%

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

   3. Taylor expanded in x around -inf 45.7%

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

      1. +-commutative 45.7%

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

      2. mul-1-neg 45.7%

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

      3. unsub-neg 45.7%

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

      4. unpow2 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in l around inf 18.0%

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

      1. associate-*r/ 18.0%

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

      2. metadata-eval 18.0%

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

      3. associate-*r/ 18.0%

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

      4. metadata-eval 18.0%

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

      5. unpow2 18.0%

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

   8. Simplified 18.0%

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

   if 3.9999999999999996e-198 < t < 1.30000000000000002e-160

   1. Initial program 4.3%

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

   3. Taylor expanded in l around 0 63.8%

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

   4. Taylor expanded in x around inf 63.8%

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

   if 1.30000000000000002e-160 < t < 9.4999999999999995e37

   1. Initial program 69.6%

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

   3. Taylor expanded in x around -inf 90.0%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. unpow2 90.0%

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

   5. Simplified 90.0%

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

   6. Taylor expanded in l around inf 87.8%

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

      1. mul-1-neg 87.8%

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

      2. neg-sub0 87.8%

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

      3. associate-/l* 87.8%

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

      4. unpow2 87.8%

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

      5. associate-*r/ 87.8%

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

      6. metadata-eval 87.8%

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

   8. Simplified 87.8%

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

   if 9.4999999999999995e37 < t

   1. Initial program 29.1%

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

   3. Taylor expanded in l around 0 95.4%

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

   4. Taylor expanded in t around 0 95.4%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-198}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-160}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.6 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;l\_m \leq 1.55 \cdot 10^{+86}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{2 \cdot \left(l\_m \cdot \frac{l\_m}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \end{array} \]

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 (<= l_m 5.6e+71)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (if (<= l_m 1.55e+86)
      (/ (* t_m (sqrt 2.0)) (sqrt (* 2.0 (* l_m (/ l_m x)))))
      (+ 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 (l_m <= 5.6e+71) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else if (l_m <= 1.55e+86) {
		tmp = (t_m * sqrt(2.0)) / sqrt((2.0 * (l_m * (l_m / x))));
	} 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 (l_m <= 5.6d+71) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (l_m <= 1.55d+86) then
        tmp = (t_m * sqrt(2.0d0)) / sqrt((2.0d0 * (l_m * (l_m / x))))
    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 (l_m <= 5.6e+71) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (l_m <= 1.55e+86) {
		tmp = (t_m * Math.sqrt(2.0)) / Math.sqrt((2.0 * (l_m * (l_m / x))));
	} 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 l_m <= 5.6e+71:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	elif l_m <= 1.55e+86:
		tmp = (t_m * math.sqrt(2.0)) / math.sqrt((2.0 * (l_m * (l_m / x))))
	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 (l_m <= 5.6e+71)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	elseif (l_m <= 1.55e+86)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(2.0 * Float64(l_m * Float64(l_m / x)))));
	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 (l_m <= 5.6e+71)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	elseif (l_m <= 1.55e+86)
		tmp = (t_m * sqrt(2.0)) / sqrt((2.0 * (l_m * (l_m / x))));
	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[l$95$m, 5.6e+71], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.55e+86], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(l$95$m * N[(l$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 5.60000000000000004e71

   1. Initial program 41.1%

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

   3. Taylor expanded in l around 0 43.9%

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

   4. Taylor expanded in t around 0 43.9%

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

   if 5.60000000000000004e71 < l < 1.5500000000000001e86

   1. Initial program 2.4%

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

   3. Taylor expanded in x around inf 99.0%

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

      1. associate--l+ 99.0%

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

      2. associate-*r/ 99.0%

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

      3. unpow2 99.0%

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

      4. sub-neg 99.0%

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

      5. unpow2 99.0%

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

      6. unpow2 99.0%

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

      7. mul-1-neg 99.0%

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

      8. remove-double-neg 99.0%

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

      9. unpow2 99.0%

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

      10. unpow2 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0 99.0%

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

      1. unpow2 99.0%

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

      2. *-commutative 99.0%

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

      3. unpow2 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in l around inf 99.0%

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

       1. unpow2 99.0%

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

       2. associate-*r/ 99.0%

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

   11. Simplified 99.0%

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

   if 1.5500000000000001e86 < l

   1. Initial program 4.9%

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

   3. Taylor expanded in l around 0 25.6%

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

      1. expm1-log1p-u 25.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\right)\right)} \]

      2. expm1-undefine 25.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\right)} - 1} \]

   5. Applied egg-rr 25.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)} - 1} \]
   6. Step-by-step derivation

      1. sub-neg 25.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)} + \left(-1\right)} \]

      2. log1p-undefine 25.6%

         \[\leadsto e^{\color{blue}{\log \left(1 + {\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)}} + \left(-1\right) \]

      3. rem-exp-log 25.6%

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

      4. sub-neg 25.6%

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

      5. metadata-eval 25.6%

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

      6. +-commutative 25.6%

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

      7. metadata-eval 25.6%

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

      8. sub-neg 25.6%

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

      9. metadata-eval 25.6%

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

   7. Simplified 25.6%

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

   8. Taylor expanded in x around -inf 25.6%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 - 0.5 \cdot \frac{1}{x}}{x}} \]
   9. Step-by-step derivation

      1. mul-1-neg 25.6%

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

      2. unsub-neg 25.6%

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

      3. sub-neg 25.6%

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

      4. associate-*r/ 25.6%

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

      5. metadata-eval 25.6%

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

      6. distribute-neg-frac 25.6%

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

      7. metadata-eval 25.6%

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

   10. Simplified 25.6%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+86}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{l\_m \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 1.8e-198)
    (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (+ (/ 2.0 x) (/ 2.0 (* 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 tmp;
	if (t_m <= 1.8e-198) {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((2.0 / x) + (2.0 / (x * 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) :: tmp
    if (t_m <= 1.8d-198) then
        tmp = (t_m * sqrt(2.0d0)) / (l_m * sqrt(((2.0d0 / x) + (2.0d0 / (x * 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 tmp;
	if (t_m <= 1.8e-198) {
		tmp = (t_m * Math.sqrt(2.0)) / (l_m * Math.sqrt(((2.0 / x) + (2.0 / (x * 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):
	tmp = 0
	if t_m <= 1.8e-198:
		tmp = (t_m * math.sqrt(2.0)) / (l_m * math.sqrt(((2.0 / x) + (2.0 / (x * 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)
	tmp = 0.0
	if (t_m <= 1.8e-198)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * 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)
	tmp = 0.0;
	if (t_m <= 1.8e-198)
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((2.0 / x) + (2.0 / (x * 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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-198], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * 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 2 regimes

2. if t < 1.79999999999999999e-198

   1. Initial program 28.8%

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

   3. Taylor expanded in x around -inf 45.7%

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

      1. +-commutative 45.7%

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

      2. mul-1-neg 45.7%

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

      3. unsub-neg 45.7%

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

      4. unpow2 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in l around inf 18.0%

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

      1. associate-*r/ 18.0%

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

      2. metadata-eval 18.0%

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

      3. associate-*r/ 18.0%

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

      4. metadata-eval 18.0%

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

      5. unpow2 18.0%

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

   8. Simplified 18.0%

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

   if 1.79999999999999999e-198 < t

   1. Initial program 43.2%

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

   3. Taylor expanded in l around 0 85.6%

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

   4. Taylor expanded in t around 0 85.6%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-198}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% 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 35.3%

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

3. Taylor expanded in l around 0 40.7%

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

4. Taylor expanded in t around 0 40.7%

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

5. Final simplification 40.7%

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

Alternative 9: 76.7% accurate, 17.3× speedup

Math

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

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 (/ (+ (/ (+ 0.5 (/ -0.5 x)) x) -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 + ((((0.5 + (-0.5 / x)) / x) + -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 + ((((0.5d0 + ((-0.5d0) / x)) / x) + (-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 + ((((0.5 + (-0.5 / x)) / x) + -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 + ((((0.5 + (-0.5 / x)) / x) + -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(Float64(Float64(Float64(0.5 + Float64(-0.5 / x)) / x) + -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 + ((((0.5 + (-0.5 / x)) / x) + -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[(N[(N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

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

3. Taylor expanded in l around 0 40.7%

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

   1. expm1-log1p-u 40.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\right)\right)} \]

   2. expm1-undefine 40.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\right)} - 1} \]

5. Applied egg-rr 40.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)} - 1} \]
6. Step-by-step derivation

   1. sub-neg 40.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)} + \left(-1\right)} \]

   2. log1p-undefine 40.7%

      \[\leadsto e^{\color{blue}{\log \left(1 + {\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)}} + \left(-1\right) \]

   3. rem-exp-log 40.7%

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

   4. sub-neg 40.7%

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

   5. metadata-eval 40.7%

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

   6. +-commutative 40.7%

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

   7. metadata-eval 40.7%

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

   8. sub-neg 40.7%

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

   9. metadata-eval 40.7%

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

7. Simplified 40.7%

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

8. Taylor expanded in x around -inf 40.4%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}{x}} \]
9. Step-by-step derivation

   1. mul-1-neg 40.4%

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

   2. unsub-neg 40.4%

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

   3. mul-1-neg 40.4%

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

   4. unsub-neg 40.4%

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

   5. sub-neg 40.4%

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

   6. associate-*r/ 40.4%

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

   7. metadata-eval 40.4%

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

   8. distribute-neg-frac 40.4%

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

   9. metadata-eval 40.4%

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

10. Simplified 40.4%

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

11. Final simplification 40.4%

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

Alternative 10: 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 35.3%

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

3. Taylor expanded in l around 0 40.7%

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

   1. expm1-log1p-u 40.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\right)\right)} \]

   2. expm1-undefine 40.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\right)} - 1} \]

5. Applied egg-rr 40.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)} - 1} \]
6. Step-by-step derivation

   1. sub-neg 40.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)} + \left(-1\right)} \]

   2. log1p-undefine 40.7%

      \[\leadsto e^{\color{blue}{\log \left(1 + {\left(\frac{x - 1}{1 + x}\right)}^{0.5}\right)}} + \left(-1\right) \]

   3. rem-exp-log 40.7%

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

   4. sub-neg 40.7%

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

   5. metadata-eval 40.7%

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

   6. +-commutative 40.7%

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

   7. metadata-eval 40.7%

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

   8. sub-neg 40.7%

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

   9. metadata-eval 40.7%

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

7. Simplified 40.7%

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

8. Taylor expanded in x around -inf 40.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 - 0.5 \cdot \frac{1}{x}}{x}} \]
9. Step-by-step derivation

   1. mul-1-neg 40.2%

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

   2. unsub-neg 40.2%

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

   3. sub-neg 40.2%

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

   4. associate-*r/ 40.2%

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

   5. metadata-eval 40.2%

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

   6. distribute-neg-frac 40.2%

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

   7. metadata-eval 40.2%

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

10. Simplified 40.2%

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

11. Final simplification 40.2%

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

Alternative 11: 76.5% 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 35.3%

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

3. Taylor expanded in l around 0 40.7%

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

4. Taylor expanded in x around inf 39.9%

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

5. Final simplification 39.9%

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

Alternative 12: 75.9% 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 35.3%

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

3. Taylor expanded in l around 0 40.7%

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

4. Taylor expanded in x around inf 39.2%

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

Reproduce

herbie shell --seed 2024107 
(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)))))