Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 84.2%

Time: 24.4s

Alternatives: 11

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% 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: 84.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {l_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.7 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\left(\frac{1}{x} + \left({x}^{-2} + {x}^{-3}\right)\right) + \frac{1}{x + -1}} \cdot l_m}\\ \mathbf{elif}\;t_m \leq 1.5 \cdot 10^{+64}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{t_3 + t_3}{{x}^{2}} + \left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right)\right) + \frac{t_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= t_m 2.7e-180)
      (*
       (sqrt 2.0)
       (/
        t_m
        (*
         (sqrt
          (+ (+ (/ 1.0 x) (+ (pow x -2.0) (pow x -3.0))) (/ 1.0 (+ x -1.0))))
         l_m)))
      (if (<= t_m 1.5e+64)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (+
             (/ (+ t_3 t_3) (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 tmp;
	if (t_m <= 2.7e-180) {
		tmp = sqrt(2.0) * (t_m / (sqrt((((1.0 / x) + (pow(x, -2.0) + pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	} else if (t_m <= 1.5e+64) {
		tmp = t_m * (sqrt(2.0) / sqrt(((((t_3 + t_3) / 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) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l_m ** 2.0d0)
    if (t_m <= 2.7d-180) then
        tmp = sqrt(2.0d0) * (t_m / (sqrt((((1.0d0 / x) + ((x ** (-2.0d0)) + (x ** (-3.0d0)))) + (1.0d0 / (x + (-1.0d0))))) * l_m))
    else if (t_m <= 1.5d+64) then
        tmp = t_m * (sqrt(2.0d0) / sqrt(((((t_3 + t_3) / (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 tmp;
	if (t_m <= 2.7e-180) {
		tmp = Math.sqrt(2.0) * (t_m / (Math.sqrt((((1.0 / x) + (Math.pow(x, -2.0) + Math.pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	} else if (t_m <= 1.5e+64) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt(((((t_3 + t_3) / 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)
	tmp = 0
	if t_m <= 2.7e-180:
		tmp = math.sqrt(2.0) * (t_m / (math.sqrt((((1.0 / x) + (math.pow(x, -2.0) + math.pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m))
	elif t_m <= 1.5e+64:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt(((((t_3 + t_3) / 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))
	tmp = 0.0
	if (t_m <= 2.7e-180)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(sqrt(Float64(Float64(Float64(1.0 / x) + Float64((x ^ -2.0) + (x ^ -3.0))) + Float64(1.0 / Float64(x + -1.0)))) * l_m)));
	elseif (t_m <= 1.5e+64)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(Float64(t_3 + t_3) / (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);
	tmp = 0.0;
	if (t_m <= 2.7e-180)
		tmp = sqrt(2.0) * (t_m / (sqrt((((1.0 / x) + ((x ^ -2.0) + (x ^ -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	elseif (t_m <= 1.5e+64)
		tmp = t_m * (sqrt(2.0) / sqrt(((((t_3 + t_3) / (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]}, N[(t$95$s * If[LessEqual[t$95$m, 2.7e-180], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[Sqrt[N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Power[x, -2.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+64], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] / 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 3 regimes

2. if t < 2.70000000000000014e-180

   1. Initial program 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 7.8%

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

      2. sub-neg 7.8%

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

      3. metadata-eval 7.8%

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

      4. +-commutative 7.8%

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

      5. sub-neg 7.8%

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

      6. metadata-eval 7.8%

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

      7. +-commutative 7.8%

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

   7. Simplified 7.8%

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

      1. *-commutative 7.8%

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

      2. sqrt-prod 8.5%

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

      3. +-commutative 8.5%

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

      4. sub-neg 8.5%

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

      5. +-commutative 8.5%

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

      6. metadata-eval 8.5%

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

      7. +-commutative 8.5%

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

      8. unpow2 8.5%

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

      9. sqrt-prod 4.8%

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

      10. add-sqr-sqrt 7.3%

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

   9. Applied egg-rr 7.3%

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

   10. Taylor expanded in x around inf 16.7%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(\frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{3}}\right)\right)} + \frac{1}{x + -1}} \cdot \ell} \]
   11. Step-by-step derivation

       1. +-commutative 16.7%

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

       2. unpow2 16.7%

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

       3. associate-/r* 16.7%

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

       4. *-rgt-identity 16.7%

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

       5. associate-*r/ 16.7%

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

       6. unpow-1 16.7%

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

       7. unpow-1 16.7%

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

       8. pow-sqr 16.7%

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

       9. metadata-eval 16.7%

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

       10. rem-exp-log 16.7%

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

       11. log-pow 16.7%

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

       12. *-commutative 16.7%

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

       13. exp-neg 16.7%

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

       14. distribute-rgt-neg-in 16.7%

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

       15. metadata-eval 16.7%

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

       16. exp-to-pow 16.7%

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

       17. +-commutative 16.7%

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

   12. Simplified 16.7%

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

   if 2.70000000000000014e-180 < t < 1.5000000000000001e64

   1. Initial program 59.5%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in x around -inf 88.2%

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

   if 1.5000000000000001e64 < t

   1. Initial program 26.7%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in t around inf 97.5%

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

      1. associate-*l* 97.5%

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

      2. +-commutative 97.5%

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

      3. sub-neg 97.5%

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

      4. metadata-eval 97.5%

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

      5. +-commutative 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in t around 0 97.6%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(\frac{1}{x} + \left({x}^{-2} + {x}^{-3}\right)\right) + \frac{1}{x + -1}} \cdot \ell}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+64}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.7 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\left(\frac{1}{x} + \left({x}^{-2} + {x}^{-3}\right)\right) + \frac{1}{x + -1}} \cdot l_m}\\ \mathbf{elif}\;t_m \leq 2.2 \cdot 10^{+63}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_2 + {l_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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 2.7e-180)
      (*
       (sqrt 2.0)
       (/
        t_m
        (*
         (sqrt
          (+ (+ (/ 1.0 x) (+ (pow x -2.0) (pow x -3.0))) (/ 1.0 (+ x -1.0))))
         l_m)))
      (if (<= t_m 2.2e+63)
        (*
         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 <= 2.7e-180) {
		tmp = sqrt(2.0) * (t_m / (sqrt((((1.0 / x) + (pow(x, -2.0) + pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	} else if (t_m <= 2.2e+63) {
		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 <= 2.7d-180) then
        tmp = sqrt(2.0d0) * (t_m / (sqrt((((1.0d0 / x) + ((x ** (-2.0d0)) + (x ** (-3.0d0)))) + (1.0d0 / (x + (-1.0d0))))) * l_m))
    else if (t_m <= 2.2d+63) 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 <= 2.7e-180) {
		tmp = Math.sqrt(2.0) * (t_m / (Math.sqrt((((1.0 / x) + (Math.pow(x, -2.0) + Math.pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	} else if (t_m <= 2.2e+63) {
		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 <= 2.7e-180:
		tmp = math.sqrt(2.0) * (t_m / (math.sqrt((((1.0 / x) + (math.pow(x, -2.0) + math.pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m))
	elif t_m <= 2.2e+63:
		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 <= 2.7e-180)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(sqrt(Float64(Float64(Float64(1.0 / x) + Float64((x ^ -2.0) + (x ^ -3.0))) + Float64(1.0 / Float64(x + -1.0)))) * l_m)));
	elseif (t_m <= 2.2e+63)
		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 <= 2.7e-180)
		tmp = sqrt(2.0) * (t_m / (sqrt((((1.0 / x) + ((x ^ -2.0) + (x ^ -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	elseif (t_m <= 2.2e+63)
		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, 2.7e-180], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[Sqrt[N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Power[x, -2.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e+63], 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 3 regimes

2. if t < 2.70000000000000014e-180

   1. Initial program 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 7.8%

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

      2. sub-neg 7.8%

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

      3. metadata-eval 7.8%

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

      4. +-commutative 7.8%

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

      5. sub-neg 7.8%

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

      6. metadata-eval 7.8%

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

      7. +-commutative 7.8%

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

   7. Simplified 7.8%

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

      1. *-commutative 7.8%

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

      2. sqrt-prod 8.5%

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

      3. +-commutative 8.5%

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

      4. sub-neg 8.5%

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

      5. +-commutative 8.5%

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

      6. metadata-eval 8.5%

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

      7. +-commutative 8.5%

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

      8. unpow2 8.5%

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

      9. sqrt-prod 4.8%

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

      10. add-sqr-sqrt 7.3%

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

   9. Applied egg-rr 7.3%

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

   10. Taylor expanded in x around inf 16.7%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(\frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{3}}\right)\right)} + \frac{1}{x + -1}} \cdot \ell} \]
   11. Step-by-step derivation

       1. +-commutative 16.7%

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

       2. unpow2 16.7%

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

       3. associate-/r* 16.7%

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

       4. *-rgt-identity 16.7%

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

       5. associate-*r/ 16.7%

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

       6. unpow-1 16.7%

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

       7. unpow-1 16.7%

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

       8. pow-sqr 16.7%

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

       9. metadata-eval 16.7%

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

       10. rem-exp-log 16.7%

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

       11. log-pow 16.7%

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

       12. *-commutative 16.7%

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

       13. exp-neg 16.7%

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

       14. distribute-rgt-neg-in 16.7%

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

       15. metadata-eval 16.7%

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

       16. exp-to-pow 16.7%

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

       17. +-commutative 16.7%

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

   12. Simplified 16.7%

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

   if 2.70000000000000014e-180 < t < 2.1999999999999999e63

   1. Initial program 59.5%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in x around inf 87.8%

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

   if 2.1999999999999999e63 < t

   1. Initial program 26.7%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in t around inf 97.5%

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

      1. associate-*l* 97.5%

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

      2. +-commutative 97.5%

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

      3. sub-neg 97.5%

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

      4. metadata-eval 97.5%

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

      5. +-commutative 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in t around 0 97.6%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(\frac{1}{x} + \left({x}^{-2} + {x}^{-3}\right)\right) + \frac{1}{x + -1}} \cdot \ell}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+63}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.7 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\left(\frac{1}{x} + \left({x}^{-2} + {x}^{-3}\right)\right) + \frac{1}{x + -1}} \cdot l_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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 (<= t_m 3.7e-180)
    (*
     (sqrt 2.0)
     (/
      t_m
      (*
       (sqrt
        (+ (+ (/ 1.0 x) (+ (pow x -2.0) (pow x -3.0))) (/ 1.0 (+ x -1.0))))
       l_m)))
    (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 tmp;
	if (t_m <= 3.7e-180) {
		tmp = sqrt(2.0) * (t_m / (sqrt((((1.0 / x) + (pow(x, -2.0) + pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	} 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) :: tmp
    if (t_m <= 3.7d-180) then
        tmp = sqrt(2.0d0) * (t_m / (sqrt((((1.0d0 / x) + ((x ** (-2.0d0)) + (x ** (-3.0d0)))) + (1.0d0 / (x + (-1.0d0))))) * l_m))
    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 tmp;
	if (t_m <= 3.7e-180) {
		tmp = Math.sqrt(2.0) * (t_m / (Math.sqrt((((1.0 / x) + (Math.pow(x, -2.0) + Math.pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	} 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):
	tmp = 0
	if t_m <= 3.7e-180:
		tmp = math.sqrt(2.0) * (t_m / (math.sqrt((((1.0 / x) + (math.pow(x, -2.0) + math.pow(x, -3.0))) + (1.0 / (x + -1.0)))) * l_m))
	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)
	tmp = 0.0
	if (t_m <= 3.7e-180)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(sqrt(Float64(Float64(Float64(1.0 / x) + Float64((x ^ -2.0) + (x ^ -3.0))) + Float64(1.0 / Float64(x + -1.0)))) * l_m)));
	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)
	tmp = 0.0;
	if (t_m <= 3.7e-180)
		tmp = sqrt(2.0) * (t_m / (sqrt((((1.0 / x) + ((x ^ -2.0) + (x ^ -3.0))) + (1.0 / (x + -1.0)))) * l_m));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.7e-180], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[Sqrt[N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Power[x, -2.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.70000000000000016e-180

   1. Initial program 18.8%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 7.9%

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

      2. sub-neg 7.9%

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

      3. metadata-eval 7.9%

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

      4. +-commutative 7.9%

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

      5. sub-neg 7.9%

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

      6. metadata-eval 7.9%

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

      7. +-commutative 7.9%

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

   7. Simplified 7.9%

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

      1. *-commutative 7.9%

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

      2. sqrt-prod 8.6%

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

      3. +-commutative 8.6%

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

      4. sub-neg 8.6%

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

      5. +-commutative 8.6%

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

      6. metadata-eval 8.6%

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

      7. +-commutative 8.6%

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

      8. unpow2 8.6%

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

      9. sqrt-prod 4.8%

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

      10. add-sqr-sqrt 7.3%

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

   9. Applied egg-rr 7.3%

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

   10. Taylor expanded in x around inf 16.6%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(\frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{3}}\right)\right)} + \frac{1}{x + -1}} \cdot \ell} \]
   11. Step-by-step derivation

       1. +-commutative 16.6%

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

       2. unpow2 16.6%

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

       3. associate-/r* 16.6%

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

       4. *-rgt-identity 16.6%

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

       5. associate-*r/ 16.6%

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

       6. unpow-1 16.6%

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

       7. unpow-1 16.6%

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

       8. pow-sqr 16.6%

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

       9. metadata-eval 16.6%

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

       10. rem-exp-log 16.6%

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

       11. log-pow 16.6%

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

       12. *-commutative 16.6%

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

       13. exp-neg 16.6%

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

       14. distribute-rgt-neg-in 16.6%

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

       15. metadata-eval 16.6%

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

       16. exp-to-pow 16.6%

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

       17. +-commutative 16.6%

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

   12. Simplified 16.6%

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

   if 3.70000000000000016e-180 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around inf 89.6%

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

      1. associate-*l* 89.6%

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

      2. +-commutative 89.6%

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

      3. sub-neg 89.6%

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

      4. metadata-eval 89.6%

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

      5. +-commutative 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in t around 0 89.7%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(\frac{1}{x} + \left({x}^{-2} + {x}^{-3}\right)\right) + \frac{1}{x + -1}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.8 \cdot 10^{-179}:\\ \;\;\;\;\frac{t_m \cdot {\left(\frac{1}{x + -1} + \left(\frac{1}{x} + {x}^{-2}\right)\right)}^{-0.5}}{\frac{l_m}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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 (<= t_m 6.8e-179)
    (/
     (* t_m (pow (+ (/ 1.0 (+ x -1.0)) (+ (/ 1.0 x) (pow x -2.0))) -0.5))
     (/ l_m (sqrt 2.0)))
    (sqrt (/ (+ x -1.0) (+ 1.0 x))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.8e-179) {
		tmp = (t_m * pow(((1.0 / (x + -1.0)) + ((1.0 / x) + pow(x, -2.0))), -0.5)) / (l_m / sqrt(2.0));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 6.8d-179) then
        tmp = (t_m * (((1.0d0 / (x + (-1.0d0))) + ((1.0d0 / x) + (x ** (-2.0d0)))) ** (-0.5d0))) / (l_m / sqrt(2.0d0))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.8e-179) {
		tmp = (t_m * Math.pow(((1.0 / (x + -1.0)) + ((1.0 / x) + Math.pow(x, -2.0))), -0.5)) / (l_m / Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 6.8e-179:
		tmp = (t_m * math.pow(((1.0 / (x + -1.0)) + ((1.0 / x) + math.pow(x, -2.0))), -0.5)) / (l_m / math.sqrt(2.0))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 6.8e-179)
		tmp = Float64(Float64(t_m * (Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 / x) + (x ^ -2.0))) ^ -0.5)) / Float64(l_m / sqrt(2.0)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 6.8e-179)
		tmp = (t_m * (((1.0 / (x + -1.0)) + ((1.0 / x) + (x ^ -2.0))) ^ -0.5)) / (l_m / sqrt(2.0));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.8e-179], N[(N[(t$95$m * N[Power[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.7999999999999995e-179

   1. Initial program 18.8%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in l around inf 2.7%

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

      1. *-commutative 2.7%

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

      2. associate--l+ 7.1%

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

      3. sub-neg 7.1%

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

      4. metadata-eval 7.1%

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

      5. +-commutative 7.1%

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

      6. sub-neg 7.1%

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

      7. metadata-eval 7.1%

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

      8. +-commutative 7.1%

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

      9. associate-/l* 7.1%

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

   7. Simplified 7.1%

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

   8. Taylor expanded in x around inf 14.5%

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

      1. associate-*r/ 16.6%

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

      2. pow1/2 16.6%

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

      3. inv-pow 16.5%

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

      4. pow-pow 16.5%

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

      5. +-commutative 16.5%

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

      6. pow-flip 16.5%

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

      7. metadata-eval 16.5%

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

      8. metadata-eval 16.5%

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

   10. Applied egg-rr 16.5%

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

   if 6.7999999999999995e-179 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around inf 89.6%

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

      1. associate-*l* 89.6%

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

      2. +-commutative 89.6%

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

      3. sub-neg 89.6%

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

      4. metadata-eval 89.6%

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

      5. +-commutative 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in t around 0 89.7%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-179}:\\ \;\;\;\;\frac{t \cdot {\left(\frac{1}{x + -1} + \left(\frac{1}{x} + {x}^{-2}\right)\right)}^{-0.5}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.3 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{l_m \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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 (<= t_m 1.3e-178)
    (* (sqrt 2.0) (/ t_m (* l_m (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0)))))))
    (sqrt (/ (+ x -1.0) (+ 1.0 x))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.3e-178) {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.3d-178) then
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0)))))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.3e-178) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.3e-178:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.3e-178)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0)))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.3e-178)
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.3e-178], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $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 2 regimes

2. if t < 1.29999999999999999e-178

   1. Initial program 18.8%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 7.9%

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

      2. sub-neg 7.9%

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

      3. metadata-eval 7.9%

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

      4. +-commutative 7.9%

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

      5. sub-neg 7.9%

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

      6. metadata-eval 7.9%

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

      7. +-commutative 7.9%

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

   7. Simplified 7.9%

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

      1. *-commutative 7.9%

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

      2. sqrt-prod 8.6%

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

      3. +-commutative 8.6%

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

      4. sub-neg 8.6%

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

      5. +-commutative 8.6%

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

      6. metadata-eval 8.6%

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

      7. +-commutative 8.6%

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

      8. unpow2 8.6%

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

      9. sqrt-prod 4.8%

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

      10. add-sqr-sqrt 7.3%

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

   9. Applied egg-rr 7.3%

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

   10. Taylor expanded in x around inf 16.1%

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

   if 1.29999999999999999e-178 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around inf 89.6%

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

      1. associate-*l* 89.6%

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

      2. +-commutative 89.6%

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

      3. sub-neg 89.6%

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

      4. metadata-eval 89.6%

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

      5. +-commutative 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in t around 0 89.7%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.4 \cdot 10^{-179}:\\ \;\;\;\;\frac{t_m \cdot \sqrt{2}}{l_m \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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 (<= t_m 2.4e-179)
    (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0))))))
    (sqrt (/ (+ x -1.0) (+ 1.0 x))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-179) {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.4d-179) then
        tmp = (t_m * sqrt(2.0d0)) / (l_m * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0))))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-179) {
		tmp = (t_m * Math.sqrt(2.0)) / (l_m * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.4e-179:
		tmp = (t_m * math.sqrt(2.0)) / (l_m * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.4e-179)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.4e-179)
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-179], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $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 2 regimes

2. if t < 2.4e-179

   1. Initial program 18.8%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate--l+ 7.9%

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

      2. sub-neg 7.9%

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

      3. metadata-eval 7.9%

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

      4. +-commutative 7.9%

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

      5. sub-neg 7.9%

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

      6. metadata-eval 7.9%

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

      7. +-commutative 7.9%

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

   7. Simplified 7.9%

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

      1. *-commutative 7.9%

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

      2. sqrt-prod 8.6%

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

      3. +-commutative 8.6%

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

      4. sub-neg 8.6%

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

      5. +-commutative 8.6%

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

      6. metadata-eval 8.6%

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

      7. +-commutative 8.6%

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

      8. unpow2 8.6%

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

      9. sqrt-prod 4.8%

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

      10. add-sqr-sqrt 7.3%

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

   9. Applied egg-rr 7.3%

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

   10. Taylor expanded in x around inf 16.1%

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

       1. associate-*r/ 16.1%

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

   12. Applied egg-rr 16.1%

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

   if 2.4e-179 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around inf 89.6%

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

      1. associate-*l* 89.6%

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

      2. +-commutative 89.6%

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

      3. sub-neg 89.6%

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

      4. metadata-eval 89.6%

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

      5. +-commutative 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in t around 0 89.7%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-179}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.1 \cdot 10^{-180}:\\ \;\;\;\;\frac{t_m}{l_m} \cdot \left(\sqrt{2} \cdot \sqrt{x \cdot 0.5 - 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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 (<= t_m 3.1e-180)
    (* (/ t_m l_m) (* (sqrt 2.0) (sqrt (- (* x 0.5) 0.5))))
    (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 tmp;
	if (t_m <= 3.1e-180) {
		tmp = (t_m / l_m) * (sqrt(2.0) * sqrt(((x * 0.5) - 0.5)));
	} 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) :: tmp
    if (t_m <= 3.1d-180) then
        tmp = (t_m / l_m) * (sqrt(2.0d0) * sqrt(((x * 0.5d0) - 0.5d0)))
    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 tmp;
	if (t_m <= 3.1e-180) {
		tmp = (t_m / l_m) * (Math.sqrt(2.0) * Math.sqrt(((x * 0.5) - 0.5)));
	} 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):
	tmp = 0
	if t_m <= 3.1e-180:
		tmp = (t_m / l_m) * (math.sqrt(2.0) * math.sqrt(((x * 0.5) - 0.5)))
	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)
	tmp = 0.0
	if (t_m <= 3.1e-180)
		tmp = Float64(Float64(t_m / l_m) * Float64(sqrt(2.0) * sqrt(Float64(Float64(x * 0.5) - 0.5))));
	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)
	tmp = 0.0;
	if (t_m <= 3.1e-180)
		tmp = (t_m / l_m) * (sqrt(2.0) * sqrt(((x * 0.5) - 0.5)));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-180], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(x * 0.5), $MachinePrecision] - 0.5), $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 2 regimes

2. if t < 3.0999999999999999e-180

   1. Initial program 18.8%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in l around inf 2.7%

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

      1. *-commutative 2.7%

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

      2. associate--l+ 7.1%

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

      3. sub-neg 7.1%

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

      4. metadata-eval 7.1%

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

      5. +-commutative 7.1%

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

      6. sub-neg 7.1%

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

      7. metadata-eval 7.1%

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

      8. +-commutative 7.1%

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

      9. associate-/l* 7.1%

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

   7. Simplified 7.1%

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

   8. Taylor expanded in x around inf 14.5%

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

   9. Taylor expanded in t around 0 14.5%

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

       1. associate-*l/ 14.5%

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

       2. associate-*l* 14.5%

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

       3. +-commutative 14.5%

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

       4. unpow2 14.5%

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

       5. associate-/r* 14.5%

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

       6. *-rgt-identity 14.5%

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

       7. associate-*r/ 14.5%

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

       8. unpow-1 14.5%

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

       9. unpow-1 14.5%

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

       10. pow-sqr 14.5%

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

       11. metadata-eval 14.5%

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

       12. sub-neg 14.5%

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

       13. metadata-eval 14.5%

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

   11. Simplified 14.5%

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

   12. Taylor expanded in x around inf 14.6%

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

   if 3.0999999999999999e-180 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around inf 89.6%

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

      1. associate-*l* 89.6%

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

      2. +-commutative 89.6%

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

      3. sub-neg 89.6%

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

      4. metadata-eval 89.6%

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

      5. +-commutative 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in t around 0 89.7%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-180}:\\ \;\;\;\;\frac{t}{\ell} \cdot \left(\sqrt{2} \cdot \sqrt{x \cdot 0.5 - 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.15 \cdot 10^{-179}:\\ \;\;\;\;\frac{t_m}{l_m} \cdot \left(\sqrt{2} \cdot \sqrt{x \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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 (<= t_m 2.15e-179)
    (* (/ t_m l_m) (* (sqrt 2.0) (sqrt (* x 0.5))))
    (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 tmp;
	if (t_m <= 2.15e-179) {
		tmp = (t_m / l_m) * (sqrt(2.0) * sqrt((x * 0.5)));
	} 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) :: tmp
    if (t_m <= 2.15d-179) then
        tmp = (t_m / l_m) * (sqrt(2.0d0) * sqrt((x * 0.5d0)))
    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 tmp;
	if (t_m <= 2.15e-179) {
		tmp = (t_m / l_m) * (Math.sqrt(2.0) * Math.sqrt((x * 0.5)));
	} 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):
	tmp = 0
	if t_m <= 2.15e-179:
		tmp = (t_m / l_m) * (math.sqrt(2.0) * math.sqrt((x * 0.5)))
	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)
	tmp = 0.0
	if (t_m <= 2.15e-179)
		tmp = Float64(Float64(t_m / l_m) * Float64(sqrt(2.0) * sqrt(Float64(x * 0.5))));
	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)
	tmp = 0.0;
	if (t_m <= 2.15e-179)
		tmp = (t_m / l_m) * (sqrt(2.0) * sqrt((x * 0.5)));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.15e-179], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(x * 0.5), $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 2 regimes

2. if t < 2.15000000000000013e-179

   1. Initial program 18.8%

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

   3. Simplified 18.8%

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

   5. Taylor expanded in l around inf 2.7%

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

      1. *-commutative 2.7%

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

      2. associate--l+ 7.1%

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

      3. sub-neg 7.1%

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

      4. metadata-eval 7.1%

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

      5. +-commutative 7.1%

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

      6. sub-neg 7.1%

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

      7. metadata-eval 7.1%

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

      8. +-commutative 7.1%

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

      9. associate-/l* 7.1%

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

   7. Simplified 7.1%

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

   8. Taylor expanded in x around inf 14.5%

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

   9. Taylor expanded in t around 0 14.5%

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

       1. associate-*l/ 14.5%

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

       2. associate-*l* 14.5%

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

       3. +-commutative 14.5%

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

       4. unpow2 14.5%

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

       5. associate-/r* 14.5%

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

       6. *-rgt-identity 14.5%

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

       7. associate-*r/ 14.5%

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

       8. unpow-1 14.5%

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

       9. unpow-1 14.5%

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

       10. pow-sqr 14.5%

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

       11. metadata-eval 14.5%

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

       12. sub-neg 14.5%

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

       13. metadata-eval 14.5%

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

   11. Simplified 14.5%

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

   12. Taylor expanded in x around inf 14.0%

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

   if 2.15000000000000013e-179 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around inf 89.6%

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

      1. associate-*l* 89.6%

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

      2. +-commutative 89.6%

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

      3. sub-neg 89.6%

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

      4. metadata-eval 89.6%

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

      5. +-commutative 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in t around 0 89.7%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-179}:\\ \;\;\;\;\frac{t}{\ell} \cdot \left(\sqrt{2} \cdot \sqrt{x \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.6% 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 27.5%

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

3. Simplified 27.4%

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

5. Taylor expanded in t around inf 36.1%

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

   1. associate-*l* 36.1%

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

   2. +-commutative 36.1%

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

   3. sub-neg 36.1%

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

   4. metadata-eval 36.1%

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

   5. +-commutative 36.1%

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

7. Simplified 36.1%

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

8. Taylor expanded in t around 0 36.1%

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

9. Final simplification 36.1%

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

Alternative 10: 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 27.5%

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

3. Simplified 27.4%

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

5. Taylor expanded in t around inf 36.1%

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

   1. associate-*l* 36.1%

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

   2. +-commutative 36.1%

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

   3. sub-neg 36.1%

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

   4. metadata-eval 36.1%

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

   5. +-commutative 36.1%

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

7. Simplified 36.1%

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

8. Taylor expanded in x around inf 35.9%

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

9. Final simplification 35.9%

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

Alternative 11: 75.5% 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 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 27.5%

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

3. Simplified 27.4%

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

5. Taylor expanded in t around inf 36.1%

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

   1. associate-*l* 36.1%

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

   2. +-commutative 36.1%

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

   3. sub-neg 36.1%

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

   4. metadata-eval 36.1%

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

   5. +-commutative 36.1%

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

7. Simplified 36.1%

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

8. Taylor expanded in x around inf 35.5%

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

9. Final simplification 35.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024013 
(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)))))