Toniolo and Linder, Equation (7)

Percentage Accurate: 33.6% → 83.2%

Time: 20.1s

Alternatives: 10

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.6% 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: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{2}{x}}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{1}{l\_m} \cdot \frac{t\_m}{t\_2}\right)\\ \mathbf{elif}\;t\_m \leq 7.6 \cdot 10^{-246}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.35 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{l\_m \cdot t\_2} \cdot \left(t\_m \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_3 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_3 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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_3 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 9e-273)
      (* (sqrt 2.0) (* (/ 1.0 l_m) (/ t_m t_2)))
      (if (<= t_m 7.6e-246)
        1.0
        (if (<= t_m 2.35e-180)
          (* (/ 1.0 (* l_m t_2)) (* t_m (sqrt 2.0)))
          (if (<= t_m 1.15e-17)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_3 (/ (pow l_m 2.0) x)))
                (/ (+ t_3 (pow l_m 2.0)) x)))))
            (sqrt (/ (+ x -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) {
	double t_2 = sqrt((2.0 / x));
	double t_3 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 9e-273) {
		tmp = sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	} else if (t_m <= 7.6e-246) {
		tmp = 1.0;
	} else if (t_m <= 2.35e-180) {
		tmp = (1.0 / (l_m * t_2)) * (t_m * sqrt(2.0));
	} else if (t_m <= 1.15e-17) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_3 + (pow(l_m, 2.0) / x))) + ((t_3 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = sqrt((2.0d0 / x))
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 9d-273) then
        tmp = sqrt(2.0d0) * ((1.0d0 / l_m) * (t_m / t_2))
    else if (t_m <= 7.6d-246) then
        tmp = 1.0d0
    else if (t_m <= 2.35d-180) then
        tmp = (1.0d0 / (l_m * t_2)) * (t_m * sqrt(2.0d0))
    else if (t_m <= 1.15d-17) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_3 + ((l_m ** 2.0d0) / x))) + ((t_3 + (l_m ** 2.0d0)) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt((2.0 / x));
	double t_3 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 9e-273) {
		tmp = Math.sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	} else if (t_m <= 7.6e-246) {
		tmp = 1.0;
	} else if (t_m <= 2.35e-180) {
		tmp = (1.0 / (l_m * t_2)) * (t_m * Math.sqrt(2.0));
	} else if (t_m <= 1.15e-17) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_3 + (Math.pow(l_m, 2.0) / x))) + ((t_3 + Math.pow(l_m, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt((2.0 / x))
	t_3 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 9e-273:
		tmp = math.sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2))
	elif t_m <= 7.6e-246:
		tmp = 1.0
	elif t_m <= 2.35e-180:
		tmp = (1.0 / (l_m * t_2)) * (t_m * math.sqrt(2.0))
	elif t_m <= 1.15e-17:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_3 + (math.pow(l_m, 2.0) / x))) + ((t_3 + math.pow(l_m, 2.0)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(2.0 / x))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 9e-273)
		tmp = Float64(sqrt(2.0) * Float64(Float64(1.0 / l_m) * Float64(t_m / t_2)));
	elseif (t_m <= 7.6e-246)
		tmp = 1.0;
	elseif (t_m <= 2.35e-180)
		tmp = Float64(Float64(1.0 / Float64(l_m * t_2)) * Float64(t_m * sqrt(2.0)));
	elseif (t_m <= 1.15e-17)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_3 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_3 + (l_m ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt((2.0 / x));
	t_3 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 9e-273)
		tmp = sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	elseif (t_m <= 7.6e-246)
		tmp = 1.0;
	elseif (t_m <= 2.35e-180)
		tmp = (1.0 / (l_m * t_2)) * (t_m * sqrt(2.0));
	elseif (t_m <= 1.15e-17)
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_3 + ((l_m ^ 2.0) / x))) + ((t_3 + (l_m ^ 2.0)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-273], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.6e-246], 1.0, If[LessEqual[t$95$m, 2.35e-180], N[(N[(1.0 / N[(l$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.15e-17], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if t < 8.99999999999999921e-273

   1. Initial program 30.6%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in l around inf 2.9%

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

      1. associate--l+ 8.8%

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

      2. sub-neg 8.8%

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

      3. metadata-eval 8.8%

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

      4. +-commutative 8.8%

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

      5. sub-neg 8.8%

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

      6. metadata-eval 8.8%

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

      7. +-commutative 8.8%

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

   7. Simplified 8.8%

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

   8. Taylor expanded in x around inf 14.7%

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

      1. add-cbrt-cube 9.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}}} \]

      2. pow1/3 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\color{blue}{\left({\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-*l* 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]

      5. sqrt-div 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. metadata-eval 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. un-div-inv 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 9.8%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 9.8%

          \[\leadsto \sqrt{2} \cdot \frac{\color{blue}{1 \cdot t}}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}} \]

       2. unpow1/3 9.9%

          \[\leadsto \sqrt{2} \cdot \frac{1 \cdot t}{\color{blue}{\sqrt[3]{{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}}}} \]

       3. rem-cbrt-cube 14.6%

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

       4. times-frac 14.6%

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

       5. sqrt-undiv 14.6%

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

   12. Applied egg-rr 14.6%

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

   if 8.99999999999999921e-273 < t < 7.59999999999999951e-246

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.5%

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

      1. associate-*l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 7.59999999999999951e-246 < t < 2.34999999999999988e-180

   1. Initial program 1.7%

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

   3. Simplified 1.7%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. associate--l+ 30.5%

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

      2. sub-neg 30.5%

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

      3. metadata-eval 30.5%

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

      4. +-commutative 30.5%

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

      5. sub-neg 30.5%

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

      6. metadata-eval 30.5%

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

      7. +-commutative 30.5%

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

   7. Simplified 30.5%

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

   8. Taylor expanded in x around inf 56.6%

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

      1. add-cbrt-cube 56.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}}} \]

      2. pow1/3 53.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 53.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\color{blue}{\left({\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-*l* 53.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]

      5. sqrt-div 53.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. metadata-eval 53.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. un-div-inv 53.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 53.1%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 56.8%

          \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}}}} \]

       2. rem-cbrt-cube 56.7%

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

       3. associate-*r/ 56.7%

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

       4. clear-num 56.6%

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

       5. sqrt-undiv 56.7%

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

   12. Applied egg-rr 56.7%

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

       1. associate-/r/ 56.9%

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

       2. *-commutative 56.9%

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

   14. Simplified 56.9%

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

   if 2.34999999999999988e-180 < t < 1.15000000000000004e-17

   1. Initial program 52.5%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in x around inf 84.1%

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

   if 1.15000000000000004e-17 < t

   1. Initial program 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in l around 0 93.2%

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

      1. associate-*l* 93.2%

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

      2. +-commutative 93.2%

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

      3. sub-neg 93.2%

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

      4. metadata-eval 93.2%

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

      5. +-commutative 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in t around 0 93.5%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{1}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-246}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \left(t \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{2}{x}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{1}{l\_m} \cdot \frac{t\_m}{t\_2}\right)\\ \mathbf{elif}\;t\_m \leq 7.6 \cdot 10^{-246}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{l\_m \cdot t\_2} \cdot \left(t\_m \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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 9e-273)
      (* (sqrt 2.0) (* (/ 1.0 l_m) (/ t_m t_2)))
      (if (<= t_m 7.6e-246)
        1.0
        (if (<= t_m 2.8e-171)
          (* (/ 1.0 (* l_m t_2)) (* t_m (sqrt 2.0)))
          (sqrt (/ (+ x -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) {
	double t_2 = sqrt((2.0 / x));
	double tmp;
	if (t_m <= 9e-273) {
		tmp = sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	} else if (t_m <= 7.6e-246) {
		tmp = 1.0;
	} else if (t_m <= 2.8e-171) {
		tmp = (1.0 / (l_m * t_2)) * (t_m * sqrt(2.0));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt((2.0d0 / x))
    if (t_m <= 9d-273) then
        tmp = sqrt(2.0d0) * ((1.0d0 / l_m) * (t_m / t_2))
    else if (t_m <= 7.6d-246) then
        tmp = 1.0d0
    else if (t_m <= 2.8d-171) then
        tmp = (1.0d0 / (l_m * t_2)) * (t_m * sqrt(2.0d0))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt((2.0 / x));
	double tmp;
	if (t_m <= 9e-273) {
		tmp = Math.sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	} else if (t_m <= 7.6e-246) {
		tmp = 1.0;
	} else if (t_m <= 2.8e-171) {
		tmp = (1.0 / (l_m * t_2)) * (t_m * Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt((2.0 / x))
	tmp = 0
	if t_m <= 9e-273:
		tmp = math.sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2))
	elif t_m <= 7.6e-246:
		tmp = 1.0
	elif t_m <= 2.8e-171:
		tmp = (1.0 / (l_m * t_2)) * (t_m * math.sqrt(2.0))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(2.0 / x))
	tmp = 0.0
	if (t_m <= 9e-273)
		tmp = Float64(sqrt(2.0) * Float64(Float64(1.0 / l_m) * Float64(t_m / t_2)));
	elseif (t_m <= 7.6e-246)
		tmp = 1.0;
	elseif (t_m <= 2.8e-171)
		tmp = Float64(Float64(1.0 / Float64(l_m * t_2)) * Float64(t_m * sqrt(2.0)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt((2.0 / x));
	tmp = 0.0;
	if (t_m <= 9e-273)
		tmp = sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	elseif (t_m <= 7.6e-246)
		tmp = 1.0;
	elseif (t_m <= 2.8e-171)
		tmp = (1.0 / (l_m * t_2)) * (t_m * sqrt(2.0));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-273], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.6e-246], 1.0, If[LessEqual[t$95$m, 2.8e-171], N[(N[(1.0 / N[(l$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 8.99999999999999921e-273

   1. Initial program 30.6%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in l around inf 2.9%

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

      1. associate--l+ 8.8%

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

      2. sub-neg 8.8%

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

      3. metadata-eval 8.8%

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

      4. +-commutative 8.8%

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

      5. sub-neg 8.8%

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

      6. metadata-eval 8.8%

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

      7. +-commutative 8.8%

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

   7. Simplified 8.8%

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

   8. Taylor expanded in x around inf 14.7%

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

      1. add-cbrt-cube 9.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}}} \]

      2. pow1/3 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\color{blue}{\left({\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-*l* 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]

      5. sqrt-div 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. metadata-eval 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. un-div-inv 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 9.8%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 9.8%

          \[\leadsto \sqrt{2} \cdot \frac{\color{blue}{1 \cdot t}}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}} \]

       2. unpow1/3 9.9%

          \[\leadsto \sqrt{2} \cdot \frac{1 \cdot t}{\color{blue}{\sqrt[3]{{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}}}} \]

       3. rem-cbrt-cube 14.6%

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

       4. times-frac 14.6%

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

       5. sqrt-undiv 14.6%

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

   12. Applied egg-rr 14.6%

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

   if 8.99999999999999921e-273 < t < 7.59999999999999951e-246

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.5%

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

      1. associate-*l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 7.59999999999999951e-246 < t < 2.80000000000000023e-171

   1. Initial program 1.7%

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

   3. Simplified 1.7%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. associate--l+ 28.0%

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

      2. sub-neg 28.0%

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

      3. metadata-eval 28.0%

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

      4. +-commutative 28.0%

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

      5. sub-neg 28.0%

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

      6. metadata-eval 28.0%

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

      7. +-commutative 28.0%

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

   7. Simplified 28.0%

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

   8. Taylor expanded in x around inf 51.5%

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

      1. add-cbrt-cube 51.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}}} \]

      2. pow1/3 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\color{blue}{\left({\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-*l* 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]

      5. sqrt-div 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. metadata-eval 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. un-div-inv 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 47.8%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 51.6%

          \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}}}} \]

       2. rem-cbrt-cube 51.6%

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

       3. associate-*r/ 51.6%

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

       4. clear-num 51.5%

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

       5. sqrt-undiv 51.6%

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

   12. Applied egg-rr 51.6%

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

       1. associate-/r/ 51.7%

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

       2. *-commutative 51.7%

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

   14. Simplified 51.7%

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

   if 2.80000000000000023e-171 < t

   1. Initial program 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in l around 0 88.3%

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

      1. associate-*l* 88.3%

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

      2. +-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. metadata-eval 88.3%

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

      5. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in t around 0 88.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{1}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-246}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{\ell \cdot \sqrt{\frac{2}{x}}} \cdot \left(t \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot \left(\frac{t\_m}{l\_m} \cdot \sqrt{\frac{x}{2}}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-273}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{-170}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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) (* (/ t_m l_m) (sqrt (/ x 2.0))))))
   (*
    t_s
    (if (<= t_m 9e-273)
      t_2
      (if (<= t_m 1.3e-245)
        1.0
        (if (<= t_m 2.8e-170) t_2 (sqrt (/ (+ x -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) {
	double t_2 = sqrt(2.0) * ((t_m / l_m) * sqrt((x / 2.0)));
	double tmp;
	if (t_m <= 9e-273) {
		tmp = t_2;
	} else if (t_m <= 1.3e-245) {
		tmp = 1.0;
	} else if (t_m <= 2.8e-170) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * ((t_m / l_m) * sqrt((x / 2.0d0)))
    if (t_m <= 9d-273) then
        tmp = t_2
    else if (t_m <= 1.3d-245) then
        tmp = 1.0d0
    else if (t_m <= 2.8d-170) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) * ((t_m / l_m) * Math.sqrt((x / 2.0)));
	double tmp;
	if (t_m <= 9e-273) {
		tmp = t_2;
	} else if (t_m <= 1.3e-245) {
		tmp = 1.0;
	} else if (t_m <= 2.8e-170) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) * ((t_m / l_m) * math.sqrt((x / 2.0)))
	tmp = 0
	if t_m <= 9e-273:
		tmp = t_2
	elif t_m <= 1.3e-245:
		tmp = 1.0
	elif t_m <= 2.8e-170:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * Float64(Float64(t_m / l_m) * sqrt(Float64(x / 2.0))))
	tmp = 0.0
	if (t_m <= 9e-273)
		tmp = t_2;
	elseif (t_m <= 1.3e-245)
		tmp = 1.0;
	elseif (t_m <= 2.8e-170)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) * ((t_m / l_m) * sqrt((x / 2.0)));
	tmp = 0.0;
	if (t_m <= 9e-273)
		tmp = t_2;
	elseif (t_m <= 1.3e-245)
		tmp = 1.0;
	elseif (t_m <= 2.8e-170)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-273], t$95$2, If[LessEqual[t$95$m, 1.3e-245], 1.0, If[LessEqual[t$95$m, 2.8e-170], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.99999999999999921e-273 or 1.30000000000000003e-245 < t < 2.79999999999999995e-170

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 10.1%

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

      2. sub-neg 10.1%

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

      3. metadata-eval 10.1%

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

      4. +-commutative 10.1%

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

      5. sub-neg 10.1%

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

      6. metadata-eval 10.1%

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

      7. +-commutative 10.1%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in x around inf 17.1%

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

      1. add-cbrt-cube 12.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}}} \]

      2. pow1/3 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\color{blue}{\left({\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-*l* 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]

      5. sqrt-div 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. metadata-eval 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. un-div-inv 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 12.4%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 12.4%

          \[\leadsto \sqrt{2} \cdot \frac{\color{blue}{1 \cdot t}}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}} \]

       2. unpow1/3 12.7%

          \[\leadsto \sqrt{2} \cdot \frac{1 \cdot t}{\color{blue}{\sqrt[3]{{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}}}} \]

       3. rem-cbrt-cube 17.1%

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

       4. *-commutative 17.1%

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

       5. times-frac 15.1%

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

       6. clear-num 15.1%

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

       7. sqrt-undiv 15.1%

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

   12. Applied egg-rr 15.1%

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

       1. *-commutative 15.1%

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

   14. Simplified 15.1%

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

   if 8.99999999999999921e-273 < t < 1.30000000000000003e-245

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.5%

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

      1. associate-*l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 2.79999999999999995e-170 < t

   1. Initial program 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in l around 0 88.3%

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

      1. associate-*l* 88.3%

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

      2. +-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. metadata-eval 88.3%

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

      5. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in t around 0 88.6%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{\ell} \cdot \sqrt{\frac{x}{2}}\right)\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{\ell} \cdot \sqrt{\frac{x}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\sqrt{2}}{l\_m \cdot \sqrt{\frac{2}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-273}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{-170}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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 8.5e-273)
      t_2
      (if (<= t_m 2e-245)
        1.0
        (if (<= t_m 5.6e-170) t_2 (sqrt (/ (+ x -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) {
	double t_2 = t_m * (sqrt(2.0) / (l_m * sqrt((2.0 / x))));
	double tmp;
	if (t_m <= 8.5e-273) {
		tmp = t_2;
	} else if (t_m <= 2e-245) {
		tmp = 1.0;
	} else if (t_m <= 5.6e-170) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (sqrt(2.0d0) / (l_m * sqrt((2.0d0 / x))))
    if (t_m <= 8.5d-273) then
        tmp = t_2
    else if (t_m <= 2d-245) then
        tmp = 1.0d0
    else if (t_m <= 5.6d-170) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * (Math.sqrt(2.0) / (l_m * Math.sqrt((2.0 / x))));
	double tmp;
	if (t_m <= 8.5e-273) {
		tmp = t_2;
	} else if (t_m <= 2e-245) {
		tmp = 1.0;
	} else if (t_m <= 5.6e-170) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * (math.sqrt(2.0) / (l_m * math.sqrt((2.0 / x))))
	tmp = 0
	if t_m <= 8.5e-273:
		tmp = t_2
	elif t_m <= 2e-245:
		tmp = 1.0
	elif t_m <= 5.6e-170:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * Float64(sqrt(2.0) / Float64(l_m * sqrt(Float64(2.0 / x)))))
	tmp = 0.0
	if (t_m <= 8.5e-273)
		tmp = t_2;
	elseif (t_m <= 2e-245)
		tmp = 1.0;
	elseif (t_m <= 5.6e-170)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * (sqrt(2.0) / (l_m * sqrt((2.0 / x))));
	tmp = 0.0;
	if (t_m <= 8.5e-273)
		tmp = t_2;
	elseif (t_m <= 2e-245)
		tmp = 1.0;
	elseif (t_m <= 5.6e-170)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.5e-273], t$95$2, If[LessEqual[t$95$m, 2e-245], 1.0, If[LessEqual[t$95$m, 5.6e-170], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.5000000000000008e-273 or 1.9999999999999999e-245 < t < 5.59999999999999991e-170

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 10.1%

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

      2. sub-neg 10.1%

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

      3. metadata-eval 10.1%

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

      4. +-commutative 10.1%

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

      5. sub-neg 10.1%

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

      6. metadata-eval 10.1%

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

      7. +-commutative 10.1%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in x around inf 17.1%

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

      1. add-cbrt-cube 12.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}}} \]

      2. pow1/3 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\color{blue}{\left({\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-*l* 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]

      5. sqrt-div 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. metadata-eval 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. un-div-inv 12.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 12.4%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 12.7%

          \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}}}} \]

       2. rem-cbrt-cube 17.1%

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

       3. clear-num 16.7%

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

       4. un-div-inv 16.7%

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

       5. sqrt-undiv 16.7%

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

   12. Applied egg-rr 16.7%

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

       1. associate-/r/ 17.1%

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

   14. Simplified 17.1%

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

   if 8.5000000000000008e-273 < t < 1.9999999999999999e-245

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.5%

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

      1. associate-*l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 5.59999999999999991e-170 < t

   1. Initial program 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in l around 0 88.3%

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

      1. associate-*l* 88.3%

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

      2. +-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. metadata-eval 88.3%

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

      5. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in t around 0 88.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-273}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-170}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{2}{x}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{1}{l\_m} \cdot \frac{t\_m}{t\_2}\right)\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{-244}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{-164}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{l\_m \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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 7.2e-273)
      (* (sqrt 2.0) (* (/ 1.0 l_m) (/ t_m t_2)))
      (if (<= t_m 1.8e-244)
        1.0
        (if (<= t_m 3.6e-164)
          (* t_m (/ (sqrt 2.0) (* l_m t_2)))
          (sqrt (/ (+ x -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) {
	double t_2 = sqrt((2.0 / x));
	double tmp;
	if (t_m <= 7.2e-273) {
		tmp = sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	} else if (t_m <= 1.8e-244) {
		tmp = 1.0;
	} else if (t_m <= 3.6e-164) {
		tmp = t_m * (sqrt(2.0) / (l_m * t_2));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt((2.0d0 / x))
    if (t_m <= 7.2d-273) then
        tmp = sqrt(2.0d0) * ((1.0d0 / l_m) * (t_m / t_2))
    else if (t_m <= 1.8d-244) then
        tmp = 1.0d0
    else if (t_m <= 3.6d-164) then
        tmp = t_m * (sqrt(2.0d0) / (l_m * t_2))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt((2.0 / x));
	double tmp;
	if (t_m <= 7.2e-273) {
		tmp = Math.sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	} else if (t_m <= 1.8e-244) {
		tmp = 1.0;
	} else if (t_m <= 3.6e-164) {
		tmp = t_m * (Math.sqrt(2.0) / (l_m * t_2));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt((2.0 / x))
	tmp = 0
	if t_m <= 7.2e-273:
		tmp = math.sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2))
	elif t_m <= 1.8e-244:
		tmp = 1.0
	elif t_m <= 3.6e-164:
		tmp = t_m * (math.sqrt(2.0) / (l_m * t_2))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = sqrt(Float64(2.0 / x))
	tmp = 0.0
	if (t_m <= 7.2e-273)
		tmp = Float64(sqrt(2.0) * Float64(Float64(1.0 / l_m) * Float64(t_m / t_2)));
	elseif (t_m <= 1.8e-244)
		tmp = 1.0;
	elseif (t_m <= 3.6e-164)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(l_m * t_2)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt((2.0 / x));
	tmp = 0.0;
	if (t_m <= 7.2e-273)
		tmp = sqrt(2.0) * ((1.0 / l_m) * (t_m / t_2));
	elseif (t_m <= 1.8e-244)
		tmp = 1.0;
	elseif (t_m <= 3.6e-164)
		tmp = t_m * (sqrt(2.0) / (l_m * t_2));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-273], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e-244], 1.0, If[LessEqual[t$95$m, 3.6e-164], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 7.19999999999999986e-273

   1. Initial program 30.6%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in l around inf 2.9%

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

      1. associate--l+ 8.8%

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

      2. sub-neg 8.8%

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

      3. metadata-eval 8.8%

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

      4. +-commutative 8.8%

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

      5. sub-neg 8.8%

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

      6. metadata-eval 8.8%

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

      7. +-commutative 8.8%

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

   7. Simplified 8.8%

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

   8. Taylor expanded in x around inf 14.7%

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

      1. add-cbrt-cube 9.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}}} \]

      2. pow1/3 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\color{blue}{\left({\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-*l* 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]

      5. sqrt-div 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. metadata-eval 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. un-div-inv 9.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 9.8%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 9.8%

          \[\leadsto \sqrt{2} \cdot \frac{\color{blue}{1 \cdot t}}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}} \]

       2. unpow1/3 9.9%

          \[\leadsto \sqrt{2} \cdot \frac{1 \cdot t}{\color{blue}{\sqrt[3]{{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}}}} \]

       3. rem-cbrt-cube 14.6%

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

       4. times-frac 14.6%

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

       5. sqrt-undiv 14.6%

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

   12. Applied egg-rr 14.6%

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

   if 7.19999999999999986e-273 < t < 1.79999999999999987e-244

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.5%

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

      1. associate-*l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 1.79999999999999987e-244 < t < 3.59999999999999994e-164

   1. Initial program 1.7%

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

   3. Simplified 1.7%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. associate--l+ 28.0%

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

      2. sub-neg 28.0%

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

      3. metadata-eval 28.0%

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

      4. +-commutative 28.0%

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

      5. sub-neg 28.0%

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

      6. metadata-eval 28.0%

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

      7. +-commutative 28.0%

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

   7. Simplified 28.0%

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

   8. Taylor expanded in x around inf 51.5%

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

      1. add-cbrt-cube 51.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}}} \]

      2. pow1/3 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\left(\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right) \cdot \left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\color{blue}{\left({\left(\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. associate-*l* 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]

      5. sqrt-div 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. metadata-eval 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. un-div-inv 47.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}\right)}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 47.8%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left({\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 51.6%

          \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt[3]{{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{x}}\right)}^{3}}}} \]

       2. rem-cbrt-cube 51.6%

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

       3. clear-num 51.7%

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

       4. un-div-inv 51.7%

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

       5. sqrt-undiv 51.6%

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

   12. Applied egg-rr 51.6%

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

       1. associate-/r/ 51.7%

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

   14. Simplified 51.7%

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

   if 3.59999999999999994e-164 < t

   1. Initial program 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in l around 0 88.3%

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

      1. associate-*l* 88.3%

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

      2. +-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. metadata-eval 88.3%

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

      5. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in t around 0 88.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{1}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-244}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{l\_m} \cdot \sqrt{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-273}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 2.85 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{-171}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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 l_m) (sqrt x))))
   (*
    t_s
    (if (<= t_m 8.5e-273)
      t_2
      (if (<= t_m 2.85e-245)
        1.0
        (if (<= t_m 2.8e-171) t_2 (sqrt (/ (+ x -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) {
	double t_2 = (t_m / l_m) * sqrt(x);
	double tmp;
	if (t_m <= 8.5e-273) {
		tmp = t_2;
	} else if (t_m <= 2.85e-245) {
		tmp = 1.0;
	} else if (t_m <= 2.8e-171) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l_m) * sqrt(x)
    if (t_m <= 8.5d-273) then
        tmp = t_2
    else if (t_m <= 2.85d-245) then
        tmp = 1.0d0
    else if (t_m <= 2.8d-171) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / l_m) * Math.sqrt(x);
	double tmp;
	if (t_m <= 8.5e-273) {
		tmp = t_2;
	} else if (t_m <= 2.85e-245) {
		tmp = 1.0;
	} else if (t_m <= 2.8e-171) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m / l_m) * math.sqrt(x)
	tmp = 0
	if t_m <= 8.5e-273:
		tmp = t_2
	elif t_m <= 2.85e-245:
		tmp = 1.0
	elif t_m <= 2.8e-171:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m / l_m) * sqrt(x))
	tmp = 0.0
	if (t_m <= 8.5e-273)
		tmp = t_2;
	elseif (t_m <= 2.85e-245)
		tmp = 1.0;
	elseif (t_m <= 2.8e-171)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m / l_m) * sqrt(x);
	tmp = 0.0;
	if (t_m <= 8.5e-273)
		tmp = t_2;
	elseif (t_m <= 2.85e-245)
		tmp = 1.0;
	elseif (t_m <= 2.8e-171)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.5e-273], t$95$2, If[LessEqual[t$95$m, 2.85e-245], 1.0, If[LessEqual[t$95$m, 2.8e-171], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.5000000000000008e-273 or 2.85e-245 < t < 2.80000000000000023e-171

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 10.1%

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

      2. sub-neg 10.1%

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

      3. metadata-eval 10.1%

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

      4. +-commutative 10.1%

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

      5. sub-neg 10.1%

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

      6. metadata-eval 10.1%

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

      7. +-commutative 10.1%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in x around inf 17.1%

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

   9. Taylor expanded in t around 0 15.1%

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

   if 8.5000000000000008e-273 < t < 2.85e-245

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.5%

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

      1. associate-*l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 2.80000000000000023e-171 < t

   1. Initial program 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in l around 0 88.3%

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

      1. associate-*l* 88.3%

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

      2. +-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. metadata-eval 88.3%

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

      5. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in t around 0 88.6%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-273}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{l\_m} \cdot \sqrt{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-273}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \end{array} \end{array} \]

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 l_m) (sqrt x))))
   (*
    t_s
    (if (<= t_m 9e-273)
      t_2
      (if (<= t_m 5.6e-245)
        1.0
        (if (<= t_m 2.7e-171) t_2 (+ 1.0 (/ (+ -1.0 (/ 0.5 x)) x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / l_m) * sqrt(x);
	double tmp;
	if (t_m <= 9e-273) {
		tmp = t_2;
	} else if (t_m <= 5.6e-245) {
		tmp = 1.0;
	} else if (t_m <= 2.7e-171) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l_m) * sqrt(x)
    if (t_m <= 9d-273) then
        tmp = t_2
    else if (t_m <= 5.6d-245) then
        tmp = 1.0d0
    else if (t_m <= 2.7d-171) then
        tmp = t_2
    else
        tmp = 1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / l_m) * Math.sqrt(x);
	double tmp;
	if (t_m <= 9e-273) {
		tmp = t_2;
	} else if (t_m <= 5.6e-245) {
		tmp = 1.0;
	} else if (t_m <= 2.7e-171) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m / l_m) * math.sqrt(x)
	tmp = 0
	if t_m <= 9e-273:
		tmp = t_2
	elif t_m <= 5.6e-245:
		tmp = 1.0
	elif t_m <= 2.7e-171:
		tmp = t_2
	else:
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m / l_m) * sqrt(x))
	tmp = 0.0
	if (t_m <= 9e-273)
		tmp = t_2;
	elseif (t_m <= 5.6e-245)
		tmp = 1.0;
	elseif (t_m <= 2.7e-171)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m / l_m) * sqrt(x);
	tmp = 0.0;
	if (t_m <= 9e-273)
		tmp = t_2;
	elseif (t_m <= 5.6e-245)
		tmp = 1.0;
	elseif (t_m <= 2.7e-171)
		tmp = t_2;
	else
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-273], t$95$2, If[LessEqual[t$95$m, 5.6e-245], 1.0, If[LessEqual[t$95$m, 2.7e-171], t$95$2, N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.99999999999999921e-273 or 5.6000000000000003e-245 < t < 2.70000000000000014e-171

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 10.1%

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

      2. sub-neg 10.1%

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

      3. metadata-eval 10.1%

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

      4. +-commutative 10.1%

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

      5. sub-neg 10.1%

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

      6. metadata-eval 10.1%

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

      7. +-commutative 10.1%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in x around inf 17.1%

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

   9. Taylor expanded in t around 0 15.1%

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

   if 8.99999999999999921e-273 < t < 5.6000000000000003e-245

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 99.5%

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

      1. associate-*l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 2.70000000000000014e-171 < t

   1. Initial program 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in l around 0 88.3%

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

      1. associate-*l* 88.3%

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

      2. +-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. metadata-eval 88.3%

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

      5. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in t around 0 88.6%

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

   9. Taylor expanded in x around inf 87.8%

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

   11. Simplified 87.8%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-273}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-245}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 + (0.5 / x)) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + ((-1.0 + (0.5 / x)) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + ((-1.0 + (0.5 / x)) / x));
end

Wolfram

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

Derivation

1. Initial program 34.2%

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

3. Simplified 34.1%

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

5. Taylor expanded in l around 0 39.0%

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

   1. associate-*l* 39.0%

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

   2. +-commutative 39.0%

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

   3. sub-neg 39.0%

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

   4. metadata-eval 39.0%

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

   5. +-commutative 39.0%

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

7. Simplified 39.0%

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

8. Taylor expanded in t around 0 39.1%

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

9. Taylor expanded in x around inf 38.8%

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

11. Simplified 38.8%

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

12. Final simplification 38.8%

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

Alternative 9: 76.3% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.2%

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

3. Simplified 34.1%

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

5. Taylor expanded in l around 0 39.0%

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

   1. associate-*l* 39.0%

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

   2. +-commutative 39.0%

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

   3. sub-neg 39.0%

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

   4. metadata-eval 39.0%

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

   5. +-commutative 39.0%

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

7. Simplified 39.0%

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

8. Taylor expanded in x around inf 38.7%

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

9. Final simplification 38.7%

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

Alternative 10: 75.6% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.2%

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

3. Simplified 34.1%

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

5. Taylor expanded in l around 0 39.0%

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

   1. associate-*l* 39.0%

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

   2. +-commutative 39.0%

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

   3. sub-neg 39.0%

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

   4. metadata-eval 39.0%

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

   5. +-commutative 39.0%

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

7. Simplified 39.0%

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

8. Taylor expanded in x around inf 38.4%

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

9. Final simplification 38.4%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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)))))