Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 85.6%

Time: 20.5s

Alternatives: 9

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {l\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{l\_m}}{{x}^{-0.5}}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_3 + t\_3}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sqrt{t\_2}}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))) (t_3 (+ t_2 (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 3.5e-221)
      (* (/ t_m (sqrt 2.0)) (/ (/ (sqrt 2.0) l_m) (pow x -0.5)))
      (if (<= t_m 1.6e-162)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x))))
           (* t_m (sqrt 2.0)))))
        (if (<= t_m 6e+92)
          (/
           (sqrt t_2)
           (sqrt
            (+
             (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
             (/ t_3 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 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 3.5e-221) {
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / pow(x, -0.5));
	} else if (t_m <= 1.6e-162) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 6e+92) {
		tmp = sqrt(t_2) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + (t_3 / x)));
	} else {
		tmp = 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 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    if (t_m <= 3.5d-221) then
        tmp = (t_m / sqrt(2.0d0)) * ((sqrt(2.0d0) / l_m) / (x ** (-0.5d0)))
    else if (t_m <= 1.6d-162) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * ((t_3 + t_3) / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 6d+92) then
        tmp = sqrt(t_2) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + (t_3 / x)))
    else
        tmp = 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 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 3.5e-221) {
		tmp = (t_m / Math.sqrt(2.0)) * ((Math.sqrt(2.0) / l_m) / Math.pow(x, -0.5));
	} else if (t_m <= 1.6e-162) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 6e+92) {
		tmp = Math.sqrt(t_2) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + (t_3 / x)));
	} else {
		tmp = 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 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 3.5e-221:
		tmp = (t_m / math.sqrt(2.0)) * ((math.sqrt(2.0) / l_m) / math.pow(x, -0.5))
	elif t_m <= 1.6e-162:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 6e+92:
		tmp = math.sqrt(t_2) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + (t_3 / x)))
	else:
		tmp = 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(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.5e-221)
		tmp = Float64(Float64(t_m / sqrt(2.0)) * Float64(Float64(sqrt(2.0) / l_m) / (x ^ -0.5)));
	elseif (t_m <= 1.6e-162)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 6e+92)
		tmp = Float64(sqrt(t_2) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x))));
	else
		tmp = Float64(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 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.5e-221)
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / (x ^ -0.5));
	elseif (t_m <= 1.6e-162)
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 6e+92)
		tmp = sqrt(t_2) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + (t_3 / x)));
	else
		tmp = 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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-221], N[(N[(t$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e-162], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+92], N[(N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.4999999999999999e-221

   1. Initial program 36.2%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-prod 0.6%

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

      3. sqrt-prod 0.6%

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

      4. pow1/2 0.6%

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

      5. pow2 0.6%

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

   4. Applied egg-rr 0.6%

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

      1. unpow1/2 0.6%

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

   6. Simplified 0.6%

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

   7. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 8.2%

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

      2. sub-neg 8.2%

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

      3. metadata-eval 8.2%

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

      4. sub-neg 8.2%

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

      5. sub-neg 8.2%

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

      6. metadata-eval 8.2%

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

      7. metadata-eval 8.2%

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

   9. Simplified 8.2%

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

   10. Taylor expanded in x around inf 12.1%

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

       1. *-un-lft-identity 12.1%

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

       2. associate-/r* 11.5%

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

       3. *-commutative 11.5%

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

       4. sqrt-prod 11.5%

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

       5. sqrt-pow1 17.8%

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

       6. metadata-eval 17.8%

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

       7. pow1 17.8%

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

       8. associate-/l* 17.8%

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

       9. *-commutative 17.8%

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

       10. inv-pow 17.8%

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

       11. sqrt-pow1 17.8%

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

       12. metadata-eval 17.8%

           \[\leadsto 1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{\color{blue}{-0.5}}} \]

   12. Applied egg-rr 17.8%

       \[\leadsto \color{blue}{1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{-0.5}}} \]
   13. Step-by-step derivation

       1. *-lft-identity 17.8%

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

       2. times-frac 19.3%

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

   14. Simplified 19.3%

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

   if 3.4999999999999999e-221 < t < 1.59999999999999988e-162

   1. Initial program 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in x around inf 75.2%

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

   if 1.59999999999999988e-162 < t < 6.00000000000000026e92

   1. Initial program 58.4%

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

      1. add-sqr-sqrt 58.2%

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

      2. sqrt-prod 59.2%

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

      3. sqrt-prod 59.6%

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

      4. pow1/2 59.6%

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

      5. pow2 59.6%

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

   4. Applied egg-rr 59.6%

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

      1. unpow1/2 59.6%

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

   6. Simplified 59.6%

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

   7. Taylor expanded in x around inf 89.1%

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

   if 6.00000000000000026e92 < t

   1. Initial program 24.5%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in t around inf 93.2%

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

   6. Taylor expanded in x around inf 93.6%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{t}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sqrt{2 \cdot {t}^{2}}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {l\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{l\_m}}{{x}^{-0.5}}\\ \mathbf{elif}\;t\_m \leq 6.8 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_3 + t\_3}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))) (t_3 (+ t_2 (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 2.5e-221)
      (* (/ t_m (sqrt 2.0)) (/ (/ (sqrt 2.0) l_m) (pow x -0.5)))
      (if (<= t_m 6.8e-158)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x))))
           (* t_m (sqrt 2.0)))))
        (if (<= t_m 6.2e+92)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ t_3 x)))))
          (+ 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 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 2.5e-221) {
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / pow(x, -0.5));
	} else if (t_m <= 6.8e-158) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 6.2e+92) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = 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 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    if (t_m <= 2.5d-221) then
        tmp = (t_m / sqrt(2.0d0)) * ((sqrt(2.0d0) / l_m) / (x ** (-0.5d0)))
    else if (t_m <= 6.8d-158) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * ((t_3 + t_3) / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 6.2d+92) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + (t_3 / x))))
    else
        tmp = 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 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 2.5e-221) {
		tmp = (t_m / Math.sqrt(2.0)) * ((Math.sqrt(2.0) / l_m) / Math.pow(x, -0.5));
	} else if (t_m <= 6.8e-158) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 6.2e+92) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = 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 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 2.5e-221:
		tmp = (t_m / math.sqrt(2.0)) * ((math.sqrt(2.0) / l_m) / math.pow(x, -0.5))
	elif t_m <= 6.8e-158:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 6.2e+92:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + (t_3 / x))))
	else:
		tmp = 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(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.5e-221)
		tmp = Float64(Float64(t_m / sqrt(2.0)) * Float64(Float64(sqrt(2.0) / l_m) / (x ^ -0.5)));
	elseif (t_m <= 6.8e-158)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 6.2e+92)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x)))));
	else
		tmp = Float64(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 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.5e-221)
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / (x ^ -0.5));
	elseif (t_m <= 6.8e-158)
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 6.2e+92)
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + (t_3 / x))));
	else
		tmp = 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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.5e-221], N[(N[(t$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.8e-158], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+92], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.49999999999999998e-221

   1. Initial program 36.2%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-prod 0.6%

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

      3. sqrt-prod 0.6%

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

      4. pow1/2 0.6%

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

      5. pow2 0.6%

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

   4. Applied egg-rr 0.6%

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

      1. unpow1/2 0.6%

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

   6. Simplified 0.6%

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

   7. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 8.2%

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

      2. sub-neg 8.2%

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

      3. metadata-eval 8.2%

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

      4. sub-neg 8.2%

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

      5. sub-neg 8.2%

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

      6. metadata-eval 8.2%

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

      7. metadata-eval 8.2%

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

   9. Simplified 8.2%

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

   10. Taylor expanded in x around inf 12.1%

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

       1. *-un-lft-identity 12.1%

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

       2. associate-/r* 11.5%

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

       3. *-commutative 11.5%

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

       4. sqrt-prod 11.5%

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

       5. sqrt-pow1 17.8%

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

       6. metadata-eval 17.8%

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

       7. pow1 17.8%

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

       8. associate-/l* 17.8%

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

       9. *-commutative 17.8%

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

       10. inv-pow 17.8%

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

       11. sqrt-pow1 17.8%

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

       12. metadata-eval 17.8%

           \[\leadsto 1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{\color{blue}{-0.5}}} \]

   12. Applied egg-rr 17.8%

       \[\leadsto \color{blue}{1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{-0.5}}} \]
   13. Step-by-step derivation

       1. *-lft-identity 17.8%

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

       2. times-frac 19.3%

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

   14. Simplified 19.3%

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

   if 2.49999999999999998e-221 < t < 6.7999999999999999e-158

   1. Initial program 9.8%

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

   3. Simplified 9.8%

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

   5. Taylor expanded in x around inf 76.9%

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

   if 6.7999999999999999e-158 < t < 6.2000000000000004e92

   1. Initial program 58.5%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in x around inf 88.4%

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

   if 6.2000000000000004e92 < t

   1. Initial program 24.5%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in t around inf 93.2%

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

   6. Taylor expanded in x around inf 93.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{t}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-221}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{l\_m}}{{x}^{-0.5}}\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_2}{t\_2 - -0.5 \cdot \frac{{l\_m}^{2} - {t\_m}^{2} \cdot -2}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_3 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_3 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

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))) (t_3 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 3e-221)
      (* (/ t_m (sqrt 2.0)) (/ (/ (sqrt 2.0) l_m) (pow x -0.5)))
      (if (<= t_m 4.2e-159)
        (/
         t_2
         (-
          t_2
          (*
           -0.5
           (/
            (- (pow l_m 2.0) (* (pow t_m 2.0) -2.0))
            (* t_m (* (sqrt 2.0) x))))))
        (if (<= t_m 6.5e+92)
          (*
           (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)))))
          (+ 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);
	double t_3 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 3e-221) {
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / pow(x, -0.5));
	} else if (t_m <= 4.2e-159) {
		tmp = t_2 / (t_2 - (-0.5 * ((pow(l_m, 2.0) - (pow(t_m, 2.0) * -2.0)) / (t_m * (sqrt(2.0) * x)))));
	} else if (t_m <= 6.5e+92) {
		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 = 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 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 3d-221) then
        tmp = (t_m / sqrt(2.0d0)) * ((sqrt(2.0d0) / l_m) / (x ** (-0.5d0)))
    else if (t_m <= 4.2d-159) then
        tmp = t_2 / (t_2 - ((-0.5d0) * (((l_m ** 2.0d0) - ((t_m ** 2.0d0) * (-2.0d0))) / (t_m * (sqrt(2.0d0) * x)))))
    else if (t_m <= 6.5d+92) 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 = 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);
	double t_3 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 3e-221) {
		tmp = (t_m / Math.sqrt(2.0)) * ((Math.sqrt(2.0) / l_m) / Math.pow(x, -0.5));
	} else if (t_m <= 4.2e-159) {
		tmp = t_2 / (t_2 - (-0.5 * ((Math.pow(l_m, 2.0) - (Math.pow(t_m, 2.0) * -2.0)) / (t_m * (Math.sqrt(2.0) * x)))));
	} else if (t_m <= 6.5e+92) {
		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 = 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)
	t_3 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 3e-221:
		tmp = (t_m / math.sqrt(2.0)) * ((math.sqrt(2.0) / l_m) / math.pow(x, -0.5))
	elif t_m <= 4.2e-159:
		tmp = t_2 / (t_2 - (-0.5 * ((math.pow(l_m, 2.0) - (math.pow(t_m, 2.0) * -2.0)) / (t_m * (math.sqrt(2.0) * x)))))
	elif t_m <= 6.5e+92:
		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 = 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 * sqrt(2.0))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 3e-221)
		tmp = Float64(Float64(t_m / sqrt(2.0)) * Float64(Float64(sqrt(2.0) / l_m) / (x ^ -0.5)));
	elseif (t_m <= 4.2e-159)
		tmp = Float64(t_2 / Float64(t_2 - Float64(-0.5 * Float64(Float64((l_m ^ 2.0) - Float64((t_m ^ 2.0) * -2.0)) / Float64(t_m * Float64(sqrt(2.0) * x))))));
	elseif (t_m <= 6.5e+92)
		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 = Float64(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);
	t_3 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3e-221)
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / (x ^ -0.5));
	elseif (t_m <= 4.2e-159)
		tmp = t_2 / (t_2 - (-0.5 * (((l_m ^ 2.0) - ((t_m ^ 2.0) * -2.0)) / (t_m * (sqrt(2.0) * x)))));
	elseif (t_m <= 6.5e+92)
		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 = 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[Sqrt[2.0], $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, 3e-221], N[(N[(t$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e-159], N[(t$95$2 / N[(t$95$2 - N[(-0.5 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] - N[(N[Power[t$95$m, 2.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e+92], 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[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.0000000000000002e-221

   1. Initial program 36.2%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-prod 0.6%

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

      3. sqrt-prod 0.6%

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

      4. pow1/2 0.6%

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

      5. pow2 0.6%

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

   4. Applied egg-rr 0.6%

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

      1. unpow1/2 0.6%

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

   6. Simplified 0.6%

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

   7. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 8.2%

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

      2. sub-neg 8.2%

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

      3. metadata-eval 8.2%

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

      4. sub-neg 8.2%

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

      5. sub-neg 8.2%

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

      6. metadata-eval 8.2%

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

      7. metadata-eval 8.2%

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

   9. Simplified 8.2%

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

   10. Taylor expanded in x around inf 12.1%

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

       1. *-un-lft-identity 12.1%

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

       2. associate-/r* 11.5%

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

       3. *-commutative 11.5%

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

       4. sqrt-prod 11.5%

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

       5. sqrt-pow1 17.8%

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

       6. metadata-eval 17.8%

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

       7. pow1 17.8%

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

       8. associate-/l* 17.8%

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

       9. *-commutative 17.8%

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

       10. inv-pow 17.8%

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

       11. sqrt-pow1 17.8%

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

       12. metadata-eval 17.8%

           \[\leadsto 1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{\color{blue}{-0.5}}} \]

   12. Applied egg-rr 17.8%

       \[\leadsto \color{blue}{1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{-0.5}}} \]
   13. Step-by-step derivation

       1. *-lft-identity 17.8%

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

       2. times-frac 19.3%

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

   14. Simplified 19.3%

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

   if 3.0000000000000002e-221 < t < 4.1999999999999998e-159

   1. Initial program 9.8%

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

   3. Taylor expanded in x around inf 7.1%

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

   4. Taylor expanded in x around -inf 77.2%

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

   if 4.1999999999999998e-159 < t < 6.49999999999999999e92

   1. Initial program 58.5%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in x around inf 88.4%

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

   if 6.49999999999999999e92 < t

   1. Initial program 24.5%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in t around inf 93.2%

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

   6. Taylor expanded in x around inf 93.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-221}:\\ \;\;\;\;\frac{t}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} - -0.5 \cdot \frac{{\ell}^{2} - {t}^{2} \cdot -2}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-221}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{l\_m}}{{x}^{-0.5}}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{t\_2}{t\_2 - -0.5 \cdot \frac{{l\_m}^{2} - {t\_m}^{2} \cdot -2}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \end{array} \end{array} \]

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 (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 4.4e-221)
      (* (/ t_m (sqrt 2.0)) (/ (/ (sqrt 2.0) l_m) (pow x -0.5)))
      (if (<= t_m 4.8e+23)
        (/
         t_2
         (-
          t_2
          (*
           -0.5
           (/
            (- (pow l_m 2.0) (* (pow t_m 2.0) -2.0))
            (* t_m (* (sqrt 2.0) x))))))
        (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 4.4e-221) {
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / pow(x, -0.5));
	} else if (t_m <= 4.8e+23) {
		tmp = t_2 / (t_2 - (-0.5 * ((pow(l_m, 2.0) - (pow(t_m, 2.0) * -2.0)) / (t_m * (sqrt(2.0) * x)))));
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    if (t_m <= 4.4d-221) then
        tmp = (t_m / sqrt(2.0d0)) * ((sqrt(2.0d0) / l_m) / (x ** (-0.5d0)))
    else if (t_m <= 4.8d+23) then
        tmp = t_2 / (t_2 - ((-0.5d0) * (((l_m ** 2.0d0) - ((t_m ** 2.0d0) * (-2.0d0))) / (t_m * (sqrt(2.0d0) * x)))))
    else
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 4.4e-221) {
		tmp = (t_m / Math.sqrt(2.0)) * ((Math.sqrt(2.0) / l_m) / Math.pow(x, -0.5));
	} else if (t_m <= 4.8e+23) {
		tmp = t_2 / (t_2 - (-0.5 * ((Math.pow(l_m, 2.0) - (Math.pow(t_m, 2.0) * -2.0)) / (t_m * (Math.sqrt(2.0) * x)))));
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 4.4e-221:
		tmp = (t_m / math.sqrt(2.0)) * ((math.sqrt(2.0) / l_m) / math.pow(x, -0.5))
	elif t_m <= 4.8e+23:
		tmp = t_2 / (t_2 - (-0.5 * ((math.pow(l_m, 2.0) - (math.pow(t_m, 2.0) * -2.0)) / (t_m * (math.sqrt(2.0) * x)))))
	else:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 4.4e-221)
		tmp = Float64(Float64(t_m / sqrt(2.0)) * Float64(Float64(sqrt(2.0) / l_m) / (x ^ -0.5)));
	elseif (t_m <= 4.8e+23)
		tmp = Float64(t_2 / Float64(t_2 - Float64(-0.5 * Float64(Float64((l_m ^ 2.0) - Float64((t_m ^ 2.0) * -2.0)) / Float64(t_m * Float64(sqrt(2.0) * x))))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 4.4e-221)
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / (x ^ -0.5));
	elseif (t_m <= 4.8e+23)
		tmp = t_2 / (t_2 - (-0.5 * (((l_m ^ 2.0) - ((t_m ^ 2.0) * -2.0)) / (t_m * (sqrt(2.0) * x)))));
	else
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.4e-221], N[(N[(t$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e+23], N[(t$95$2 / N[(t$95$2 - N[(-0.5 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] - N[(N[Power[t$95$m, 2.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.40000000000000003e-221

   1. Initial program 36.2%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-prod 0.6%

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

      3. sqrt-prod 0.6%

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

      4. pow1/2 0.6%

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

      5. pow2 0.6%

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

   4. Applied egg-rr 0.6%

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

      1. unpow1/2 0.6%

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

   6. Simplified 0.6%

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

   7. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 8.2%

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

      2. sub-neg 8.2%

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

      3. metadata-eval 8.2%

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

      4. sub-neg 8.2%

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

      5. sub-neg 8.2%

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

      6. metadata-eval 8.2%

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

      7. metadata-eval 8.2%

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

   9. Simplified 8.2%

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

   10. Taylor expanded in x around inf 12.1%

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

       1. *-un-lft-identity 12.1%

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

       2. associate-/r* 11.5%

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

       3. *-commutative 11.5%

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

       4. sqrt-prod 11.5%

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

       5. sqrt-pow1 17.8%

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

       6. metadata-eval 17.8%

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

       7. pow1 17.8%

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

       8. associate-/l* 17.8%

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

       9. *-commutative 17.8%

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

       10. inv-pow 17.8%

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

       11. sqrt-pow1 17.8%

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

       12. metadata-eval 17.8%

           \[\leadsto 1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{\color{blue}{-0.5}}} \]

   12. Applied egg-rr 17.8%

       \[\leadsto \color{blue}{1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{-0.5}}} \]
   13. Step-by-step derivation

       1. *-lft-identity 17.8%

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

       2. times-frac 19.3%

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

   14. Simplified 19.3%

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

   if 4.40000000000000003e-221 < t < 4.8e23

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf 43.3%

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

   4. Taylor expanded in x around -inf 82.1%

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

   if 4.8e23 < t

   1. Initial program 35.8%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in t around inf 90.8%

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

   6. Taylor expanded in t around 0 91.1%

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

   7. Taylor expanded in x around inf 91.1%

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

   8. Taylor expanded in x around -inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. unsub-neg 91.1%

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

      3. sub-neg 91.1%

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

      4. associate-*r/ 91.1%

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

      5. metadata-eval 91.1%

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

      6. distribute-neg-frac 91.1%

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

      7. metadata-eval 91.1%

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

   10. Simplified 91.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-221}:\\ \;\;\;\;\frac{t}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} - -0.5 \cdot \frac{{\ell}^{2} - {t}^{2} \cdot -2}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{+194}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{l\_m}}{{x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 3e+194)
    (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x))
    (* (/ t_m (sqrt 2.0)) (/ (/ (sqrt 2.0) l_m) (pow x -0.5))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3e+194) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else {
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / pow(x, -0.5));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 3d+194) then
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    else
        tmp = (t_m / sqrt(2.0d0)) * ((sqrt(2.0d0) / l_m) / (x ** (-0.5d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3e+194) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else {
		tmp = (t_m / Math.sqrt(2.0)) * ((Math.sqrt(2.0) / l_m) / Math.pow(x, -0.5));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 3e+194:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	else:
		tmp = (t_m / math.sqrt(2.0)) * ((math.sqrt(2.0) / l_m) / math.pow(x, -0.5))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 3e+194)
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	else
		tmp = Float64(Float64(t_m / sqrt(2.0)) * Float64(Float64(sqrt(2.0) / l_m) / (x ^ -0.5)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 3e+194)
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	else
		tmp = (t_m / sqrt(2.0)) * ((sqrt(2.0) / l_m) / (x ^ -0.5));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 3e+194], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.0000000000000003e194

   1. Initial program 40.3%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in t around inf 39.3%

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

   6. Taylor expanded in t around 0 39.4%

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

   7. Taylor expanded in x around inf 39.4%

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

   8. Taylor expanded in x around -inf 39.4%

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

      1. mul-1-neg 39.4%

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

      2. unsub-neg 39.4%

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

      3. sub-neg 39.4%

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

      4. associate-*r/ 39.4%

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

      5. metadata-eval 39.4%

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

      6. distribute-neg-frac 39.4%

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

      7. metadata-eval 39.4%

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

   10. Simplified 39.4%

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

   if 3.0000000000000003e194 < l

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 0.0%

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

      3. sqrt-prod 0.0%

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

      4. pow1/2 0.0%

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

      5. pow2 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in l around inf 7.4%

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

      1. associate--l+ 43.4%

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

      2. sub-neg 43.4%

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

      3. metadata-eval 43.4%

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

      4. sub-neg 43.4%

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

      5. sub-neg 43.4%

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

      6. metadata-eval 43.4%

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

      7. metadata-eval 43.4%

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

   9. Simplified 43.4%

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

   10. Taylor expanded in x around inf 53.7%

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

       1. *-un-lft-identity 53.7%

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

       2. associate-/r* 52.5%

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

       3. *-commutative 52.5%

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

       4. sqrt-prod 52.5%

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

       5. sqrt-pow1 81.1%

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

       6. metadata-eval 81.1%

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

       7. pow1 81.1%

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

       8. associate-/l* 81.2%

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

       9. *-commutative 81.2%

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

       10. inv-pow 81.2%

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

       11. sqrt-pow1 81.3%

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

       12. metadata-eval 81.3%

           \[\leadsto 1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{\color{blue}{-0.5}}} \]

   12. Applied egg-rr 81.3%

       \[\leadsto \color{blue}{1 \cdot \frac{t \cdot \frac{\sqrt{2}}{\ell}}{\sqrt{2} \cdot {x}^{-0.5}}} \]
   13. Step-by-step derivation

       1. *-lft-identity 81.3%

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

       2. times-frac 82.4%

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

   14. Simplified 82.4%

       \[\leadsto \color{blue}{\frac{t}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{\ell}}{{x}^{-0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+194}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{2}} \cdot \frac{\frac{\sqrt{2}}{\ell}}{{x}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 3.5e+194)
    (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x))
    (* (/ t_m l_m) (sqrt x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3.5e+194) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else {
		tmp = (t_m / l_m) * sqrt(x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 3.5d+194) then
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    else
        tmp = (t_m / l_m) * sqrt(x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3.5e+194) {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	} else {
		tmp = (t_m / l_m) * Math.sqrt(x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 3.5e+194:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	else:
		tmp = (t_m / l_m) * math.sqrt(x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 3.5e+194)
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	else
		tmp = Float64(Float64(t_m / l_m) * sqrt(x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 3.5e+194)
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	else
		tmp = (t_m / l_m) * sqrt(x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 3.5e+194], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.4999999999999997e194

   1. Initial program 40.3%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in t around inf 39.3%

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

   6. Taylor expanded in t around 0 39.4%

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

   7. Taylor expanded in x around inf 39.4%

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

   8. Taylor expanded in x around -inf 39.4%

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

      1. mul-1-neg 39.4%

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

      2. unsub-neg 39.4%

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

      3. sub-neg 39.4%

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

      4. associate-*r/ 39.4%

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

      5. metadata-eval 39.4%

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

      6. distribute-neg-frac 39.4%

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

      7. metadata-eval 39.4%

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

   10. Simplified 39.4%

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

   if 3.4999999999999997e194 < l

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 0.0%

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

      3. sqrt-prod 0.0%

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

      4. pow1/2 0.0%

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

      5. pow2 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. unpow1/2 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in l around inf 7.4%

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

      1. associate--l+ 43.4%

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

      2. sub-neg 43.4%

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

      3. metadata-eval 43.4%

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

      4. sub-neg 43.4%

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

      5. sub-neg 43.4%

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

      6. metadata-eval 43.4%

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

      7. metadata-eval 43.4%

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

   9. Simplified 43.4%

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

   10. Taylor expanded in x around inf 53.7%

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

   11. Taylor expanded in t around 0 81.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% 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 37.7%

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

3. Simplified 31.2%

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

5. Taylor expanded in t around inf 37.7%

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

6. Taylor expanded in t around 0 37.8%

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

7. Taylor expanded in x around inf 37.8%

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

8. Taylor expanded in x around -inf 37.8%

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

   1. mul-1-neg 37.8%

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

   2. unsub-neg 37.8%

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

   3. sub-neg 37.8%

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

   4. associate-*r/ 37.8%

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

   5. metadata-eval 37.8%

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

   6. distribute-neg-frac 37.8%

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

   7. metadata-eval 37.8%

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

10. Simplified 37.8%

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

11. Final simplification 37.8%

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

Alternative 8: 77.3% 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 37.7%

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

3. Simplified 31.2%

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

5. Taylor expanded in t around inf 37.7%

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

6. Taylor expanded in x around inf 37.7%

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

7. Final simplification 37.7%

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

Alternative 9: 76.5% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

3. Simplified 31.2%

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

5. Taylor expanded in t around inf 37.7%

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

6. Taylor expanded in x around inf 37.3%

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

Reproduce

herbie shell --seed 2024191 
(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)))))