Toniolo and Linder, Equation (7)

Percentage Accurate: 32.9% → 84.7%

Time: 21.7s

Alternatives: 13

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: 32.9% 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: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{{\ell}^{2}}{x}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := 2 \cdot {t}^{2}\\ t_4 := {\ell}^{2} + t_3\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+43}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(t_1 + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)\right)}{2}}}\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-170}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{2 \cdot \left(t \cdot x\right)}, t\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(\frac{t_4 + t_4}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(t_1 + t_3\right)\right)\right) + \frac{t_4}{x}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (pow l 2.0) x))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_3 (* 2.0 (pow t 2.0)))
        (t_4 (+ (pow l 2.0) t_3)))
   (if (<= t -5.2e+43)
     (- t_2)
     (if (<= t -8.2e-159)
       (/ t (sqrt (/ (+ t_1 (+ t_1 (* (pow t 2.0) (+ 2.0 (/ 2.0 x))))) 2.0)))
       (if (<= t -1.45e-170)
         -1.0
         (if (<= t 9.5e-304)
           (/ t (* l (sqrt (/ 1.0 x))))
           (if (<= t 4.2e-160)
             (/
              t
              (fma
               0.5
               (/ (fma 2.0 (pow t 2.0) (fma l l (pow l 2.0))) (* 2.0 (* t x)))
               t))
             (if (<= t 8.8e+98)
               (/
                t
                (/
                 (sqrt
                  (+
                   (+
                    (/ (+ t_4 t_4) (pow x 2.0))
                    (+ (* 2.0 (/ (pow t 2.0) x)) (+ t_1 t_3)))
                   (/ t_4 x)))
                 (sqrt 2.0)))
               t_2))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = pow(l, 2.0) / x;
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_3 = 2.0 * pow(t, 2.0);
	double t_4 = pow(l, 2.0) + t_3;
	double tmp;
	if (t <= -5.2e+43) {
		tmp = -t_2;
	} else if (t <= -8.2e-159) {
		tmp = t / sqrt(((t_1 + (t_1 + (pow(t, 2.0) * (2.0 + (2.0 / x))))) / 2.0));
	} else if (t <= -1.45e-170) {
		tmp = -1.0;
	} else if (t <= 9.5e-304) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else if (t <= 4.2e-160) {
		tmp = t / fma(0.5, (fma(2.0, pow(t, 2.0), fma(l, l, pow(l, 2.0))) / (2.0 * (t * x))), t);
	} else if (t <= 8.8e+98) {
		tmp = t / (sqrt(((((t_4 + t_4) / pow(x, 2.0)) + ((2.0 * (pow(t, 2.0) / x)) + (t_1 + t_3))) + (t_4 / x))) / sqrt(2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64((l ^ 2.0) / x)
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_3 = Float64(2.0 * (t ^ 2.0))
	t_4 = Float64((l ^ 2.0) + t_3)
	tmp = 0.0
	if (t <= -5.2e+43)
		tmp = Float64(-t_2);
	elseif (t <= -8.2e-159)
		tmp = Float64(t / sqrt(Float64(Float64(t_1 + Float64(t_1 + Float64((t ^ 2.0) * Float64(2.0 + Float64(2.0 / x))))) / 2.0)));
	elseif (t <= -1.45e-170)
		tmp = -1.0;
	elseif (t <= 9.5e-304)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	elseif (t <= 4.2e-160)
		tmp = Float64(t / fma(0.5, Float64(fma(2.0, (t ^ 2.0), fma(l, l, (l ^ 2.0))) / Float64(2.0 * Float64(t * x))), t));
	elseif (t <= 8.8e+98)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(Float64(Float64(t_4 + t_4) / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(t_1 + t_3))) + Float64(t_4 / x))) / sqrt(2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[l, 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t, -5.2e+43], (-t$95$2), If[LessEqual[t, -8.2e-159], N[(t / N[Sqrt[N[(N[(t$95$1 + N[(t$95$1 + N[(N[Power[t, 2.0], $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.45e-170], -1.0, If[LessEqual[t, 9.5e-304], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-160], N[(t / N[(0.5 * N[(N[(2.0 * N[Power[t, 2.0], $MachinePrecision] + N[(l * l + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.8e+98], N[(t / N[(N[Sqrt[N[(N[(N[(N[(t$95$4 + t$95$4), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -5.20000000000000042e43

   1. Initial program 24.2%

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

   3. Simplified 24.1%

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

   5. Applied egg-rr 76.6%

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

   6. Taylor expanded in t around -inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

   8. Simplified 94.5%

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

   if -5.20000000000000042e43 < t < -8.20000000000000029e-159

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in x around inf 88.3%

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

   5. Taylor expanded in t around 0 88.2%

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

      1. expm1-log1p-u 86.9%

         \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{{\ell}^{2}}{x}}}{\sqrt{2}}\right)\right)}} \]

      2. expm1-udef 35.5%

         \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{{\ell}^{2}}{x}}}{\sqrt{2}}\right)} - 1}} \]

   7. Applied egg-rr 35.5%

      \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2}}{x}}{2}}\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 87.3%

         \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2}}{x}}{2}}\right)\right)}} \]

      2. expm1-log1p 88.8%

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

   9. Simplified 88.8%

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

   if -8.20000000000000029e-159 < t < -1.45e-170

   1. Initial program 22.6%

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

   3. Simplified 22.6%

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

   5. Applied egg-rr 67.8%

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

   6. Taylor expanded in t around inf 1.6%

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

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 0.0%

         \[\leadsto \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

      2. rem-square-sqrt 100.0%

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

   9. Simplified 100.0%

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

   if -1.45e-170 < t < 9.50000000000000023e-304

   1. Initial program 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in x around inf 60.3%

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

   5. Taylor expanded in t around 0 60.3%

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

   6. Taylor expanded in l around inf 59.6%

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

   if 9.50000000000000023e-304 < t < 4.2000000000000001e-160

   1. Initial program 2.3%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in x around inf 37.7%

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

   5. Taylor expanded in t around 0 37.7%

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

   6. Taylor expanded in x around inf 93.2%

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

      1. +-commutative 93.2%

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

      2. fma-def 93.2%

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

      3. associate--l+ 93.2%

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

      4. fma-def 93.2%

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

      5. unpow2 93.2%

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

      6. fma-neg 93.2%

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

      7. mul-1-neg 93.2%

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

      8. remove-double-neg 93.2%

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

      9. *-commutative 93.2%

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

      10. *-commutative 93.2%

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

      11. unpow2 93.2%

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

      12. rem-square-sqrt 93.2%

          \[\leadsto \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{\left(\color{blue}{2} \cdot x\right) \cdot t}, t\right)} \]

      13. associate-*l* 93.2%

          \[\leadsto \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{\color{blue}{2 \cdot \left(x \cdot t\right)}}, t\right)} \]

   8. Simplified 93.2%

      \[\leadsto \frac{t}{\color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{2 \cdot \left(x \cdot t\right)}, t\right)}} \]

   if 4.2000000000000001e-160 < t < 8.80000000000000034e98

   1. Initial program 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in x around -inf 83.2%

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

   if 8.80000000000000034e98 < t

   1. Initial program 12.2%

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

   3. Simplified 12.1%

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

   5. Applied egg-rr 84.6%

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

   6. Taylor expanded in t around inf 96.2%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+43}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{{\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)\right)}{2}}}\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-170}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{2 \cdot \left(t \cdot x\right)}, t\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(\frac{\left({\ell}^{2} + 2 \cdot {t}^{2}\right) + \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right) + \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{{\ell}^{2}}{x}\\ t_2 := \frac{t}{\sqrt{\frac{t_1 + \left(t_1 + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)\right)}{2}}}\\ t_3 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+43}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-168}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-302}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{2 \cdot \left(t \cdot x\right)}, t\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (pow l 2.0) x))
        (t_2
         (/
          t
          (sqrt (/ (+ t_1 (+ t_1 (* (pow t 2.0) (+ 2.0 (/ 2.0 x))))) 2.0))))
        (t_3 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1e+43)
     (- t_3)
     (if (<= t -8e-159)
       t_2
       (if (<= t -5.2e-168)
         -1.0
         (if (<= t 7.6e-302)
           (/ t (* l (sqrt (/ 1.0 x))))
           (if (<= t 4.2e-161)
             (/
              t
              (fma
               0.5
               (/ (fma 2.0 (pow t 2.0) (fma l l (pow l 2.0))) (* 2.0 (* t x)))
               t))
             (if (<= t 1.6e+100) t_2 t_3))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = pow(l, 2.0) / x;
	double t_2 = t / sqrt(((t_1 + (t_1 + (pow(t, 2.0) * (2.0 + (2.0 / x))))) / 2.0));
	double t_3 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1e+43) {
		tmp = -t_3;
	} else if (t <= -8e-159) {
		tmp = t_2;
	} else if (t <= -5.2e-168) {
		tmp = -1.0;
	} else if (t <= 7.6e-302) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else if (t <= 4.2e-161) {
		tmp = t / fma(0.5, (fma(2.0, pow(t, 2.0), fma(l, l, pow(l, 2.0))) / (2.0 * (t * x))), t);
	} else if (t <= 1.6e+100) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64((l ^ 2.0) / x)
	t_2 = Float64(t / sqrt(Float64(Float64(t_1 + Float64(t_1 + Float64((t ^ 2.0) * Float64(2.0 + Float64(2.0 / x))))) / 2.0)))
	t_3 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1e+43)
		tmp = Float64(-t_3);
	elseif (t <= -8e-159)
		tmp = t_2;
	elseif (t <= -5.2e-168)
		tmp = -1.0;
	elseif (t <= 7.6e-302)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	elseif (t <= 4.2e-161)
		tmp = Float64(t / fma(0.5, Float64(fma(2.0, (t ^ 2.0), fma(l, l, (l ^ 2.0))) / Float64(2.0 * Float64(t * x))), t));
	elseif (t <= 1.6e+100)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Sqrt[N[(N[(t$95$1 + N[(t$95$1 + N[(N[Power[t, 2.0], $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1e+43], (-t$95$3), If[LessEqual[t, -8e-159], t$95$2, If[LessEqual[t, -5.2e-168], -1.0, If[LessEqual[t, 7.6e-302], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-161], N[(t / N[(0.5 * N[(N[(2.0 * N[Power[t, 2.0], $MachinePrecision] + N[(l * l + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e+100], t$95$2, t$95$3]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.00000000000000001e43

   1. Initial program 24.2%

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

   3. Simplified 24.1%

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

   5. Applied egg-rr 76.6%

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

   6. Taylor expanded in t around -inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

   8. Simplified 94.5%

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

   if -1.00000000000000001e43 < t < -7.99999999999999991e-159 or 4.2000000000000001e-161 < t < 1.5999999999999999e100

   1. Initial program 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in x around inf 85.3%

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

   5. Taylor expanded in t around 0 85.0%

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

      1. expm1-log1p-u 83.1%

         \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{{\ell}^{2}}{x}}}{\sqrt{2}}\right)\right)}} \]

      2. expm1-udef 38.6%

         \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{{\ell}^{2}}{x}}}{\sqrt{2}}\right)} - 1}} \]

   7. Applied egg-rr 38.6%

      \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2}}{x}}{2}}\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 83.5%

         \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2}}{x}}{2}}\right)\right)}} \]

      2. expm1-log1p 85.7%

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

   9. Simplified 85.7%

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

   if -7.99999999999999991e-159 < t < -5.2000000000000002e-168

   1. Initial program 22.6%

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

   3. Simplified 22.6%

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

   5. Applied egg-rr 67.8%

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

   6. Taylor expanded in t around inf 1.6%

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

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 0.0%

         \[\leadsto \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

      2. rem-square-sqrt 100.0%

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

   9. Simplified 100.0%

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

   if -5.2000000000000002e-168 < t < 7.5999999999999999e-302

   1. Initial program 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in x around inf 60.3%

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

   5. Taylor expanded in t around 0 60.3%

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

   6. Taylor expanded in l around inf 59.6%

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

   if 7.5999999999999999e-302 < t < 4.2000000000000001e-161

   1. Initial program 2.3%

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

   3. Simplified 2.3%

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

   4. Taylor expanded in x around inf 37.7%

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

   5. Taylor expanded in t around 0 37.7%

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

   6. Taylor expanded in x around inf 93.2%

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

      1. +-commutative 93.2%

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

      2. fma-def 93.2%

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

      3. associate--l+ 93.2%

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

      4. fma-def 93.2%

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

      5. unpow2 93.2%

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

      6. fma-neg 93.2%

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

      7. mul-1-neg 93.2%

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

      8. remove-double-neg 93.2%

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

      9. *-commutative 93.2%

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

      10. *-commutative 93.2%

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

      11. unpow2 93.2%

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

      12. rem-square-sqrt 93.2%

          \[\leadsto \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{\left(\color{blue}{2} \cdot x\right) \cdot t}, t\right)} \]

      13. associate-*l* 93.2%

          \[\leadsto \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{\color{blue}{2 \cdot \left(x \cdot t\right)}}, t\right)} \]

   8. Simplified 93.2%

      \[\leadsto \frac{t}{\color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{2 \cdot \left(x \cdot t\right)}, t\right)}} \]

   if 1.5999999999999999e100 < t

   1. Initial program 12.2%

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

   3. Simplified 12.1%

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

   5. Applied egg-rr 84.6%

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

   6. Taylor expanded in t around inf 96.2%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+43}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-159}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{{\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)\right)}{2}}}\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-168}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-302}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)\right)}{2 \cdot \left(t \cdot x\right)}, t\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{{\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{{\ell}^{2}}{x}\\ t_2 := \frac{t}{\sqrt{\frac{t_1 + \left(t_1 + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)\right)}{2}}}\\ t_3 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.96 \cdot 10^{+43}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -8.3 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.1 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-205}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+99}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (pow l 2.0) x))
        (t_2
         (/
          t
          (sqrt (/ (+ t_1 (+ t_1 (* (pow t 2.0) (+ 2.0 (/ 2.0 x))))) 2.0))))
        (t_3 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.96e+43)
     (- t_3)
     (if (<= t -8.3e-159)
       t_2
       (if (<= t -5.1e-169)
         -1.0
         (if (<= t 2.7e-205)
           (/ t (* l (sqrt (/ 1.0 x))))
           (if (<= t 1.75e-162) 1.0 (if (<= t 7.2e+99) t_2 t_3))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = pow(l, 2.0) / x;
	double t_2 = t / sqrt(((t_1 + (t_1 + (pow(t, 2.0) * (2.0 + (2.0 / x))))) / 2.0));
	double t_3 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.96e+43) {
		tmp = -t_3;
	} else if (t <= -8.3e-159) {
		tmp = t_2;
	} else if (t <= -5.1e-169) {
		tmp = -1.0;
	} else if (t <= 2.7e-205) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else if (t <= 1.75e-162) {
		tmp = 1.0;
	} else if (t <= 7.2e+99) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (l ** 2.0d0) / x
    t_2 = t / sqrt(((t_1 + (t_1 + ((t ** 2.0d0) * (2.0d0 + (2.0d0 / x))))) / 2.0d0))
    t_3 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-1.96d+43)) then
        tmp = -t_3
    else if (t <= (-8.3d-159)) then
        tmp = t_2
    else if (t <= (-5.1d-169)) then
        tmp = -1.0d0
    else if (t <= 2.7d-205) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else if (t <= 1.75d-162) then
        tmp = 1.0d0
    else if (t <= 7.2d+99) then
        tmp = t_2
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.pow(l, 2.0) / x;
	double t_2 = t / Math.sqrt(((t_1 + (t_1 + (Math.pow(t, 2.0) * (2.0 + (2.0 / x))))) / 2.0));
	double t_3 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.96e+43) {
		tmp = -t_3;
	} else if (t <= -8.3e-159) {
		tmp = t_2;
	} else if (t <= -5.1e-169) {
		tmp = -1.0;
	} else if (t <= 2.7e-205) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else if (t <= 1.75e-162) {
		tmp = 1.0;
	} else if (t <= 7.2e+99) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.pow(l, 2.0) / x
	t_2 = t / math.sqrt(((t_1 + (t_1 + (math.pow(t, 2.0) * (2.0 + (2.0 / x))))) / 2.0))
	t_3 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -1.96e+43:
		tmp = -t_3
	elif t <= -8.3e-159:
		tmp = t_2
	elif t <= -5.1e-169:
		tmp = -1.0
	elif t <= 2.7e-205:
		tmp = t / (l * math.sqrt((1.0 / x)))
	elif t <= 1.75e-162:
		tmp = 1.0
	elif t <= 7.2e+99:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64((l ^ 2.0) / x)
	t_2 = Float64(t / sqrt(Float64(Float64(t_1 + Float64(t_1 + Float64((t ^ 2.0) * Float64(2.0 + Float64(2.0 / x))))) / 2.0)))
	t_3 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.96e+43)
		tmp = Float64(-t_3);
	elseif (t <= -8.3e-159)
		tmp = t_2;
	elseif (t <= -5.1e-169)
		tmp = -1.0;
	elseif (t <= 2.7e-205)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	elseif (t <= 1.75e-162)
		tmp = 1.0;
	elseif (t <= 7.2e+99)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (l ^ 2.0) / x;
	t_2 = t / sqrt(((t_1 + (t_1 + ((t ^ 2.0) * (2.0 + (2.0 / x))))) / 2.0));
	t_3 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1.96e+43)
		tmp = -t_3;
	elseif (t <= -8.3e-159)
		tmp = t_2;
	elseif (t <= -5.1e-169)
		tmp = -1.0;
	elseif (t <= 2.7e-205)
		tmp = t / (l * sqrt((1.0 / x)));
	elseif (t <= 1.75e-162)
		tmp = 1.0;
	elseif (t <= 7.2e+99)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Sqrt[N[(N[(t$95$1 + N[(t$95$1 + N[(N[Power[t, 2.0], $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.96e+43], (-t$95$3), If[LessEqual[t, -8.3e-159], t$95$2, If[LessEqual[t, -5.1e-169], -1.0, If[LessEqual[t, 2.7e-205], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.75e-162], 1.0, If[LessEqual[t, 7.2e+99], t$95$2, t$95$3]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.9600000000000001e43

   1. Initial program 24.2%

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

   3. Simplified 24.1%

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

   5. Applied egg-rr 76.6%

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

   6. Taylor expanded in t around -inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

   8. Simplified 94.5%

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

   if -1.9600000000000001e43 < t < -8.30000000000000047e-159 or 1.74999999999999995e-162 < t < 7.2000000000000003e99

   1. Initial program 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in x around inf 85.3%

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

   5. Taylor expanded in t around 0 85.0%

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

      1. expm1-log1p-u 83.1%

         \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{{\ell}^{2}}{x}}}{\sqrt{2}}\right)\right)}} \]

      2. expm1-udef 38.6%

         \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{{\ell}^{2}}{x}}}{\sqrt{2}}\right)} - 1}} \]

   7. Applied egg-rr 38.6%

      \[\leadsto \frac{t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2}}{x}}{2}}\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 83.5%

         \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2}}{x}}{2}}\right)\right)}} \]

      2. expm1-log1p 85.7%

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

   9. Simplified 85.7%

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

   if -8.30000000000000047e-159 < t < -5.09999999999999997e-169

   1. Initial program 22.6%

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

   3. Simplified 22.6%

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

   5. Applied egg-rr 67.8%

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

   6. Taylor expanded in t around inf 1.6%

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

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 0.0%

         \[\leadsto \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

      2. rem-square-sqrt 100.0%

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

   9. Simplified 100.0%

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

   if -5.09999999999999997e-169 < t < 2.7000000000000001e-205

   1. Initial program 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

   5. Taylor expanded in t around 0 63.8%

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

   6. Taylor expanded in l around inf 63.2%

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

   if 2.7000000000000001e-205 < t < 1.74999999999999995e-162

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Applied egg-rr 61.8%

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

   6. Taylor expanded in t around inf 97.3%

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

   7. Taylor expanded in x around inf 97.3%

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

   if 7.2000000000000003e99 < t

   1. Initial program 12.2%

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

   3. Simplified 12.1%

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

   5. Applied egg-rr 84.6%

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

   6. Taylor expanded in t around inf 96.2%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.96 \cdot 10^{+43}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -8.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{{\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)\right)}{2}}}\\ \mathbf{elif}\;t \leq -5.1 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-205}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{{\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-168}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-168)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 1.5e-209)
     (/ t (* l (sqrt (/ 1.0 x))))
     (if (<= t 7e-88)
       (+ 1.0 (/ -1.0 x))
       (if (<= t 1.05e-48)
         (* (sqrt 2.0) (/ t (* l (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0)))))))
         (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-168) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.5e-209) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else if (t <= 7e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 1.05e-48) {
		tmp = sqrt(2.0) * (t / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	} else {
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-168)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 1.5d-209) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else if (t <= 7d-88) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t <= 1.05d-48) then
        tmp = sqrt(2.0d0) * (t / (l * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0)))))))
    else
        tmp = 1.0d0 / sqrt(((x + 1.0d0) / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-168) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.5e-209) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else if (t <= 7e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 1.05e-48) {
		tmp = Math.sqrt(2.0) * (t / (l * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	} else {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-168:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 1.5e-209:
		tmp = t / (l * math.sqrt((1.0 / x)))
	elif t <= 7e-88:
		tmp = 1.0 + (-1.0 / x)
	elif t <= 1.05e-48:
		tmp = math.sqrt(2.0) * (t / (l * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))))
	else:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-168)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 1.5e-209)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	elseif (t <= 7e-88)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t <= 1.05e-48)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(l * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0)))))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-168)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 1.5e-209)
		tmp = t / (l * sqrt((1.0 / x)));
	elseif (t <= 7e-88)
		tmp = 1.0 + (-1.0 / x);
	elseif (t <= 1.05e-48)
		tmp = sqrt(2.0) * (t / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	else
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-168], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 1.5e-209], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e-88], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-48], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(l * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.00000000000000001e-168

   1. Initial program 38.2%

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

   3. Simplified 38.1%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. sub-neg 87.2%

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

      3. metadata-eval 87.2%

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

   8. Simplified 87.2%

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

   if -5.00000000000000001e-168 < t < 1.4999999999999999e-209

   1. Initial program 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

   5. Taylor expanded in t around 0 63.8%

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

   6. Taylor expanded in l around inf 63.2%

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

   if 1.4999999999999999e-209 < t < 7.0000000000000002e-88

   1. Initial program 35.0%

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

   3. Simplified 34.8%

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

   5. Applied egg-rr 52.4%

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

   6. Taylor expanded in t around inf 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if 7.0000000000000002e-88 < t < 1.04999999999999994e-48

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in l around inf 2.3%

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

      1. associate--l+ 12.9%

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

      2. sub-neg 12.9%

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

      3. metadata-eval 12.9%

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

      4. +-commutative 12.9%

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

      5. sub-neg 12.9%

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

      6. metadata-eval 12.9%

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

      7. +-commutative 12.9%

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

   6. Simplified 12.9%

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

   7. Taylor expanded in x around inf 58.1%

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

   if 1.04999999999999994e-48 < t

   1. Initial program 31.3%

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

   3. Simplified 31.2%

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

   5. Applied egg-rr 75.5%

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

   6. Taylor expanded in t around inf 89.5%

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

      1. clear-num 89.5%

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

      2. sqrt-div 89.5%

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

      3. metadata-eval 89.5%

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

      4. +-commutative 89.5%

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

      5. sub-neg 89.5%

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

      6. metadata-eval 89.5%

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

   8. Applied egg-rr 89.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-168}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 5: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-170}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.4e-170)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 7e-212)
     (/ t (* l (sqrt (/ 1.0 x))))
     (if (<= t 7.2e-88)
       (+ 1.0 (/ -1.0 x))
       (if (<= t 4.8e-50)
         (/ t (* (/ l (sqrt 2.0)) (sqrt (/ 2.0 x))))
         (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.4e-170) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 7e-212) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else if (t <= 7.2e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 4.8e-50) {
		tmp = t / ((l / sqrt(2.0)) * sqrt((2.0 / x)));
	} else {
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.4d-170)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 7d-212) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else if (t <= 7.2d-88) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t <= 4.8d-50) then
        tmp = t / ((l / sqrt(2.0d0)) * sqrt((2.0d0 / x)))
    else
        tmp = 1.0d0 / sqrt(((x + 1.0d0) / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.4e-170) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 7e-212) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else if (t <= 7.2e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 4.8e-50) {
		tmp = t / ((l / Math.sqrt(2.0)) * Math.sqrt((2.0 / x)));
	} else {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.4e-170:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 7e-212:
		tmp = t / (l * math.sqrt((1.0 / x)))
	elif t <= 7.2e-88:
		tmp = 1.0 + (-1.0 / x)
	elif t <= 4.8e-50:
		tmp = t / ((l / math.sqrt(2.0)) * math.sqrt((2.0 / x)))
	else:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.4e-170)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 7e-212)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	elseif (t <= 7.2e-88)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t <= 4.8e-50)
		tmp = Float64(t / Float64(Float64(l / sqrt(2.0)) * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.4e-170)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 7e-212)
		tmp = t / (l * sqrt((1.0 / x)));
	elseif (t <= 7.2e-88)
		tmp = 1.0 + (-1.0 / x);
	elseif (t <= 4.8e-50)
		tmp = t / ((l / sqrt(2.0)) * sqrt((2.0 / x)));
	else
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.4e-170], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 7e-212], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e-88], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.8e-50], N[(t / N[(N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.39999999999999998e-170

   1. Initial program 38.2%

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

   3. Simplified 38.1%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. sub-neg 87.2%

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

      3. metadata-eval 87.2%

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

   8. Simplified 87.2%

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

   if -1.39999999999999998e-170 < t < 6.9999999999999995e-212

   1. Initial program 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

   5. Taylor expanded in t around 0 63.8%

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

   6. Taylor expanded in l around inf 63.2%

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

   if 6.9999999999999995e-212 < t < 7.1999999999999999e-88

   1. Initial program 35.0%

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

   3. Simplified 34.8%

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

   5. Applied egg-rr 52.4%

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

   6. Taylor expanded in t around inf 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if 7.1999999999999999e-88 < t < 4.80000000000000004e-50

   1. Initial program 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in l around inf 2.3%

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

   5. Taylor expanded in x around inf 58.4%

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

   if 4.80000000000000004e-50 < t

   1. Initial program 31.3%

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

   3. Simplified 31.2%

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

   5. Applied egg-rr 75.5%

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

   6. Taylor expanded in t around inf 89.5%

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

      1. clear-num 89.5%

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

      2. sqrt-div 89.5%

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

      3. metadata-eval 89.5%

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

      4. +-commutative 89.5%

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

      5. sub-neg 89.5%

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

      6. metadata-eval 89.5%

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

   8. Applied egg-rr 89.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-170}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 6: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-167}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (* l (sqrt (/ 1.0 x))))))
   (if (<= t -2.1e-167)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t 3.8e-206)
       t_1
       (if (<= t 7.5e-88)
         (+ 1.0 (/ -1.0 x))
         (if (<= t 4.8e-50) t_1 (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / (l * sqrt((1.0 / x)));
	double tmp;
	if (t <= -2.1e-167) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 3.8e-206) {
		tmp = t_1;
	} else if (t <= 7.5e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 4.8e-50) {
		tmp = t_1;
	} else {
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (l * sqrt((1.0d0 / x)))
    if (t <= (-2.1d-167)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 3.8d-206) then
        tmp = t_1
    else if (t <= 7.5d-88) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t <= 4.8d-50) then
        tmp = t_1
    else
        tmp = 1.0d0 / sqrt(((x + 1.0d0) / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = t / (l * Math.sqrt((1.0 / x)));
	double tmp;
	if (t <= -2.1e-167) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 3.8e-206) {
		tmp = t_1;
	} else if (t <= 7.5e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 4.8e-50) {
		tmp = t_1;
	} else {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t / (l * math.sqrt((1.0 / x)))
	tmp = 0
	if t <= -2.1e-167:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 3.8e-206:
		tmp = t_1
	elif t <= 7.5e-88:
		tmp = 1.0 + (-1.0 / x)
	elif t <= 4.8e-50:
		tmp = t_1
	else:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / Float64(l * sqrt(Float64(1.0 / x))))
	tmp = 0.0
	if (t <= -2.1e-167)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 3.8e-206)
		tmp = t_1;
	elseif (t <= 7.5e-88)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t <= 4.8e-50)
		tmp = t_1;
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t / (l * sqrt((1.0 / x)));
	tmp = 0.0;
	if (t <= -2.1e-167)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 3.8e-206)
		tmp = t_1;
	elseif (t <= 7.5e-88)
		tmp = 1.0 + (-1.0 / x);
	elseif (t <= 4.8e-50)
		tmp = t_1;
	else
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.1e-167], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 3.8e-206], t$95$1, If[LessEqual[t, 7.5e-88], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.8e-50], t$95$1, N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.10000000000000017e-167

   1. Initial program 38.2%

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

   3. Simplified 38.1%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. sub-neg 87.2%

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

      3. metadata-eval 87.2%

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

   8. Simplified 87.2%

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

   if -2.10000000000000017e-167 < t < 3.80000000000000003e-206 or 7.50000000000000041e-88 < t < 4.80000000000000004e-50

   1. Initial program 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in x around inf 62.6%

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

   5. Taylor expanded in t around 0 62.6%

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

   6. Taylor expanded in l around inf 62.1%

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

   if 3.80000000000000003e-206 < t < 7.50000000000000041e-88

   1. Initial program 35.0%

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

   3. Simplified 34.8%

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

   5. Applied egg-rr 52.4%

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

   6. Taylor expanded in t around inf 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if 4.80000000000000004e-50 < t

   1. Initial program 31.3%

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

   3. Simplified 31.2%

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

   5. Applied egg-rr 75.5%

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

   6. Taylor expanded in t around inf 89.5%

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

      1. clear-num 89.5%

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

      2. sqrt-div 89.5%

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

      3. metadata-eval 89.5%

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

      4. +-commutative 89.5%

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

      5. sub-neg 89.5%

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

      6. metadata-eval 89.5%

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

   8. Applied egg-rr 89.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-167}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-206}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 7: 79.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-169}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (* l (sqrt (/ 1.0 x))))))
   (if (<= t -2.4e-169)
     (+ -1.0 (/ 1.0 x))
     (if (<= t 2.7e-207)
       t_1
       (if (<= t 7.5e-88)
         (+ 1.0 (/ -1.0 x))
         (if (<= t 2.3e-49) t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / (l * sqrt((1.0 / x)));
	double tmp;
	if (t <= -2.4e-169) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 2.7e-207) {
		tmp = t_1;
	} else if (t <= 7.5e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 2.3e-49) {
		tmp = t_1;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (l * sqrt((1.0d0 / x)))
    if (t <= (-2.4d-169)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 2.7d-207) then
        tmp = t_1
    else if (t <= 7.5d-88) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t <= 2.3d-49) then
        tmp = t_1
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = t / (l * Math.sqrt((1.0 / x)));
	double tmp;
	if (t <= -2.4e-169) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 2.7e-207) {
		tmp = t_1;
	} else if (t <= 7.5e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 2.3e-49) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t / (l * math.sqrt((1.0 / x)))
	tmp = 0
	if t <= -2.4e-169:
		tmp = -1.0 + (1.0 / x)
	elif t <= 2.7e-207:
		tmp = t_1
	elif t <= 7.5e-88:
		tmp = 1.0 + (-1.0 / x)
	elif t <= 2.3e-49:
		tmp = t_1
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / Float64(l * sqrt(Float64(1.0 / x))))
	tmp = 0.0
	if (t <= -2.4e-169)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 2.7e-207)
		tmp = t_1;
	elseif (t <= 7.5e-88)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t <= 2.3e-49)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t / (l * sqrt((1.0 / x)));
	tmp = 0.0;
	if (t <= -2.4e-169)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 2.7e-207)
		tmp = t_1;
	elseif (t <= 7.5e-88)
		tmp = 1.0 + (-1.0 / x);
	elseif (t <= 2.3e-49)
		tmp = t_1;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.4e-169], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e-207], t$95$1, If[LessEqual[t, 7.5e-88], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e-49], t$95$1, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.40000000000000011e-169

   1. Initial program 38.2%

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

   3. Simplified 38.1%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in t around inf 1.6%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 85.7%

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

   9. Simplified 85.7%

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

   if -2.40000000000000011e-169 < t < 2.7e-207 or 7.50000000000000041e-88 < t < 2.2999999999999999e-49

   1. Initial program 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in x around inf 62.6%

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

   5. Taylor expanded in t around 0 62.6%

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

   6. Taylor expanded in l around inf 62.1%

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

   if 2.7e-207 < t < 7.50000000000000041e-88

   1. Initial program 35.0%

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

   3. Simplified 34.8%

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

   5. Applied egg-rr 52.4%

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

   6. Taylor expanded in t around inf 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if 2.2999999999999999e-49 < t

   1. Initial program 31.3%

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

   3. Simplified 31.2%

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

   5. Applied egg-rr 75.5%

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

   6. Taylor expanded in t around inf 89.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-169}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-207}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 8: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-171}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (* l (sqrt (/ 1.0 x)))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -6.5e-171)
     (- t_2)
     (if (<= t 2.6e-207)
       t_1
       (if (<= t 7.5e-88) (+ 1.0 (/ -1.0 x)) (if (<= t 4.8e-50) t_1 t_2))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / (l * sqrt((1.0 / x)));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -6.5e-171) {
		tmp = -t_2;
	} else if (t <= 2.6e-207) {
		tmp = t_1;
	} else if (t <= 7.5e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 4.8e-50) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t / (l * sqrt((1.0d0 / x)))
    t_2 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-6.5d-171)) then
        tmp = -t_2
    else if (t <= 2.6d-207) then
        tmp = t_1
    else if (t <= 7.5d-88) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t <= 4.8d-50) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = t / (l * Math.sqrt((1.0 / x)));
	double t_2 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -6.5e-171) {
		tmp = -t_2;
	} else if (t <= 2.6e-207) {
		tmp = t_1;
	} else if (t <= 7.5e-88) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t <= 4.8e-50) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t / (l * math.sqrt((1.0 / x)))
	t_2 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -6.5e-171:
		tmp = -t_2
	elif t <= 2.6e-207:
		tmp = t_1
	elif t <= 7.5e-88:
		tmp = 1.0 + (-1.0 / x)
	elif t <= 4.8e-50:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / Float64(l * sqrt(Float64(1.0 / x))))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -6.5e-171)
		tmp = Float64(-t_2);
	elseif (t <= 2.6e-207)
		tmp = t_1;
	elseif (t <= 7.5e-88)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t <= 4.8e-50)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t / (l * sqrt((1.0 / x)));
	t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -6.5e-171)
		tmp = -t_2;
	elseif (t <= 2.6e-207)
		tmp = t_1;
	elseif (t <= 7.5e-88)
		tmp = 1.0 + (-1.0 / x);
	elseif (t <= 4.8e-50)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -6.5e-171], (-t$95$2), If[LessEqual[t, 2.6e-207], t$95$1, If[LessEqual[t, 7.5e-88], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.8e-50], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.5000000000000004e-171

   1. Initial program 38.2%

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

   3. Simplified 38.1%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. sub-neg 87.2%

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

      3. metadata-eval 87.2%

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

   8. Simplified 87.2%

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

   if -6.5000000000000004e-171 < t < 2.5999999999999999e-207 or 7.50000000000000041e-88 < t < 4.80000000000000004e-50

   1. Initial program 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in x around inf 62.6%

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

   5. Taylor expanded in t around 0 62.6%

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

   6. Taylor expanded in l around inf 62.1%

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

   if 2.5999999999999999e-207 < t < 7.50000000000000041e-88

   1. Initial program 35.0%

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

   3. Simplified 34.8%

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

   5. Applied egg-rr 52.4%

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

   6. Taylor expanded in t around inf 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if 4.80000000000000004e-50 < t

   1. Initial program 31.3%

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

   3. Simplified 31.2%

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

   5. Applied egg-rr 75.5%

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

   6. Taylor expanded in t around inf 89.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-171}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-207}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 9: 76.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-311) (+ -1.0 (/ 1.0 x)) (sqrt (/ (+ x -1.0) (+ x 1.0)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-311) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-311)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-311) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-311:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-311)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-311)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-311], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.00000000000023e-311

   1. Initial program 33.7%

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

   3. Simplified 33.6%

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

   5. Applied egg-rr 65.9%

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

   6. Taylor expanded in t around inf 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 77.9%

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

   9. Simplified 77.9%

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

   if -5.00000000000023e-311 < t

   1. Initial program 29.3%

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

   3. Simplified 29.2%

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

   5. Applied egg-rr 68.0%

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

   6. Taylor expanded in t around inf 78.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 10: 76.1% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-311) (+ -1.0 (/ 1.0 x)) 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-311) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-311)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-311) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-311:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-311)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-311)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-311], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -5.00000000000023e-311

   1. Initial program 33.7%

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

   3. Simplified 33.6%

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

   5. Applied egg-rr 65.9%

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

   6. Taylor expanded in t around inf 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 77.9%

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

   9. Simplified 77.9%

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

   if -5.00000000000023e-311 < t

   1. Initial program 29.3%

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

   3. Simplified 29.2%

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

   5. Applied egg-rr 68.0%

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

   6. Taylor expanded in t around inf 78.9%

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

   7. Taylor expanded in x around inf 77.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 76.5% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-311) (+ -1.0 (/ 1.0 x)) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-311) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-311)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-311) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-311:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-311)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-311)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-311], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.00000000000023e-311

   1. Initial program 33.7%

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

   3. Simplified 33.6%

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

   5. Applied egg-rr 65.9%

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

   6. Taylor expanded in t around inf 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 77.9%

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

   9. Simplified 77.9%

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

   if -5.00000000000023e-311 < t

   1. Initial program 29.3%

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

   3. Simplified 29.2%

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

   5. Applied egg-rr 68.0%

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

   6. Taylor expanded in t around inf 78.9%

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

   7. Taylor expanded in x around inf 78.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12: 75.8% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-311) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-311) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-311)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-311) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-311:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-311)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-311)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-311], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -5.00000000000023e-311

   1. Initial program 33.7%

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

   3. Simplified 33.6%

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

   5. Applied egg-rr 65.9%

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

   6. Taylor expanded in t around inf 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 0.0%

         \[\leadsto \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

      2. rem-square-sqrt 77.1%

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

   9. Simplified 77.1%

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

   if -5.00000000000023e-311 < t

   1. Initial program 29.3%

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

   3. Simplified 29.2%

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

   5. Applied egg-rr 68.0%

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

   6. Taylor expanded in t around inf 78.9%

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

   7. Taylor expanded in x around inf 77.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 37.6% accurate, 225.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ -1 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 -1.0)

C

l = abs(l);
double code(double x, double l, double t) {
	return -1.0;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return -1.0;
}

Python

l = abs(l)
def code(x, l, t):
	return -1.0

Julia

l = abs(l)
function code(x, l, t)
	return -1.0
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := -1.0

Derivation

1. Initial program 31.7%

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

3. Simplified 31.6%

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

5. Applied egg-rr 66.8%

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

6. Taylor expanded in t around inf 37.0%

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

7. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

   2. rem-square-sqrt 42.6%

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

9. Simplified 42.6%

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

10. Final simplification 42.6%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023313 
(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)))))