Toniolo and Linder, Equation (7)

Percentage Accurate: 33.0% → 85.2%

Time: 21.7s

Alternatives: 11

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.0% 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: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t_4 := t\_3 + {l\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{l\_m \cdot \frac{\sqrt{2}}{\sqrt{x}}}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_2}{0.5 \cdot \frac{t\_4 + t\_4}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_2}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_3 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_4}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_3 (* 2.0 (pow t_m 2.0)))
        (t_4 (+ t_3 (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 2.6e-266)
      (/ (sqrt 2.0) (/ (* l_m (/ (sqrt 2.0) (sqrt x))) t_m))
      (if (<= t_m 4.6e-162)
        (/ t_2 (+ (* 0.5 (/ (+ t_4 t_4) (* t_m (* (sqrt 2.0) x)))) t_2))
        (if (<= t_m 2.5e+72)
          (/
           t_2
           (sqrt
            (+
             (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_3 (/ (pow l_m 2.0) x)))
             (/ t_4 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 t_3 = 2.0 * pow(t_m, 2.0);
	double t_4 = t_3 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 2.6e-266) {
		tmp = sqrt(2.0) / ((l_m * (sqrt(2.0) / sqrt(x))) / t_m);
	} else if (t_m <= 4.6e-162) {
		tmp = t_2 / ((0.5 * ((t_4 + t_4) / (t_m * (sqrt(2.0) * x)))) + t_2);
	} else if (t_m <= 2.5e+72) {
		tmp = t_2 / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_3 + (pow(l_m, 2.0) / x))) + (t_4 / 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 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    t_4 = t_3 + (l_m ** 2.0d0)
    if (t_m <= 2.6d-266) then
        tmp = sqrt(2.0d0) / ((l_m * (sqrt(2.0d0) / sqrt(x))) / t_m)
    else if (t_m <= 4.6d-162) then
        tmp = t_2 / ((0.5d0 * ((t_4 + t_4) / (t_m * (sqrt(2.0d0) * x)))) + t_2)
    else if (t_m <= 2.5d+72) then
        tmp = t_2 / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_3 + ((l_m ** 2.0d0) / x))) + (t_4 / 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 t_3 = 2.0 * Math.pow(t_m, 2.0);
	double t_4 = t_3 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 2.6e-266) {
		tmp = Math.sqrt(2.0) / ((l_m * (Math.sqrt(2.0) / Math.sqrt(x))) / t_m);
	} else if (t_m <= 4.6e-162) {
		tmp = t_2 / ((0.5 * ((t_4 + t_4) / (t_m * (Math.sqrt(2.0) * x)))) + t_2);
	} else if (t_m <= 2.5e+72) {
		tmp = t_2 / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_3 + (Math.pow(l_m, 2.0) / x))) + (t_4 / 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)
	t_3 = 2.0 * math.pow(t_m, 2.0)
	t_4 = t_3 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 2.6e-266:
		tmp = math.sqrt(2.0) / ((l_m * (math.sqrt(2.0) / math.sqrt(x))) / t_m)
	elif t_m <= 4.6e-162:
		tmp = t_2 / ((0.5 * ((t_4 + t_4) / (t_m * (math.sqrt(2.0) * x)))) + t_2)
	elif t_m <= 2.5e+72:
		tmp = t_2 / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_3 + (math.pow(l_m, 2.0) / x))) + (t_4 / 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))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	t_4 = Float64(t_3 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.6e-266)
		tmp = Float64(sqrt(2.0) / Float64(Float64(l_m * Float64(sqrt(2.0) / sqrt(x))) / t_m));
	elseif (t_m <= 4.6e-162)
		tmp = Float64(t_2 / Float64(Float64(0.5 * Float64(Float64(t_4 + t_4) / Float64(t_m * Float64(sqrt(2.0) * x)))) + t_2));
	elseif (t_m <= 2.5e+72)
		tmp = Float64(t_2 / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_3 + Float64((l_m ^ 2.0) / x))) + Float64(t_4 / 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);
	t_3 = 2.0 * (t_m ^ 2.0);
	t_4 = t_3 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.6e-266)
		tmp = sqrt(2.0) / ((l_m * (sqrt(2.0) / sqrt(x))) / t_m);
	elseif (t_m <= 4.6e-162)
		tmp = t_2 / ((0.5 * ((t_4 + t_4) / (t_m * (sqrt(2.0) * x)))) + t_2);
	elseif (t_m <= 2.5e+72)
		tmp = t_2 / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_3 + ((l_m ^ 2.0) / x))) + (t_4 / 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]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-266], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e-162], N[(t$95$2 / N[(N[(0.5 * N[(N[(t$95$4 + t$95$4), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+72], N[(t$95$2 / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / 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 < 2.6e-266

   1. Initial program 33.4%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 9.5%

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

      2. sub-neg 9.5%

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

      3. metadata-eval 9.5%

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

      4. +-commutative 9.5%

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

      5. sub-neg 9.5%

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

      6. metadata-eval 9.5%

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

      7. +-commutative 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in x around inf 19.4%

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

      1. clear-num 19.4%

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

      2. un-div-inv 19.5%

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

      3. associate-*l* 19.5%

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

      4. sqrt-div 19.5%

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

      5. metadata-eval 19.5%

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

      6. un-div-inv 19.4%

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

   10. Applied egg-rr 19.4%

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

   if 2.6e-266 < t < 4.5999999999999996e-162

   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 76.7%

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

   if 4.5999999999999996e-162 < t < 2.49999999999999996e72

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf 85.9%

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

   if 2.49999999999999996e72 < t

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in t around inf 97.0%

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

   6. Taylor expanded in t around 0 97.3%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}}{t}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{l\_m \cdot \frac{\sqrt{2}}{\sqrt{x}}}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_m \cdot \frac{\sqrt{x}}{\sqrt{2} \cdot l\_m}\right)\\ \mathbf{elif}\;t\_m \leq 10^{+72}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_2 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 4.8e-227)
      (/ (sqrt 2.0) (/ (* l_m (/ (sqrt 2.0) (sqrt x))) t_m))
      (if (<= t_m 1.4e-188)
        1.0
        (if (<= t_m 3.6e-162)
          (* (sqrt 2.0) (* t_m (/ (sqrt x) (* (sqrt 2.0) l_m))))
          (if (<= t_m 1e+72)
            (/
             (* t_m (sqrt 2.0))
             (sqrt
              (+
               (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
               (/ (+ t_2 (pow l_m 2.0)) x))))
            (sqrt (/ (+ x -1.0) (+ x 1.0))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 4.8e-227) {
		tmp = sqrt(2.0) / ((l_m * (sqrt(2.0) / sqrt(x))) / t_m);
	} else if (t_m <= 1.4e-188) {
		tmp = 1.0;
	} else if (t_m <= 3.6e-162) {
		tmp = sqrt(2.0) * (t_m * (sqrt(x) / (sqrt(2.0) * l_m)));
	} else if (t_m <= 1e+72) {
		tmp = (t_m * sqrt(2.0)) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 4.8d-227) then
        tmp = sqrt(2.0d0) / ((l_m * (sqrt(2.0d0) / sqrt(x))) / t_m)
    else if (t_m <= 1.4d-188) then
        tmp = 1.0d0
    else if (t_m <= 3.6d-162) then
        tmp = sqrt(2.0d0) * (t_m * (sqrt(x) / (sqrt(2.0d0) * l_m)))
    else if (t_m <= 1d+72) then
        tmp = (t_m * sqrt(2.0d0)) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 4.8e-227) {
		tmp = Math.sqrt(2.0) / ((l_m * (Math.sqrt(2.0) / Math.sqrt(x))) / t_m);
	} else if (t_m <= 1.4e-188) {
		tmp = 1.0;
	} else if (t_m <= 3.6e-162) {
		tmp = Math.sqrt(2.0) * (t_m * (Math.sqrt(x) / (Math.sqrt(2.0) * l_m)));
	} else if (t_m <= 1e+72) {
		tmp = (t_m * Math.sqrt(2.0)) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((t_2 + Math.pow(l_m, 2.0)) / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 4.8e-227:
		tmp = math.sqrt(2.0) / ((l_m * (math.sqrt(2.0) / math.sqrt(x))) / t_m)
	elif t_m <= 1.4e-188:
		tmp = 1.0
	elif t_m <= 3.6e-162:
		tmp = math.sqrt(2.0) * (t_m * (math.sqrt(x) / (math.sqrt(2.0) * l_m)))
	elif t_m <= 1e+72:
		tmp = (t_m * math.sqrt(2.0)) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((t_2 + math.pow(l_m, 2.0)) / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 4.8e-227)
		tmp = Float64(sqrt(2.0) / Float64(Float64(l_m * Float64(sqrt(2.0) / sqrt(x))) / t_m));
	elseif (t_m <= 1.4e-188)
		tmp = 1.0;
	elseif (t_m <= 3.6e-162)
		tmp = Float64(sqrt(2.0) * Float64(t_m * Float64(sqrt(x) / Float64(sqrt(2.0) * l_m))));
	elseif (t_m <= 1e+72)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 4.8e-227)
		tmp = sqrt(2.0) / ((l_m * (sqrt(2.0) / sqrt(x))) / t_m);
	elseif (t_m <= 1.4e-188)
		tmp = 1.0;
	elseif (t_m <= 3.6e-162)
		tmp = sqrt(2.0) * (t_m * (sqrt(x) / (sqrt(2.0) * l_m)));
	elseif (t_m <= 1e+72)
		tmp = (t_m * sqrt(2.0)) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.8e-227], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e-188], 1.0, If[LessEqual[t$95$m, 3.6e-162], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+72], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 4.7999999999999999e-227

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. clear-num 20.1%

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

      2. un-div-inv 20.2%

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

      3. associate-*l* 20.2%

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

      4. sqrt-div 20.2%

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

      5. metadata-eval 20.2%

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

      6. un-div-inv 20.2%

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

   10. Applied egg-rr 20.2%

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

   if 4.7999999999999999e-227 < t < 1.4000000000000001e-188

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 1.4000000000000001e-188 < t < 3.5999999999999998e-162

   1. Initial program 2.3%

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

   3. Simplified 1.7%

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

   5. Taylor expanded in l around inf 1.3%

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

      1. associate--l+ 23.5%

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

      2. sub-neg 23.5%

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

      3. metadata-eval 23.5%

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

      4. +-commutative 23.5%

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

      5. sub-neg 23.5%

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

      6. metadata-eval 23.5%

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

      7. +-commutative 23.5%

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

   7. Simplified 23.5%

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

   8. Taylor expanded in x around inf 34.6%

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

      1. clear-num 32.0%

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

      2. inv-pow 32.0%

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

      3. associate-*l* 32.0%

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

      4. sqrt-div 32.0%

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

      5. metadata-eval 32.0%

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

      6. un-div-inv 32.0%

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

   10. Applied egg-rr 32.0%

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

       1. unpow-1 32.0%

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

       2. associate-/l* 32.0%

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

   12. Simplified 32.0%

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

   13. Taylor expanded in l around 0 32.0%

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

       1. associate-*l/ 34.6%

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

       2. associate-/l* 34.6%

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

       3. *-commutative 34.6%

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

   15. Simplified 34.6%

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

   if 3.5999999999999998e-162 < t < 9.99999999999999944e71

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf 85.9%

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

   if 9.99999999999999944e71 < t

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in t around inf 97.0%

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

   6. Taylor expanded in t around 0 97.3%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}}{t}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \frac{\sqrt{x}}{\sqrt{2} \cdot \ell}\right)\\ \mathbf{elif}\;t \leq 10^{+72}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-228}:\\ \;\;\;\;t\_m \cdot \frac{\frac{\sqrt{2}}{l\_m}}{\frac{\sqrt{2}}{\sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 6.9 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 10^{-178}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_m \cdot \frac{\sqrt{x}}{\sqrt{2} \cdot l\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 3.6e-228)
    (* t_m (/ (/ (sqrt 2.0) l_m) (/ (sqrt 2.0) (sqrt x))))
    (if (<= t_m 6.9e-187)
      1.0
      (if (<= t_m 1e-178)
        (* (sqrt 2.0) (* t_m (/ (sqrt x) (* (sqrt 2.0) l_m))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.6e-228) {
		tmp = t_m * ((sqrt(2.0) / l_m) / (sqrt(2.0) / sqrt(x)));
	} else if (t_m <= 6.9e-187) {
		tmp = 1.0;
	} else if (t_m <= 1e-178) {
		tmp = sqrt(2.0) * (t_m * (sqrt(x) / (sqrt(2.0) * l_m)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.6d-228) then
        tmp = t_m * ((sqrt(2.0d0) / l_m) / (sqrt(2.0d0) / sqrt(x)))
    else if (t_m <= 6.9d-187) then
        tmp = 1.0d0
    else if (t_m <= 1d-178) then
        tmp = sqrt(2.0d0) * (t_m * (sqrt(x) / (sqrt(2.0d0) * l_m)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.6e-228) {
		tmp = t_m * ((Math.sqrt(2.0) / l_m) / (Math.sqrt(2.0) / Math.sqrt(x)));
	} else if (t_m <= 6.9e-187) {
		tmp = 1.0;
	} else if (t_m <= 1e-178) {
		tmp = Math.sqrt(2.0) * (t_m * (Math.sqrt(x) / (Math.sqrt(2.0) * l_m)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 3.6e-228:
		tmp = t_m * ((math.sqrt(2.0) / l_m) / (math.sqrt(2.0) / math.sqrt(x)))
	elif t_m <= 6.9e-187:
		tmp = 1.0
	elif t_m <= 1e-178:
		tmp = math.sqrt(2.0) * (t_m * (math.sqrt(x) / (math.sqrt(2.0) * l_m)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 3.6e-228)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) / Float64(sqrt(2.0) / sqrt(x))));
	elseif (t_m <= 6.9e-187)
		tmp = 1.0;
	elseif (t_m <= 1e-178)
		tmp = Float64(sqrt(2.0) * Float64(t_m * Float64(sqrt(x) / Float64(sqrt(2.0) * l_m))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 3.6e-228)
		tmp = t_m * ((sqrt(2.0) / l_m) / (sqrt(2.0) / sqrt(x)));
	elseif (t_m <= 6.9e-187)
		tmp = 1.0;
	elseif (t_m <= 1e-178)
		tmp = sqrt(2.0) * (t_m * (sqrt(x) / (sqrt(2.0) * l_m)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-228], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.9e-187], 1.0, If[LessEqual[t$95$m, 1e-178], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $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 < 3.6000000000000002e-228

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. associate-*r/ 20.2%

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

      2. clear-num 20.1%

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

      3. associate-*l* 20.2%

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

      4. sqrt-div 20.1%

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

      5. metadata-eval 20.1%

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

      6. un-div-inv 20.1%

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

      7. *-commutative 20.1%

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

   10. Applied egg-rr 20.1%

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

       1. associate-/r/ 20.1%

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

       2. associate-*l/ 20.2%

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

       3. *-lft-identity 20.2%

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

       4. associate-/l* 20.2%

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

       5. associate-/r* 20.2%

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

   12. Simplified 20.2%

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

   if 3.6000000000000002e-228 < t < 6.90000000000000045e-187

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 6.90000000000000045e-187 < t < 9.9999999999999995e-179

   1. Initial program 0.7%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 65.5%

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

      2. sub-neg 65.5%

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

      3. metadata-eval 65.5%

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

      4. +-commutative 65.5%

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

      5. sub-neg 65.5%

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

      6. metadata-eval 65.5%

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

      7. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. clear-num 90.9%

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

      2. inv-pow 90.9%

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

      3. associate-*l* 90.9%

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

      4. sqrt-div 90.9%

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

      5. metadata-eval 90.9%

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

      6. un-div-inv 90.9%

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

   10. Applied egg-rr 90.9%

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

       1. unpow-1 90.9%

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

       2. associate-/l* 90.9%

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

   12. Simplified 90.9%

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

   13. Taylor expanded in l around 0 90.9%

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

       1. associate-*l/ 99.0%

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

       2. associate-/l* 99.0%

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

       3. *-commutative 99.0%

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

   15. Simplified 99.0%

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

   if 9.9999999999999995e-179 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-228}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\frac{\sqrt{2}}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 10^{-178}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \frac{\sqrt{x}}{\sqrt{2} \cdot \ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{l\_m \cdot \frac{\sqrt{2}}{\sqrt{x}}}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 8.8 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_m \cdot \frac{\sqrt{x}}{\sqrt{2} \cdot l\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 7e-228)
    (/ (sqrt 2.0) (/ (* l_m (/ (sqrt 2.0) (sqrt x))) t_m))
    (if (<= t_m 8.6e-189)
      1.0
      (if (<= t_m 8.8e-179)
        (* (sqrt 2.0) (* t_m (/ (sqrt x) (* (sqrt 2.0) l_m))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 7e-228) {
		tmp = sqrt(2.0) / ((l_m * (sqrt(2.0) / sqrt(x))) / t_m);
	} else if (t_m <= 8.6e-189) {
		tmp = 1.0;
	} else if (t_m <= 8.8e-179) {
		tmp = sqrt(2.0) * (t_m * (sqrt(x) / (sqrt(2.0) * l_m)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 7d-228) then
        tmp = sqrt(2.0d0) / ((l_m * (sqrt(2.0d0) / sqrt(x))) / t_m)
    else if (t_m <= 8.6d-189) then
        tmp = 1.0d0
    else if (t_m <= 8.8d-179) then
        tmp = sqrt(2.0d0) * (t_m * (sqrt(x) / (sqrt(2.0d0) * l_m)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 7e-228) {
		tmp = Math.sqrt(2.0) / ((l_m * (Math.sqrt(2.0) / Math.sqrt(x))) / t_m);
	} else if (t_m <= 8.6e-189) {
		tmp = 1.0;
	} else if (t_m <= 8.8e-179) {
		tmp = Math.sqrt(2.0) * (t_m * (Math.sqrt(x) / (Math.sqrt(2.0) * l_m)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 7e-228:
		tmp = math.sqrt(2.0) / ((l_m * (math.sqrt(2.0) / math.sqrt(x))) / t_m)
	elif t_m <= 8.6e-189:
		tmp = 1.0
	elif t_m <= 8.8e-179:
		tmp = math.sqrt(2.0) * (t_m * (math.sqrt(x) / (math.sqrt(2.0) * l_m)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 7e-228)
		tmp = Float64(sqrt(2.0) / Float64(Float64(l_m * Float64(sqrt(2.0) / sqrt(x))) / t_m));
	elseif (t_m <= 8.6e-189)
		tmp = 1.0;
	elseif (t_m <= 8.8e-179)
		tmp = Float64(sqrt(2.0) * Float64(t_m * Float64(sqrt(x) / Float64(sqrt(2.0) * l_m))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 7e-228)
		tmp = sqrt(2.0) / ((l_m * (sqrt(2.0) / sqrt(x))) / t_m);
	elseif (t_m <= 8.6e-189)
		tmp = 1.0;
	elseif (t_m <= 8.8e-179)
		tmp = sqrt(2.0) * (t_m * (sqrt(x) / (sqrt(2.0) * l_m)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 7e-228], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e-189], 1.0, If[LessEqual[t$95$m, 8.8e-179], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $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 < 6.9999999999999995e-228

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. clear-num 20.1%

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

      2. un-div-inv 20.2%

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

      3. associate-*l* 20.2%

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

      4. sqrt-div 20.2%

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

      5. metadata-eval 20.2%

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

      6. un-div-inv 20.2%

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

   10. Applied egg-rr 20.2%

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

   if 6.9999999999999995e-228 < t < 8.60000000000000071e-189

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 8.60000000000000071e-189 < t < 8.80000000000000018e-179

   1. Initial program 0.7%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 65.5%

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

      2. sub-neg 65.5%

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

      3. metadata-eval 65.5%

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

      4. +-commutative 65.5%

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

      5. sub-neg 65.5%

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

      6. metadata-eval 65.5%

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

      7. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. clear-num 90.9%

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

      2. inv-pow 90.9%

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

      3. associate-*l* 90.9%

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

      4. sqrt-div 90.9%

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

      5. metadata-eval 90.9%

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

      6. un-div-inv 90.9%

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

   10. Applied egg-rr 90.9%

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

       1. unpow-1 90.9%

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

       2. associate-/l* 90.9%

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

   12. Simplified 90.9%

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

   13. Taylor expanded in l around 0 90.9%

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

       1. associate-*l/ 99.0%

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

       2. associate-/l* 99.0%

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

       3. *-commutative 99.0%

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

   15. Simplified 99.0%

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

   if 8.80000000000000018e-179 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}}{t}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \frac{\sqrt{x}}{\sqrt{2} \cdot \ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-226}:\\ \;\;\;\;t\_m \cdot \frac{t\_2}{\frac{\sqrt{2}}{\sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.35 \cdot 10^{-178}:\\ \;\;\;\;t\_m \cdot \frac{t\_2}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) l_m)))
   (*
    t_s
    (if (<= t_m 9.5e-226)
      (* t_m (/ t_2 (/ (sqrt 2.0) (sqrt x))))
      (if (<= t_m 1.2e-186)
        1.0
        (if (<= t_m 2.35e-178)
          (* t_m (/ t_2 (sqrt (/ 2.0 x))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) / l_m;
	double tmp;
	if (t_m <= 9.5e-226) {
		tmp = t_m * (t_2 / (sqrt(2.0) / sqrt(x)));
	} else if (t_m <= 1.2e-186) {
		tmp = 1.0;
	} else if (t_m <= 2.35e-178) {
		tmp = t_m * (t_2 / sqrt((2.0 / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) / l_m
    if (t_m <= 9.5d-226) then
        tmp = t_m * (t_2 / (sqrt(2.0d0) / sqrt(x)))
    else if (t_m <= 1.2d-186) then
        tmp = 1.0d0
    else if (t_m <= 2.35d-178) then
        tmp = t_m * (t_2 / sqrt((2.0d0 / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) / l_m;
	double tmp;
	if (t_m <= 9.5e-226) {
		tmp = t_m * (t_2 / (Math.sqrt(2.0) / Math.sqrt(x)));
	} else if (t_m <= 1.2e-186) {
		tmp = 1.0;
	} else if (t_m <= 2.35e-178) {
		tmp = t_m * (t_2 / Math.sqrt((2.0 / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) / l_m
	tmp = 0
	if t_m <= 9.5e-226:
		tmp = t_m * (t_2 / (math.sqrt(2.0) / math.sqrt(x)))
	elif t_m <= 1.2e-186:
		tmp = 1.0
	elif t_m <= 2.35e-178:
		tmp = t_m * (t_2 / math.sqrt((2.0 / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) / l_m)
	tmp = 0.0
	if (t_m <= 9.5e-226)
		tmp = Float64(t_m * Float64(t_2 / Float64(sqrt(2.0) / sqrt(x))));
	elseif (t_m <= 1.2e-186)
		tmp = 1.0;
	elseif (t_m <= 2.35e-178)
		tmp = Float64(t_m * Float64(t_2 / sqrt(Float64(2.0 / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) / l_m;
	tmp = 0.0;
	if (t_m <= 9.5e-226)
		tmp = t_m * (t_2 / (sqrt(2.0) / sqrt(x)));
	elseif (t_m <= 1.2e-186)
		tmp = 1.0;
	elseif (t_m <= 2.35e-178)
		tmp = t_m * (t_2 / sqrt((2.0 / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-226], N[(t$95$m * N[(t$95$2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e-186], 1.0, If[LessEqual[t$95$m, 2.35e-178], N[(t$95$m * N[(t$95$2 / N[Sqrt[N[(2.0 / 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 < 9.5000000000000007e-226

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. associate-*r/ 20.2%

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

      2. clear-num 20.1%

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

      3. associate-*l* 20.2%

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

      4. sqrt-div 20.1%

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

      5. metadata-eval 20.1%

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

      6. un-div-inv 20.1%

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

      7. *-commutative 20.1%

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

   10. Applied egg-rr 20.1%

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

       1. associate-/r/ 20.1%

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

       2. associate-*l/ 20.2%

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

       3. *-lft-identity 20.2%

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

       4. associate-/l* 20.2%

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

       5. associate-/r* 20.2%

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

   12. Simplified 20.2%

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

   if 9.5000000000000007e-226 < t < 1.20000000000000002e-186

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 1.20000000000000002e-186 < t < 2.35e-178

   1. Initial program 0.7%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 65.5%

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

      2. sub-neg 65.5%

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

      3. metadata-eval 65.5%

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

      4. +-commutative 65.5%

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

      5. sub-neg 65.5%

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

      6. metadata-eval 65.5%

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

      7. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. clear-num 90.9%

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

      2. inv-pow 90.9%

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

      3. associate-*l* 90.9%

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

      4. sqrt-div 90.9%

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

      5. metadata-eval 90.9%

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

      6. un-div-inv 90.9%

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

   10. Applied egg-rr 90.9%

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

       1. unpow-1 90.9%

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

       2. associate-/l* 90.9%

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

   12. Simplified 90.9%

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

       1. un-div-inv 91.4%

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

       2. clear-num 91.4%

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

       3. un-div-inv 91.4%

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

       4. sqrt-undiv 91.4%

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

   14. Applied egg-rr 91.4%

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

       1. associate-/r/ 99.5%

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

       2. *-commutative 99.5%

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

       3. associate-*l/ 91.4%

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

       4. associate-/l* 99.5%

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

   16. Simplified 99.5%

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

   if 2.35e-178 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\frac{\sqrt{2}}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-178}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\frac{\sqrt{2}}{l\_m}}{\sqrt{\frac{2}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-224}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 9.6 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{-178}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ (/ (sqrt 2.0) l_m) (sqrt (/ 2.0 x))))))
   (*
    t_s
    (if (<= t_m 7.8e-224)
      t_2
      (if (<= t_m 9.6e-188)
        1.0
        (if (<= t_m 1.45e-178) t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * ((sqrt(2.0) / l_m) / sqrt((2.0 / x)));
	double tmp;
	if (t_m <= 7.8e-224) {
		tmp = t_2;
	} else if (t_m <= 9.6e-188) {
		tmp = 1.0;
	} else if (t_m <= 1.45e-178) {
		tmp = t_2;
	} 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) / l_m) / sqrt((2.0d0 / x)))
    if (t_m <= 7.8d-224) then
        tmp = t_2
    else if (t_m <= 9.6d-188) then
        tmp = 1.0d0
    else if (t_m <= 1.45d-178) then
        tmp = t_2
    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) / l_m) / Math.sqrt((2.0 / x)));
	double tmp;
	if (t_m <= 7.8e-224) {
		tmp = t_2;
	} else if (t_m <= 9.6e-188) {
		tmp = 1.0;
	} else if (t_m <= 1.45e-178) {
		tmp = t_2;
	} 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) / l_m) / math.sqrt((2.0 / x)))
	tmp = 0
	if t_m <= 7.8e-224:
		tmp = t_2
	elif t_m <= 9.6e-188:
		tmp = 1.0
	elif t_m <= 1.45e-178:
		tmp = t_2
	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 * Float64(Float64(sqrt(2.0) / l_m) / sqrt(Float64(2.0 / x))))
	tmp = 0.0
	if (t_m <= 7.8e-224)
		tmp = t_2;
	elseif (t_m <= 9.6e-188)
		tmp = 1.0;
	elseif (t_m <= 1.45e-178)
		tmp = t_2;
	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) / l_m) / sqrt((2.0 / x)));
	tmp = 0.0;
	if (t_m <= 7.8e-224)
		tmp = t_2;
	elseif (t_m <= 9.6e-188)
		tmp = 1.0;
	elseif (t_m <= 1.45e-178)
		tmp = t_2;
	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[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.8e-224], t$95$2, If[LessEqual[t$95$m, 9.6e-188], 1.0, If[LessEqual[t$95$m, 1.45e-178], t$95$2, 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 < 7.7999999999999996e-224 or 9.6e-188 < t < 1.4499999999999999e-178

   1. Initial program 31.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 11.9%

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

      2. sub-neg 11.9%

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

      3. metadata-eval 11.9%

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

      4. +-commutative 11.9%

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

      5. sub-neg 11.9%

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

      6. metadata-eval 11.9%

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

      7. +-commutative 11.9%

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

   7. Simplified 11.9%

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

   8. Taylor expanded in x around inf 22.0%

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

      1. clear-num 21.8%

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

      2. inv-pow 21.8%

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

      3. associate-*l* 21.8%

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

      4. sqrt-div 21.8%

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

      5. metadata-eval 21.8%

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

      6. un-div-inv 21.8%

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

   10. Applied egg-rr 21.8%

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

       1. unpow-1 21.8%

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

       2. associate-/l* 21.8%

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

   12. Simplified 21.8%

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

       1. un-div-inv 21.8%

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

       2. clear-num 21.8%

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

       3. un-div-inv 21.8%

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

       4. sqrt-undiv 21.8%

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

   14. Applied egg-rr 21.8%

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

       1. associate-/r/ 22.0%

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

       2. *-commutative 22.0%

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

       3. associate-*l/ 19.7%

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

       4. associate-/l* 22.1%

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

   16. Simplified 22.1%

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

   if 7.7999999999999996e-224 < t < 9.6e-188

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 1.4499999999999999e-178 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-224}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-178}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{2}{x}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.4 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{l\_m}{\frac{t\_m}{t\_2}}}\\ \mathbf{elif}\;t\_m \leq 1.62 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{-179}:\\ \;\;\;\;t\_m \cdot \frac{\frac{\sqrt{2}}{l\_m}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (sqrt (/ 2.0 x))))
   (*
    t_s
    (if (<= t_m 6.4e-223)
      (/ (sqrt 2.0) (/ l_m (/ t_m t_2)))
      (if (<= t_m 1.62e-187)
        1.0
        (if (<= t_m 5.6e-179)
          (* t_m (/ (/ (sqrt 2.0) l_m) t_2))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt((2.0 / x));
	double tmp;
	if (t_m <= 6.4e-223) {
		tmp = sqrt(2.0) / (l_m / (t_m / t_2));
	} else if (t_m <= 1.62e-187) {
		tmp = 1.0;
	} else if (t_m <= 5.6e-179) {
		tmp = t_m * ((sqrt(2.0) / l_m) / t_2);
	} 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 = sqrt((2.0d0 / x))
    if (t_m <= 6.4d-223) then
        tmp = sqrt(2.0d0) / (l_m / (t_m / t_2))
    else if (t_m <= 1.62d-187) then
        tmp = 1.0d0
    else if (t_m <= 5.6d-179) then
        tmp = t_m * ((sqrt(2.0d0) / l_m) / t_2)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt((2.0 / x));
	double tmp;
	if (t_m <= 6.4e-223) {
		tmp = Math.sqrt(2.0) / (l_m / (t_m / t_2));
	} else if (t_m <= 1.62e-187) {
		tmp = 1.0;
	} else if (t_m <= 5.6e-179) {
		tmp = t_m * ((Math.sqrt(2.0) / l_m) / t_2);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt((2.0 / x))
	tmp = 0
	if t_m <= 6.4e-223:
		tmp = math.sqrt(2.0) / (l_m / (t_m / t_2))
	elif t_m <= 1.62e-187:
		tmp = 1.0
	elif t_m <= 5.6e-179:
		tmp = t_m * ((math.sqrt(2.0) / l_m) / t_2)
	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 = sqrt(Float64(2.0 / x))
	tmp = 0.0
	if (t_m <= 6.4e-223)
		tmp = Float64(sqrt(2.0) / Float64(l_m / Float64(t_m / t_2)));
	elseif (t_m <= 1.62e-187)
		tmp = 1.0;
	elseif (t_m <= 5.6e-179)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) / t_2));
	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 = sqrt((2.0 / x));
	tmp = 0.0;
	if (t_m <= 6.4e-223)
		tmp = sqrt(2.0) / (l_m / (t_m / t_2));
	elseif (t_m <= 1.62e-187)
		tmp = 1.0;
	elseif (t_m <= 5.6e-179)
		tmp = t_m * ((sqrt(2.0) / l_m) / t_2);
	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[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.4e-223], N[(N[Sqrt[2.0], $MachinePrecision] / N[(l$95$m / N[(t$95$m / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.62e-187], 1.0, If[LessEqual[t$95$m, 5.6e-179], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / t$95$2), $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 < 6.4000000000000001e-223

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. clear-num 20.1%

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

      2. inv-pow 20.1%

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

      3. associate-*l* 20.2%

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

      4. sqrt-div 20.1%

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

      5. metadata-eval 20.1%

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

      6. un-div-inv 20.1%

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

   10. Applied egg-rr 20.1%

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

       1. unpow-1 20.1%

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

       2. associate-/l* 20.1%

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

   12. Simplified 20.1%

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

       1. un-div-inv 20.1%

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

       2. clear-num 20.1%

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

       3. un-div-inv 20.1%

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

       4. sqrt-undiv 20.1%

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

   14. Applied egg-rr 20.1%

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

   if 6.4000000000000001e-223 < t < 1.6200000000000001e-187

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 1.6200000000000001e-187 < t < 5.6000000000000001e-179

   1. Initial program 0.7%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 65.5%

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

      2. sub-neg 65.5%

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

      3. metadata-eval 65.5%

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

      4. +-commutative 65.5%

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

      5. sub-neg 65.5%

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

      6. metadata-eval 65.5%

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

      7. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. clear-num 90.9%

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

      2. inv-pow 90.9%

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

      3. associate-*l* 90.9%

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

      4. sqrt-div 90.9%

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

      5. metadata-eval 90.9%

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

      6. un-div-inv 90.9%

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

   10. Applied egg-rr 90.9%

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

       1. unpow-1 90.9%

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

       2. associate-/l* 90.9%

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

   12. Simplified 90.9%

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

       1. un-div-inv 91.4%

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

       2. clear-num 91.4%

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

       3. un-div-inv 91.4%

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

       4. sqrt-undiv 91.4%

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

   14. Applied egg-rr 91.4%

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

       1. associate-/r/ 99.5%

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

       2. *-commutative 99.5%

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

       3. associate-*l/ 91.4%

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

       4. associate-/l* 99.5%

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

   16. Simplified 99.5%

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

   if 5.6000000000000001e-179 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell}{\frac{t}{\sqrt{\frac{2}{x}}}}}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 46.3%

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

6. Taylor expanded in t around 0 46.4%

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

7. Final simplification 46.4%

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

Alternative 9: 76.0% 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 37.2%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 46.3%

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

6. Taylor expanded in x around -inf 0.0%

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

8. Simplified 46.2%

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

9. Final simplification 46.2%

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

Alternative 10: 75.7% 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 37.2%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 46.3%

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

6. Taylor expanded in x around inf 46.2%

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

7. Final simplification 46.2%

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

Alternative 11: 75.0% 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 37.2%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 46.3%

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

6. Taylor expanded in x around inf 45.8%

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

7. Final simplification 45.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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)))))