Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 85.3%

Time: 19.3s

Alternatives: 7

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {l_m}^{2}\\ t_4 := t_3 + t_3\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 9 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{l_m} \cdot \left(t_m \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t_m \leq 1.45 \cdot 10^{-170}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_4}{t_m \cdot \left(x \cdot \sqrt{2}\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 350000:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{t_4}{{x}^{2}} + \left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right)\right) + \frac{t_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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_4 (+ t_3 t_3)))
   (*
    t_s
    (if (<= t_m 9e-240)
      (* (/ 1.0 l_m) (* t_m (sqrt x)))
      (if (<= t_m 1.45e-170)
        (*
         t_m
         (/
          (sqrt 2.0)
          (+ (* 0.5 (/ t_4 (* t_m (* x (sqrt 2.0))))) (* t_m (sqrt 2.0)))))
        (if (<= t_m 350000.0)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (+
               (/ t_4 (pow x 2.0))
               (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x))))
              (/ t_3 x)))))
          (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 9e-240) {
		tmp = (1.0 / l_m) * (t_m * sqrt(x));
	} else if (t_m <= 1.45e-170) {
		tmp = t_m * (sqrt(2.0) / ((0.5 * (t_4 / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 350000.0) {
		tmp = t_m * (sqrt(2.0) / sqrt((((t_4 / pow(x, 2.0)) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x)))) + (t_3 / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (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) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    t_4 = t_3 + t_3
    if (t_m <= 9d-240) then
        tmp = (1.0d0 / l_m) * (t_m * sqrt(x))
    else if (t_m <= 1.45d-170) then
        tmp = t_m * (sqrt(2.0d0) / ((0.5d0 * (t_4 / (t_m * (x * sqrt(2.0d0))))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 350000.0d0) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((t_4 / (x ** 2.0d0)) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x)))) + (t_3 / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 9e-240) {
		tmp = (1.0 / l_m) * (t_m * Math.sqrt(x));
	} else if (t_m <= 1.45e-170) {
		tmp = t_m * (Math.sqrt(2.0) / ((0.5 * (t_4 / (t_m * (x * Math.sqrt(2.0))))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 350000.0) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((t_4 / Math.pow(x, 2.0)) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x)))) + (t_3 / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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)
	t_4 = t_3 + t_3
	tmp = 0
	if t_m <= 9e-240:
		tmp = (1.0 / l_m) * (t_m * math.sqrt(x))
	elif t_m <= 1.45e-170:
		tmp = t_m * (math.sqrt(2.0) / ((0.5 * (t_4 / (t_m * (x * math.sqrt(2.0))))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 350000.0:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((t_4 / math.pow(x, 2.0)) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x)))) + (t_3 / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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))
	t_4 = Float64(t_3 + t_3)
	tmp = 0.0
	if (t_m <= 9e-240)
		tmp = Float64(Float64(1.0 / l_m) * Float64(t_m * sqrt(x)));
	elseif (t_m <= 1.45e-170)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(x * sqrt(2.0))))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 350000.0)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(t_4 / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x)))) + Float64(t_3 / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(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);
	t_4 = t_3 + t_3;
	tmp = 0.0;
	if (t_m <= 9e-240)
		tmp = (1.0 / l_m) * (t_m * sqrt(x));
	elseif (t_m <= 1.45e-170)
		tmp = t_m * (sqrt(2.0) / ((0.5 * (t_4 / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 350000.0)
		tmp = t_m * (sqrt(2.0) / sqrt((((t_4 / (x ^ 2.0)) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x)))) + (t_3 / x))));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(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]}, Block[{t$95$4 = N[(t$95$3 + t$95$3), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-240], N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e-170], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 350000.0], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(t$95$4 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 9.0000000000000003e-240

   1. Initial program 28.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in l around inf 2.7%

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

      1. *-commutative 2.7%

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

      2. associate--l+ 11.4%

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

      3. sub-neg 11.4%

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

      4. metadata-eval 11.4%

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

      5. +-commutative 11.4%

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

      6. sub-neg 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

   6. Simplified 11.4%

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

      1. *-un-lft-identity 11.4%

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

      2. associate-/r* 11.4%

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

      3. sqrt-undiv 11.4%

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

      4. +-commutative 11.4%

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

      5. sub-neg 11.4%

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

      6. +-commutative 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

   8. Applied egg-rr 11.4%

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

   9. Taylor expanded in x around inf 18.4%

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

       1. clear-num 18.4%

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

       2. inv-pow 18.4%

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

   11. Applied egg-rr 18.4%

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

       1. unpow-1 18.4%

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

       2. associate-/l/ 19.7%

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

       3. associate-/r/ 19.9%

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

       4. *-commutative 19.9%

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

   13. Simplified 19.9%

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

   if 9.0000000000000003e-240 < t < 1.45e-170

   1. Initial program 5.9%

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

   3. Simplified 5.9%

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

   4. Taylor expanded in x around inf 59.1%

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

   if 1.45e-170 < t < 3.5e5

   1. Initial program 47.8%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in x around -inf 89.6%

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

   if 3.5e5 < t

   1. Initial program 37.4%

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

   3. Simplified 37.4%

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

   4. Taylor expanded in t around inf 95.5%

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

   5. Taylor expanded in t around 0 95.7%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\ell} \cdot \left(t \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-170}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 350000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 2: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 10^{-240}:\\ \;\;\;\;\frac{1}{l_m} \cdot \left(t_m \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t_m \leq 6.8 \cdot 10^{-176}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_m \leq 760000:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_2 + {l_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 1e-240)
      (* (/ 1.0 l_m) (* t_m (sqrt x)))
      (if (<= t_m 6.8e-176)
        1.0
        (if (<= t_m 760000.0)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ (+ t_2 (pow l_m 2.0)) x)))))
          (sqrt (/ (+ x -1.0) (+ 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 tmp;
	if (t_m <= 1e-240) {
		tmp = (1.0 / l_m) * (t_m * sqrt(x));
	} else if (t_m <= 6.8e-176) {
		tmp = 1.0;
	} else if (t_m <= 760000.0) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 1d-240) then
        tmp = (1.0d0 / l_m) * (t_m * sqrt(x))
    else if (t_m <= 6.8d-176) then
        tmp = 1.0d0
    else if (t_m <= 760000.0d0) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (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 tmp;
	if (t_m <= 1e-240) {
		tmp = (1.0 / l_m) * (t_m * Math.sqrt(x));
	} else if (t_m <= 6.8e-176) {
		tmp = 1.0;
	} else if (t_m <= 760000.0) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((t_2 + Math.pow(l_m, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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)
	tmp = 0
	if t_m <= 1e-240:
		tmp = (1.0 / l_m) * (t_m * math.sqrt(x))
	elif t_m <= 6.8e-176:
		tmp = 1.0
	elif t_m <= 760000.0:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((t_2 + math.pow(l_m, 2.0)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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))
	tmp = 0.0
	if (t_m <= 1e-240)
		tmp = Float64(Float64(1.0 / l_m) * Float64(t_m * sqrt(x)));
	elseif (t_m <= 6.8e-176)
		tmp = 1.0;
	elseif (t_m <= 760000.0)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(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);
	tmp = 0.0;
	if (t_m <= 1e-240)
		tmp = (1.0 / l_m) * (t_m * sqrt(x));
	elseif (t_m <= 6.8e-176)
		tmp = 1.0;
	elseif (t_m <= 760000.0)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (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]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-240], N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.8e-176], 1.0, If[LessEqual[t$95$m, 760000.0], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 9.9999999999999997e-241

   1. Initial program 28.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in l around inf 2.7%

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

      1. *-commutative 2.7%

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

      2. associate--l+ 11.4%

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

      3. sub-neg 11.4%

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

      4. metadata-eval 11.4%

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

      5. +-commutative 11.4%

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

      6. sub-neg 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

   6. Simplified 11.4%

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

      1. *-un-lft-identity 11.4%

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

      2. associate-/r* 11.4%

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

      3. sqrt-undiv 11.4%

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

      4. +-commutative 11.4%

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

      5. sub-neg 11.4%

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

      6. +-commutative 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

   8. Applied egg-rr 11.4%

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

   9. Taylor expanded in x around inf 18.4%

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

       1. clear-num 18.4%

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

       2. inv-pow 18.4%

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

   11. Applied egg-rr 18.4%

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

       1. unpow-1 18.4%

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

       2. associate-/l/ 19.7%

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

       3. associate-/r/ 19.9%

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

       4. *-commutative 19.9%

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

   13. Simplified 19.9%

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

   if 9.9999999999999997e-241 < t < 6.7999999999999994e-176

   1. Initial program 7.4%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in l around 0 61.7%

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

      1. +-commutative 61.7%

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

      2. sub-neg 61.7%

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

      3. metadata-eval 61.7%

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

      4. +-commutative 61.7%

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

   6. Simplified 61.7%

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

   7. Taylor expanded in x around inf 61.7%

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

   if 6.7999999999999994e-176 < t < 7.6e5

   1. Initial program 43.7%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in x around inf 85.1%

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

   if 7.6e5 < t

   1. Initial program 37.4%

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

   3. Simplified 37.4%

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

   4. Taylor expanded in t around inf 95.5%

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

   5. Taylor expanded in t around 0 95.7%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-240}:\\ \;\;\;\;\frac{1}{\ell} \cdot \left(t \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-176}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 760000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 3: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {l_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.1 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{l_m} \cdot \left(t_m \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t_m \leq 1.2 \cdot 10^{-171}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_3 + t_3}{t_m \cdot \left(x \cdot \sqrt{2}\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 680000:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 3.1e-240)
      (* (/ 1.0 l_m) (* t_m (sqrt x)))
      (if (<= t_m 1.2e-171)
        (*
         t_m
         (/
          (sqrt 2.0)
          (+
           (* 0.5 (/ (+ t_3 t_3) (* t_m (* x (sqrt 2.0)))))
           (* t_m (sqrt 2.0)))))
        (if (<= t_m 680000.0)
          (*
           t_m
           (/
            (sqrt 2.0)
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ t_3 x)))))
          (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 3.1e-240) {
		tmp = (1.0 / l_m) * (t_m * sqrt(x));
	} else if (t_m <= 1.2e-171) {
		tmp = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 680000.0) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (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 <= 3.1d-240) then
        tmp = (1.0d0 / l_m) * (t_m * sqrt(x))
    else if (t_m <= 1.2d-171) then
        tmp = t_m * (sqrt(2.0d0) / ((0.5d0 * ((t_3 + t_3) / (t_m * (x * sqrt(2.0d0))))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 680000.0d0) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + (t_3 / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double tmp;
	if (t_m <= 3.1e-240) {
		tmp = (1.0 / l_m) * (t_m * Math.sqrt(x));
	} else if (t_m <= 1.2e-171) {
		tmp = t_m * (Math.sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (x * Math.sqrt(2.0))))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 680000.0) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	tmp = 0
	if t_m <= 3.1e-240:
		tmp = (1.0 / l_m) * (t_m * math.sqrt(x))
	elif t_m <= 1.2e-171:
		tmp = t_m * (math.sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (x * math.sqrt(2.0))))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 680000.0:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + (t_3 / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.1e-240)
		tmp = Float64(Float64(1.0 / l_m) * Float64(t_m * sqrt(x)));
	elseif (t_m <= 1.2e-171)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(x * sqrt(2.0))))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 680000.0)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.1e-240)
		tmp = (1.0 / l_m) * (t_m * sqrt(x));
	elseif (t_m <= 1.2e-171)
		tmp = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 680000.0)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + (t_3 / x))));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(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, 3.1e-240], N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e-171], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 680000.0], N[(t$95$m * N[(N[Sqrt[2.0], $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]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.10000000000000017e-240

   1. Initial program 28.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in l around inf 2.7%

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

      1. *-commutative 2.7%

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

      2. associate--l+ 11.4%

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

      3. sub-neg 11.4%

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

      4. metadata-eval 11.4%

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

      5. +-commutative 11.4%

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

      6. sub-neg 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

   6. Simplified 11.4%

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

      1. *-un-lft-identity 11.4%

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

      2. associate-/r* 11.4%

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

      3. sqrt-undiv 11.4%

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

      4. +-commutative 11.4%

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

      5. sub-neg 11.4%

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

      6. +-commutative 11.4%

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

      7. metadata-eval 11.4%

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

      8. +-commutative 11.4%

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

   8. Applied egg-rr 11.4%

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

   9. Taylor expanded in x around inf 18.4%

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

       1. clear-num 18.4%

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

       2. inv-pow 18.4%

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

   11. Applied egg-rr 18.4%

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

       1. unpow-1 18.4%

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

       2. associate-/l/ 19.7%

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

       3. associate-/r/ 19.9%

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

       4. *-commutative 19.9%

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

   13. Simplified 19.9%

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

   if 3.10000000000000017e-240 < t < 1.19999999999999993e-171

   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%

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

   4. Taylor expanded in x around inf 63.5%

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

   if 1.19999999999999993e-171 < t < 6.8e5

   1. Initial program 46.3%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in x around inf 87.2%

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

   if 6.8e5 < t

   1. Initial program 37.4%

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

   3. Simplified 37.4%

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

   4. Taylor expanded in t around inf 95.5%

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

   5. Taylor expanded in t around 0 95.7%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\ell} \cdot \left(t \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-171}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 680000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 4: 80.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \cdot l_m \leq 10^{+298}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{x}}{l_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 1e+298)
    (sqrt (/ (+ x -1.0) (+ 1.0 x)))
    (* t_m (/ (sqrt x) l_m)))))

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 * l_m) <= 1e+298) {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	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 * l_m) <= 1d+298) then
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    else
        tmp = t_m * (sqrt(x) / l_m)
    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 * l_m) <= 1e+298) {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	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 * l_m) <= 1e+298:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	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 (Float64(l_m * l_m) <= 1e+298)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	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 * l_m) <= 1e+298)
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	else
		tmp = t_m * (sqrt(x) / l_m);
	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[N[(l$95$m * l$95$m), $MachinePrecision], 1e+298], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999996e297

   1. Initial program 39.8%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around inf 48.7%

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

   5. Taylor expanded in t around 0 48.7%

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

   if 9.9999999999999996e297 < (*.f64 l l)

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in l around inf 5.1%

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

      1. *-commutative 5.1%

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

      2. associate--l+ 34.4%

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

      3. sub-neg 34.4%

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

      4. metadata-eval 34.4%

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

      5. +-commutative 34.4%

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

      6. sub-neg 34.4%

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

      7. metadata-eval 34.4%

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

      8. +-commutative 34.4%

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

   6. Simplified 34.4%

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

      1. *-un-lft-identity 34.4%

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

      2. associate-/r* 34.4%

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

      3. sqrt-undiv 34.4%

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

      4. +-commutative 34.4%

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

      5. sub-neg 34.4%

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

      6. +-commutative 34.4%

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

      7. metadata-eval 34.4%

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

      8. +-commutative 34.4%

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

   8. Applied egg-rr 34.4%

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

   9. Taylor expanded in x around inf 48.2%

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

       1. associate-/r/ 48.6%

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

   11. Applied egg-rr 48.6%

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{\sqrt{x}}{\ell} \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+298}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 5: 76.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * sqrt(((x + -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 * sqrt(((x + (-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 * Math.sqrt(((x + -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 * math.sqrt(((x + -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 * sqrt(Float64(Float64(x + -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 * sqrt(((x + -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[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.6%

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

3. Simplified 32.6%

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

4. Taylor expanded in t around inf 43.5%

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

5. Taylor expanded in t around 0 43.6%

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

6. Final simplification 43.6%

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

Alternative 6: 76.1% 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 1 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 32.6%

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

3. Simplified 32.6%

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

4. Taylor expanded in l around 0 43.5%

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

   1. +-commutative 43.5%

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

   2. sub-neg 43.5%

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

   3. metadata-eval 43.5%

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

   4. +-commutative 43.5%

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

6. Simplified 43.5%

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

7. Taylor expanded in x around inf 43.3%

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

8. Final simplification 43.3%

   \[\leadsto 1 + \frac{-1}{x} \]

Alternative 7: 75.3% 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 1 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 32.6%

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

3. Simplified 32.6%

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

4. Taylor expanded in l around 0 43.5%

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

   1. +-commutative 43.5%

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

   2. sub-neg 43.5%

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

   3. metadata-eval 43.5%

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

   4. +-commutative 43.5%

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

6. Simplified 43.5%

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

7. Taylor expanded in x around inf 43.0%

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

8. Final simplification 43.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023319 
(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)))))