Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 84.7%

Time: 23.5s

Alternatives: 11

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.9% 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, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+28}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-206}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             (/ (* l l) x)
             (+
              (* 2.0 (+ (* t t) (/ (* t t) x)))
              (/ (fma 2.0 (* t t) (* l l)) x))))))))
   (if (<= t -1.6e+28)
     (- -1.0 (- (/ 0.5 (* x x)) (/ 1.0 x)))
     (if (<= t -1.45e-162)
       t_1
       (if (<= t 2.7e-206)
         (*
          t
          (* (/ (sqrt 2.0) l) (sqrt (/ 1.0 (+ (/ 2.0 (* x x)) (/ 2.0 x))))))
         (if (<= t 1.15e+32) t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(2.0, (t * t), (l * l)) / x)))));
	double tmp;
	if (t <= -1.6e+28) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -1.45e-162) {
		tmp = t_1;
	} else if (t <= 2.7e-206) {
		tmp = t * ((sqrt(2.0) / l) * sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))));
	} else if (t <= 1.15e+32) {
		tmp = t_1;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))))))
	tmp = 0.0
	if (t <= -1.6e+28)
		tmp = Float64(-1.0 - Float64(Float64(0.5 / Float64(x * x)) - Float64(1.0 / x)));
	elseif (t <= -1.45e-162)
		tmp = t_1;
	elseif (t <= 2.7e-206)
		tmp = Float64(t * Float64(Float64(sqrt(2.0) / l) * sqrt(Float64(1.0 / Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x))))));
	elseif (t <= 1.15e+32)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.6e+28], N[(-1.0 - N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.45e-162], t$95$1, If[LessEqual[t, 2.7e-206], N[(t * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e+32], t$95$1, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.6e28

   1. Initial program 39.3%

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

      1. associate-*r/ 39.2%

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

      2. fma-neg 39.2%

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

      3. remove-double-neg 39.2%

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

      4. fma-neg 39.2%

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

      5. sub-neg 39.2%

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

      6. metadata-eval 39.2%

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

      7. remove-double-neg 39.2%

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

      8. +-commutative 39.2%

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

      9. fma-def 39.2%

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

   3. Simplified 39.2%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in t around -inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. sub-neg 97.2%

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

      3. metadata-eval 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in x around inf 97.2%

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

       1. associate--l+ 97.2%

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

       2. associate-*r/ 97.2%

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

       3. metadata-eval 97.2%

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

       4. unpow2 97.2%

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

   11. Simplified 97.2%

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

   if -1.6e28 < t < -1.4500000000000001e-162 or 2.7000000000000001e-206 < t < 1.15e32

   1. Initial program 39.5%

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

      1. associate-*l/ 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in x around inf 91.1%

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

      1. associate--l+ 91.1%

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

      2. unpow2 91.1%

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

      3. distribute-lft-out 91.1%

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

      4. unpow2 91.1%

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

      5. unpow2 91.1%

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

      6. associate-*r/ 91.1%

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

      7. mul-1-neg 91.1%

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

      8. unpow2 91.1%

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

      9. +-commutative 91.1%

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

      10. unpow2 91.1%

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

      11. fma-udef 91.1%

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

   6. Simplified 91.1%

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

   if -1.4500000000000001e-162 < t < 2.7000000000000001e-206

   1. Initial program 3.2%

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

      1. associate-*l/ 3.2%

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

   3. Simplified 3.2%

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

   4. Taylor expanded in l around inf 2.8%

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

   5. Taylor expanded in x around inf 49.7%

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

      1. associate-*r/ 49.7%

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

      2. metadata-eval 49.7%

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

      3. unpow2 49.7%

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

      4. associate-*r/ 49.7%

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

      5. metadata-eval 49.7%

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

   7. Simplified 49.7%

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

   if 1.15e32 < t

   1. Initial program 37.1%

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

      1. associate-*r/ 37.0%

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

      2. fma-neg 37.0%

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

      3. remove-double-neg 37.0%

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

      4. fma-neg 37.0%

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

      5. sub-neg 37.0%

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

      6. metadata-eval 37.0%

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

      7. remove-double-neg 37.0%

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

      8. +-commutative 37.0%

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

      9. fma-def 37.0%

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

   3. Simplified 37.0%

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

   5. Applied egg-rr 82.7%

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

   6. Taylor expanded in t around inf 93.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+28}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-206}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+32}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 2: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \sqrt{2} \cdot \frac{t}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-201}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+36}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x))
        (t_2
         (*
          (sqrt 2.0)
          (/ t (sqrt (+ t_1 (+ t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))))))))
   (if (<= t -3.7e+25)
     (- -1.0 (- (/ 0.5 (* x x)) (/ 1.0 x)))
     (if (<= t -2.2e-162)
       t_2
       (if (<= t 1.25e-201)
         (*
          t
          (* (/ (sqrt 2.0) l) (sqrt (/ 1.0 (+ (/ 2.0 (* x x)) (/ 2.0 x))))))
         (if (<= t 1.35e+36) t_2 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double t_2 = sqrt(2.0) * (t / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -3.7e+25) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -2.2e-162) {
		tmp = t_2;
	} else if (t <= 1.25e-201) {
		tmp = t * ((sqrt(2.0) / l) * sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))));
	} else if (t <= 1.35e+36) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (l * l) / x
    t_2 = sqrt(2.0d0) * (t / sqrt((t_1 + (t_1 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    if (t <= (-3.7d+25)) then
        tmp = (-1.0d0) - ((0.5d0 / (x * x)) - (1.0d0 / x))
    else if (t <= (-2.2d-162)) then
        tmp = t_2
    else if (t <= 1.25d-201) then
        tmp = t * ((sqrt(2.0d0) / l) * sqrt((1.0d0 / ((2.0d0 / (x * x)) + (2.0d0 / x)))))
    else if (t <= 1.35d+36) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-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 t_1 = (l * l) / x;
	double t_2 = Math.sqrt(2.0) * (t / Math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -3.7e+25) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -2.2e-162) {
		tmp = t_2;
	} else if (t <= 1.25e-201) {
		tmp = t * ((Math.sqrt(2.0) / l) * Math.sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))));
	} else if (t <= 1.35e+36) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (l * l) / x
	t_2 = math.sqrt(2.0) * (t / math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))))
	tmp = 0
	if t <= -3.7e+25:
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x))
	elif t <= -2.2e-162:
		tmp = t_2
	elif t <= 1.25e-201:
		tmp = t * ((math.sqrt(2.0) / l) * math.sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))))
	elif t <= 1.35e+36:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	t_2 = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(t_1 + Float64(t_1 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))))
	tmp = 0.0
	if (t <= -3.7e+25)
		tmp = Float64(-1.0 - Float64(Float64(0.5 / Float64(x * x)) - Float64(1.0 / x)));
	elseif (t <= -2.2e-162)
		tmp = t_2;
	elseif (t <= 1.25e-201)
		tmp = Float64(t * Float64(Float64(sqrt(2.0) / l) * sqrt(Float64(1.0 / Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x))))));
	elseif (t <= 1.35e+36)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (l * l) / x;
	t_2 = sqrt(2.0) * (t / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	tmp = 0.0;
	if (t <= -3.7e+25)
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	elseif (t <= -2.2e-162)
		tmp = t_2;
	elseif (t <= 1.25e-201)
		tmp = t * ((sqrt(2.0) / l) * sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))));
	elseif (t <= 1.35e+36)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	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[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(t$95$1 + N[(t$95$1 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.7e+25], N[(-1.0 - N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.2e-162], t$95$2, If[LessEqual[t, 1.25e-201], N[(t * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+36], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.6999999999999999e25

   1. Initial program 39.3%

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

      1. associate-*r/ 39.2%

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

      2. fma-neg 39.2%

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

      3. remove-double-neg 39.2%

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

      4. fma-neg 39.2%

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

      5. sub-neg 39.2%

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

      6. metadata-eval 39.2%

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

      7. remove-double-neg 39.2%

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

      8. +-commutative 39.2%

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

      9. fma-def 39.2%

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

   3. Simplified 39.2%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in t around -inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. sub-neg 97.2%

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

      3. metadata-eval 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in x around inf 97.2%

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

       1. associate--l+ 97.2%

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

       2. associate-*r/ 97.2%

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

       3. metadata-eval 97.2%

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

       4. unpow2 97.2%

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

   11. Simplified 97.2%

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

   if -3.6999999999999999e25 < t < -2.1999999999999999e-162 or 1.25e-201 < t < 1.35e36

   1. Initial program 40.5%

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

      1. associate-*r/ 40.3%

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

      2. fma-neg 40.4%

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

      3. remove-double-neg 40.4%

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

      4. fma-neg 40.3%

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

      5. sub-neg 40.3%

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

      6. metadata-eval 40.3%

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

      7. remove-double-neg 40.3%

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

      8. +-commutative 40.3%

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

      9. fma-def 40.3%

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

   3. Simplified 40.3%

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

   4. Taylor expanded in x around inf 90.6%

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

      1. associate--l+ 90.9%

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

      2. unpow2 90.9%

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

      3. distribute-lft-out 90.9%

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

      4. unpow2 90.9%

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

      5. unpow2 90.9%

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

      6. associate-*r/ 90.9%

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

      7. mul-1-neg 90.9%

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

      8. unpow2 90.9%

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

      9. +-commutative 90.9%

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

      10. unpow2 90.9%

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

      11. fma-udef 90.9%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in t around 0 90.6%

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

      1. unpow2 90.6%

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

      2. associate-*r/ 90.6%

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

      3. mul-1-neg 90.6%

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

   9. Simplified 90.6%

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

   if -2.1999999999999999e-162 < t < 1.25e-201

   1. Initial program 3.2%

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

      1. associate-*l/ 3.2%

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

   3. Simplified 3.2%

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

   4. Taylor expanded in l around inf 2.8%

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

   5. Taylor expanded in x around inf 49.7%

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

      1. associate-*r/ 49.7%

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

      2. metadata-eval 49.7%

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

      3. unpow2 49.7%

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

      4. associate-*r/ 49.7%

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

      5. metadata-eval 49.7%

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

   7. Simplified 49.7%

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

   if 1.35e36 < t

   1. Initial program 37.1%

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

      1. associate-*r/ 37.0%

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

      2. fma-neg 37.0%

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

      3. remove-double-neg 37.0%

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

      4. fma-neg 37.0%

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

      5. sub-neg 37.0%

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

      6. metadata-eval 37.0%

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

      7. remove-double-neg 37.0%

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

      8. +-commutative 37.0%

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

      9. fma-def 37.0%

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

   3. Simplified 37.0%

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

   5. Applied egg-rr 82.7%

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

   6. Taylor expanded in t around inf 93.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-201}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-150}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-231}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (* (/ (sqrt 2.0) l) (sqrt (/ 1.0 (+ (/ 2.0 (* x x)) (/ 2.0 x))))))))
   (if (<= t -1.6e-150)
     (- -1.0 (- (/ 0.5 (* x x)) (/ 1.0 x)))
     (if (<= t -1.5e-194)
       t_1
       (if (<= t -3e-231)
         (+ (/ 1.0 x) -1.0)
         (if (<= t 4e-203) t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * ((sqrt(2.0) / l) * sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))));
	double tmp;
	if (t <= -1.6e-150) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -1.5e-194) {
		tmp = t_1;
	} else if (t <= -3e-231) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 4e-203) {
		tmp = t_1;
	} else {
		tmp = sqrt(((x + -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) :: t_1
    real(8) :: tmp
    t_1 = t * ((sqrt(2.0d0) / l) * sqrt((1.0d0 / ((2.0d0 / (x * x)) + (2.0d0 / x)))))
    if (t <= (-1.6d-150)) then
        tmp = (-1.0d0) - ((0.5d0 / (x * x)) - (1.0d0 / x))
    else if (t <= (-1.5d-194)) then
        tmp = t_1
    else if (t <= (-3d-231)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 4d-203) then
        tmp = t_1
    else
        tmp = sqrt(((x + (-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 t_1 = t * ((Math.sqrt(2.0) / l) * Math.sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))));
	double tmp;
	if (t <= -1.6e-150) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -1.5e-194) {
		tmp = t_1;
	} else if (t <= -3e-231) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 4e-203) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * ((math.sqrt(2.0) / l) * math.sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))))
	tmp = 0
	if t <= -1.6e-150:
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x))
	elif t <= -1.5e-194:
		tmp = t_1
	elif t <= -3e-231:
		tmp = (1.0 / x) + -1.0
	elif t <= 4e-203:
		tmp = t_1
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(Float64(sqrt(2.0) / l) * sqrt(Float64(1.0 / Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x))))))
	tmp = 0.0
	if (t <= -1.6e-150)
		tmp = Float64(-1.0 - Float64(Float64(0.5 / Float64(x * x)) - Float64(1.0 / x)));
	elseif (t <= -1.5e-194)
		tmp = t_1;
	elseif (t <= -3e-231)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 4e-203)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t * ((sqrt(2.0) / l) * sqrt((1.0 / ((2.0 / (x * x)) + (2.0 / x)))));
	tmp = 0.0;
	if (t <= -1.6e-150)
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	elseif (t <= -1.5e-194)
		tmp = t_1;
	elseif (t <= -3e-231)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 4e-203)
		tmp = t_1;
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	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[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.6e-150], N[(-1.0 - N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.5e-194], t$95$1, If[LessEqual[t, -3e-231], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 4e-203], t$95$1, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.5999999999999999e-150

   1. Initial program 40.9%

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

      1. associate-*r/ 40.8%

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

      2. fma-neg 40.8%

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

      3. remove-double-neg 40.8%

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

      4. fma-neg 40.8%

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

      5. sub-neg 40.8%

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

      6. metadata-eval 40.8%

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

      7. remove-double-neg 40.8%

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

      8. +-commutative 40.8%

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

      9. fma-def 40.8%

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

   3. Simplified 40.8%

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

   5. Applied egg-rr 73.0%

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

   6. Taylor expanded in t around -inf 88.2%

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

      1. mul-1-neg 88.2%

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

      2. sub-neg 88.2%

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

      3. metadata-eval 88.2%

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

   8. Simplified 88.2%

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

   9. Taylor expanded in x around inf 88.2%

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

       1. associate--l+ 88.2%

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

       2. associate-*r/ 88.2%

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

       3. metadata-eval 88.2%

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

       4. unpow2 88.2%

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

   11. Simplified 88.2%

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

   if -1.5999999999999999e-150 < t < -1.5e-194 or -3.0000000000000003e-231 < t < 4.0000000000000001e-203

   1. Initial program 4.2%

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

      1. associate-*l/ 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in l around inf 4.2%

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

   5. Taylor expanded in x around inf 56.1%

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

      1. associate-*r/ 56.1%

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

      2. metadata-eval 56.1%

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

      3. unpow2 56.1%

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

      4. associate-*r/ 56.1%

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

      5. metadata-eval 56.1%

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

   7. Simplified 56.1%

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

   if -1.5e-194 < t < -3.0000000000000003e-231

   1. Initial program 3.0%

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

      1. associate-*r/ 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. +-commutative 3.0%

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

      9. fma-def 3.0%

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

   3. Simplified 3.0%

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

   5. Applied egg-rr 41.9%

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

   6. Taylor expanded in t around -inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. sub-neg 79.3%

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

      3. metadata-eval 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in x around inf 79.3%

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

   if 4.0000000000000001e-203 < t

   1. Initial program 38.3%

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

      1. associate-*r/ 38.2%

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

      2. fma-neg 38.2%

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

      3. remove-double-neg 38.2%

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

      4. fma-neg 38.2%

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

      5. sub-neg 38.2%

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

      6. metadata-eval 38.2%

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

      7. remove-double-neg 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-def 38.2%

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

   3. Simplified 38.2%

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

   5. Applied egg-rr 68.3%

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

   6. Taylor expanded in t around inf 86.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-150}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-194}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-231}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-203}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 4: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-150}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) (/ t (sqrt (* 2.0 (* l (/ l x))))))))
   (if (<= t -1.55e-150)
     (- -1.0 (- (/ 0.5 (* x x)) (/ 1.0 x)))
     (if (<= t -1.5e-194)
       t_1
       (if (<= t -6.5e-233)
         (+ (/ 1.0 x) -1.0)
         (if (<= t 1.15e-201) t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) * (t / sqrt((2.0 * (l * (l / x)))));
	double tmp;
	if (t <= -1.55e-150) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -1.5e-194) {
		tmp = t_1;
	} else if (t <= -6.5e-233) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.15e-201) {
		tmp = t_1;
	} else {
		tmp = sqrt(((x + -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) :: t_1
    real(8) :: tmp
    t_1 = sqrt(2.0d0) * (t / sqrt((2.0d0 * (l * (l / x)))))
    if (t <= (-1.55d-150)) then
        tmp = (-1.0d0) - ((0.5d0 / (x * x)) - (1.0d0 / x))
    else if (t <= (-1.5d-194)) then
        tmp = t_1
    else if (t <= (-6.5d-233)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 1.15d-201) then
        tmp = t_1
    else
        tmp = sqrt(((x + (-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 t_1 = Math.sqrt(2.0) * (t / Math.sqrt((2.0 * (l * (l / x)))));
	double tmp;
	if (t <= -1.55e-150) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -1.5e-194) {
		tmp = t_1;
	} else if (t <= -6.5e-233) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.15e-201) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(2.0) * (t / math.sqrt((2.0 * (l * (l / x)))))
	tmp = 0
	if t <= -1.55e-150:
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x))
	elif t <= -1.5e-194:
		tmp = t_1
	elif t <= -6.5e-233:
		tmp = (1.0 / x) + -1.0
	elif t <= 1.15e-201:
		tmp = t_1
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(2.0 * Float64(l * Float64(l / x))))))
	tmp = 0.0
	if (t <= -1.55e-150)
		tmp = Float64(-1.0 - Float64(Float64(0.5 / Float64(x * x)) - Float64(1.0 / x)));
	elseif (t <= -1.5e-194)
		tmp = t_1;
	elseif (t <= -6.5e-233)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 1.15e-201)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(2.0) * (t / sqrt((2.0 * (l * (l / x)))));
	tmp = 0.0;
	if (t <= -1.55e-150)
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	elseif (t <= -1.5e-194)
		tmp = t_1;
	elseif (t <= -6.5e-233)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 1.15e-201)
		tmp = t_1;
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	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[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.55e-150], N[(-1.0 - N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.5e-194], t$95$1, If[LessEqual[t, -6.5e-233], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 1.15e-201], t$95$1, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.54999999999999999e-150

   1. Initial program 40.9%

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

      1. associate-*r/ 40.8%

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

      2. fma-neg 40.8%

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

      3. remove-double-neg 40.8%

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

      4. fma-neg 40.8%

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

      5. sub-neg 40.8%

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

      6. metadata-eval 40.8%

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

      7. remove-double-neg 40.8%

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

      8. +-commutative 40.8%

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

      9. fma-def 40.8%

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

   3. Simplified 40.8%

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

   5. Applied egg-rr 73.0%

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

   6. Taylor expanded in t around -inf 88.2%

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

      1. mul-1-neg 88.2%

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

      2. sub-neg 88.2%

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

      3. metadata-eval 88.2%

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

   8. Simplified 88.2%

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

   9. Taylor expanded in x around inf 88.2%

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

       1. associate--l+ 88.2%

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

       2. associate-*r/ 88.2%

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

       3. metadata-eval 88.2%

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

       4. unpow2 88.2%

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

   11. Simplified 88.2%

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

   if -1.54999999999999999e-150 < t < -1.5e-194 or -6.49999999999999989e-233 < t < 1.14999999999999993e-201

   1. Initial program 4.2%

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

      1. associate-*r/ 4.2%

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

      2. fma-neg 4.2%

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

      3. remove-double-neg 4.2%

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

      4. fma-neg 4.2%

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

      5. sub-neg 4.2%

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

      6. metadata-eval 4.2%

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

      7. remove-double-neg 4.2%

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

      8. +-commutative 4.2%

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

      9. fma-def 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in x around inf 65.6%

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

      1. associate--l+ 65.6%

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

      2. unpow2 65.6%

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

      3. distribute-lft-out 65.6%

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

      4. unpow2 65.6%

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

      5. unpow2 65.6%

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

      6. associate-*r/ 65.6%

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

      7. mul-1-neg 65.6%

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

      8. unpow2 65.6%

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

      9. +-commutative 65.6%

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

      10. unpow2 65.6%

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

      11. fma-udef 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in t around 0 62.9%

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

      1. cancel-sign-sub-inv 62.9%

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

      2. metadata-eval 62.9%

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

      3. distribute-rgt1-in 62.9%

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

      4. metadata-eval 62.9%

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

      5. unpow2 62.9%

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

      6. associate-*l/ 69.7%

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

      7. *-commutative 69.7%

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

   9. Simplified 69.7%

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

   if -1.5e-194 < t < -6.49999999999999989e-233

   1. Initial program 3.0%

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

      1. associate-*r/ 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. +-commutative 3.0%

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

      9. fma-def 3.0%

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

   3. Simplified 3.0%

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

   5. Applied egg-rr 41.9%

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

   6. Taylor expanded in t around -inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. sub-neg 79.3%

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

      3. metadata-eval 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in x around inf 79.3%

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

   if 1.14999999999999993e-201 < t

   1. Initial program 38.3%

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

      1. associate-*r/ 38.2%

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

      2. fma-neg 38.2%

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

      3. remove-double-neg 38.2%

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

      4. fma-neg 38.2%

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

      5. sub-neg 38.2%

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

      6. metadata-eval 38.2%

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

      7. remove-double-neg 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-def 38.2%

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

   3. Simplified 38.2%

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

   5. Applied egg-rr 68.3%

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

   6. Taylor expanded in t around inf 86.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-150}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-150}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-203}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2.7e-150)
   (- -1.0 (- (/ 0.5 (* x x)) (/ 1.0 x)))
   (if (<= t -1.5e-194)
     (* (sqrt 2.0) (/ t (sqrt (* 2.0 (* l (/ l x))))))
     (if (<= t -1.05e-235)
       (+ (/ 1.0 x) -1.0)
       (if (<= t 1.35e-203)
         (* t (* (/ (sqrt 2.0) l) (sqrt (/ 1.0 (/ 2.0 x)))))
         (sqrt (/ (+ x -1.0) (+ 1.0 x))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.7e-150) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -1.5e-194) {
		tmp = sqrt(2.0) * (t / sqrt((2.0 * (l * (l / x)))));
	} else if (t <= -1.05e-235) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.35e-203) {
		tmp = t * ((sqrt(2.0) / l) * sqrt((1.0 / (2.0 / x))));
	} else {
		tmp = sqrt(((x + -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 <= (-2.7d-150)) then
        tmp = (-1.0d0) - ((0.5d0 / (x * x)) - (1.0d0 / x))
    else if (t <= (-1.5d-194)) then
        tmp = sqrt(2.0d0) * (t / sqrt((2.0d0 * (l * (l / x)))))
    else if (t <= (-1.05d-235)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 1.35d-203) then
        tmp = t * ((sqrt(2.0d0) / l) * sqrt((1.0d0 / (2.0d0 / x))))
    else
        tmp = sqrt(((x + (-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 <= -2.7e-150) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else if (t <= -1.5e-194) {
		tmp = Math.sqrt(2.0) * (t / Math.sqrt((2.0 * (l * (l / x)))));
	} else if (t <= -1.05e-235) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.35e-203) {
		tmp = t * ((Math.sqrt(2.0) / l) * Math.sqrt((1.0 / (2.0 / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2.7e-150:
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x))
	elif t <= -1.5e-194:
		tmp = math.sqrt(2.0) * (t / math.sqrt((2.0 * (l * (l / x)))))
	elif t <= -1.05e-235:
		tmp = (1.0 / x) + -1.0
	elif t <= 1.35e-203:
		tmp = t * ((math.sqrt(2.0) / l) * math.sqrt((1.0 / (2.0 / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2.7e-150)
		tmp = Float64(-1.0 - Float64(Float64(0.5 / Float64(x * x)) - Float64(1.0 / x)));
	elseif (t <= -1.5e-194)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(2.0 * Float64(l * Float64(l / x))))));
	elseif (t <= -1.05e-235)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 1.35e-203)
		tmp = Float64(t * Float64(Float64(sqrt(2.0) / l) * sqrt(Float64(1.0 / Float64(2.0 / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -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 <= -2.7e-150)
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	elseif (t <= -1.5e-194)
		tmp = sqrt(2.0) * (t / sqrt((2.0 * (l * (l / x)))));
	elseif (t <= -1.05e-235)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 1.35e-203)
		tmp = t * ((sqrt(2.0) / l) * sqrt((1.0 / (2.0 / x))));
	else
		tmp = sqrt(((x + -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, -2.7e-150], N[(-1.0 - N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.5e-194], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.05e-235], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 1.35e-203], N[(t * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.7000000000000001e-150

   1. Initial program 40.9%

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

      1. associate-*r/ 40.8%

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

      2. fma-neg 40.8%

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

      3. remove-double-neg 40.8%

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

      4. fma-neg 40.8%

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

      5. sub-neg 40.8%

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

      6. metadata-eval 40.8%

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

      7. remove-double-neg 40.8%

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

      8. +-commutative 40.8%

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

      9. fma-def 40.8%

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

   3. Simplified 40.8%

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

   5. Applied egg-rr 73.0%

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

   6. Taylor expanded in t around -inf 88.2%

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

      1. mul-1-neg 88.2%

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

      2. sub-neg 88.2%

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

      3. metadata-eval 88.2%

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

   8. Simplified 88.2%

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

   9. Taylor expanded in x around inf 88.2%

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

       1. associate--l+ 88.2%

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

       2. associate-*r/ 88.2%

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

       3. metadata-eval 88.2%

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

       4. unpow2 88.2%

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

   11. Simplified 88.2%

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

   if -2.7000000000000001e-150 < t < -1.5e-194

   1. Initial program 5.1%

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

      1. associate-*r/ 5.1%

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

      2. fma-neg 5.4%

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

      3. remove-double-neg 5.4%

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

      4. fma-neg 5.1%

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

      5. sub-neg 5.1%

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

      6. metadata-eval 5.1%

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

      7. remove-double-neg 5.1%

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

      8. +-commutative 5.1%

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

      9. fma-def 5.1%

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

   3. Simplified 5.1%

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

   4. Taylor expanded in x around inf 67.0%

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

      1. associate--l+ 67.1%

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

      2. unpow2 67.1%

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

      3. distribute-lft-out 67.1%

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

      4. unpow2 67.1%

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

      5. unpow2 67.1%

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

      6. associate-*r/ 67.1%

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

      7. mul-1-neg 67.1%

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

      8. unpow2 67.1%

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

      9. +-commutative 67.1%

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

      10. unpow2 67.1%

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

      11. fma-udef 67.1%

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

   6. Simplified 67.0%

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

   7. Taylor expanded in t around 0 58.5%

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

      1. cancel-sign-sub-inv 58.5%

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

      2. metadata-eval 58.5%

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

      3. distribute-rgt1-in 58.5%

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

      4. metadata-eval 58.5%

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

      5. unpow2 58.5%

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

      6. associate-*l/ 81.1%

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

      7. *-commutative 81.1%

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

   9. Simplified 81.1%

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

   if -1.5e-194 < t < -1.05e-235

   1. Initial program 3.0%

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

      1. associate-*r/ 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. +-commutative 3.0%

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

      9. fma-def 3.0%

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

   3. Simplified 3.0%

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

   5. Applied egg-rr 41.9%

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

   6. Taylor expanded in t around -inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. sub-neg 79.3%

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

      3. metadata-eval 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in x around inf 79.3%

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

   if -1.05e-235 < t < 1.34999999999999999e-203

   1. Initial program 3.8%

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

      1. associate-*l/ 3.8%

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

   3. Simplified 3.8%

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

   4. Taylor expanded in l around inf 3.5%

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

   5. Taylor expanded in x around inf 50.4%

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

   if 1.34999999999999999e-203 < t

   1. Initial program 38.3%

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

      1. associate-*r/ 38.2%

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

      2. fma-neg 38.2%

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

      3. remove-double-neg 38.2%

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

      4. fma-neg 38.2%

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

      5. sub-neg 38.2%

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

      6. metadata-eval 38.2%

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

      7. remove-double-neg 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-def 38.2%

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

   3. Simplified 38.2%

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

   5. Applied egg-rr 68.3%

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

   6. Taylor expanded in t around inf 86.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-150}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-203}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 6: 77.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310)
   (- -1.0 (- (/ 0.5 (* x x)) (/ 1.0 x)))
   (sqrt (/ (+ x -1.0) (+ 1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else {
		tmp = sqrt(((x + -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-310)) then
        tmp = (-1.0d0) - ((0.5d0 / (x * x)) - (1.0d0 / x))
    else
        tmp = sqrt(((x + (-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-310) {
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 - Float64(Float64(0.5 / Float64(x * x)) - Float64(1.0 / x)));
	else
		tmp = sqrt(Float64(Float64(x + -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-310)
		tmp = -1.0 - ((0.5 / (x * x)) - (1.0 / x));
	else
		tmp = sqrt(((x + -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-310], N[(-1.0 - N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 32.0%

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

      1. associate-*r/ 31.9%

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

      2. fma-neg 31.9%

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

      3. remove-double-neg 31.9%

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

      4. fma-neg 31.9%

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

      5. sub-neg 31.9%

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

      6. metadata-eval 31.9%

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

      7. remove-double-neg 31.9%

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

      8. +-commutative 31.9%

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

      9. fma-def 31.9%

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

   3. Simplified 31.9%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in t around -inf 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

   8. Simplified 75.4%

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

   9. Taylor expanded in x around inf 75.4%

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

       1. associate--l+ 75.4%

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

       2. associate-*r/ 75.4%

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

       3. metadata-eval 75.4%

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

       4. unpow2 75.4%

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

   11. Simplified 75.4%

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

   if -4.999999999999985e-310 < t

   1. Initial program 33.9%

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

      1. associate-*r/ 33.8%

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

      2. fma-neg 33.8%

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

      3. remove-double-neg 33.8%

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

      4. fma-neg 33.8%

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

      5. sub-neg 33.8%

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

      6. metadata-eval 33.8%

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

      7. remove-double-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. fma-def 33.8%

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

   3. Simplified 33.8%

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

   5. Applied egg-rr 63.3%

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

   6. Taylor expanded in t around inf 79.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 7: 76.6% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ (/ 1.0 x) -1.0) (+ (+ 1.0 (/ 0.5 (* x x))) (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-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-310)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else
        tmp = (1.0d0 + (0.5d0 / (x * x))) + ((-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-310) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = (1.0 / x) + -1.0
	else:
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) + 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-310)
		tmp = (1.0 / x) + -1.0;
	else
		tmp = (1.0 + (0.5 / (x * x))) + (-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-310], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 32.0%

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

      1. associate-*r/ 31.9%

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

      2. fma-neg 31.9%

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

      3. remove-double-neg 31.9%

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

      4. fma-neg 31.9%

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

      5. sub-neg 31.9%

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

      6. metadata-eval 31.9%

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

      7. remove-double-neg 31.9%

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

      8. +-commutative 31.9%

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

      9. fma-def 31.9%

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

   3. Simplified 31.9%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in t around -inf 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

   8. Simplified 75.4%

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

   9. Taylor expanded in x around inf 75.3%

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

   if -4.999999999999985e-310 < t

   1. Initial program 33.9%

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

      1. associate-*l/ 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in t around inf 32.2%

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

      1. associate-/l* 36.2%

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

      2. +-commutative 36.2%

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

      3. sub-neg 36.2%

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

      4. metadata-eval 36.2%

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

      5. +-commutative 36.2%

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

      6. unpow2 36.2%

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

   6. Simplified 36.2%

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

      1. add-sqr-sqrt 35.9%

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

      2. sqrt-prod 35.7%

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

      3. times-frac 35.9%

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

      4. pow1/2 35.9%

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

      5. sqrt-pow1 35.9%

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

      6. metadata-eval 35.9%

         \[\leadsto \left(\frac{{2}^{\color{blue}{0.25}}}{\sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      7. pow1/2 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{\sqrt{\color{blue}{{2}^{0.5}}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      8. sqrt-pow1 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      9. metadata-eval 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{\color{blue}{0.25}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      10. associate-/r/ 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\color{blue}{\frac{x + 1}{-1 + x} \cdot \left(t \cdot t\right)}}}\right) \cdot t \]

      11. +-commutative 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}} \cdot \left(t \cdot t\right)}}\right) \cdot t \]

      12. sqrt-prod 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \sqrt{t \cdot t}}}\right) \cdot t \]

      13. sqrt-prod 78.7%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}}\right) \cdot t \]

      14. add-sqr-sqrt 78.9%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \color{blue}{t}}\right) \cdot t \]

   8. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot t}\right)} \cdot t \]
   9. Step-by-step derivation

      1. associate-*r/ 78.6%

         \[\leadsto \color{blue}{\frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot t}} \cdot t \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{\color{blue}{t \cdot \sqrt{\frac{x + 1}{x + -1}}}} \cdot t \]

      3. metadata-eval 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{x + 1}{x + \color{blue}{\left(-1\right)}}}} \cdot t \]

      4. sub-neg 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{x + 1}{\color{blue}{x - 1}}}} \cdot t \]

      5. +-commutative 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{\color{blue}{1 + x}}{x - 1}}} \cdot t \]

      6. sub-neg 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      7. metadata-eval 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \cdot t \]

   10. Simplified 78.6%

       \[\leadsto \color{blue}{\frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{x + -1}}}} \cdot t \]

   11. Taylor expanded in x around inf 79.3%

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

       1. sub-neg 79.3%

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

       2. associate-*r/ 79.3%

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

       3. metadata-eval 79.3%

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

       4. unpow2 79.3%

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

       5. distribute-neg-frac 79.3%

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

       6. metadata-eval 79.3%

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

   13. Simplified 79.3%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 8: 76.8% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 - \left(t_1 - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t_1\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -5e-310) (- -1.0 (- t_1 (/ 1.0 x))) (+ (+ 1.0 t_1) (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 - (t_1 - (1.0 / x));
	} else {
		tmp = (1.0 + t_1) + (-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) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) - (t_1 - (1.0d0 / x))
    else
        tmp = (1.0d0 + t_1) + ((-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 t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 - (t_1 - (1.0 / x));
	} else {
		tmp = (1.0 + t_1) + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 - (t_1 - (1.0 / x))
	else:
		tmp = (1.0 + t_1) + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 - Float64(t_1 - Float64(1.0 / x)));
	else
		tmp = Float64(Float64(1.0 + t_1) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 - (t_1 - (1.0 / x));
	else
		tmp = (1.0 + t_1) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e-310], N[(-1.0 - N[(t$95$1 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 32.0%

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

      1. associate-*r/ 31.9%

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

      2. fma-neg 31.9%

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

      3. remove-double-neg 31.9%

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

      4. fma-neg 31.9%

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

      5. sub-neg 31.9%

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

      6. metadata-eval 31.9%

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

      7. remove-double-neg 31.9%

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

      8. +-commutative 31.9%

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

      9. fma-def 31.9%

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

   3. Simplified 31.9%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in t around -inf 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

   8. Simplified 75.4%

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

   9. Taylor expanded in x around inf 75.4%

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

       1. associate--l+ 75.4%

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

       2. associate-*r/ 75.4%

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

       3. metadata-eval 75.4%

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

       4. unpow2 75.4%

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

   11. Simplified 75.4%

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

   if -4.999999999999985e-310 < t

   1. Initial program 33.9%

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

      1. associate-*l/ 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in t around inf 32.2%

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

      1. associate-/l* 36.2%

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

      2. +-commutative 36.2%

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

      3. sub-neg 36.2%

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

      4. metadata-eval 36.2%

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

      5. +-commutative 36.2%

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

      6. unpow2 36.2%

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

   6. Simplified 36.2%

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

      1. add-sqr-sqrt 35.9%

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

      2. sqrt-prod 35.7%

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

      3. times-frac 35.9%

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

      4. pow1/2 35.9%

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

      5. sqrt-pow1 35.9%

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

      6. metadata-eval 35.9%

         \[\leadsto \left(\frac{{2}^{\color{blue}{0.25}}}{\sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      7. pow1/2 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{\sqrt{\color{blue}{{2}^{0.5}}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      8. sqrt-pow1 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      9. metadata-eval 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{\color{blue}{0.25}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      10. associate-/r/ 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\color{blue}{\frac{x + 1}{-1 + x} \cdot \left(t \cdot t\right)}}}\right) \cdot t \]

      11. +-commutative 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}} \cdot \left(t \cdot t\right)}}\right) \cdot t \]

      12. sqrt-prod 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \sqrt{t \cdot t}}}\right) \cdot t \]

      13. sqrt-prod 78.7%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}}\right) \cdot t \]

      14. add-sqr-sqrt 78.9%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \color{blue}{t}}\right) \cdot t \]

   8. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot t}\right)} \cdot t \]
   9. Step-by-step derivation

      1. associate-*r/ 78.6%

         \[\leadsto \color{blue}{\frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot t}} \cdot t \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{\color{blue}{t \cdot \sqrt{\frac{x + 1}{x + -1}}}} \cdot t \]

      3. metadata-eval 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{x + 1}{x + \color{blue}{\left(-1\right)}}}} \cdot t \]

      4. sub-neg 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{x + 1}{\color{blue}{x - 1}}}} \cdot t \]

      5. +-commutative 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{\color{blue}{1 + x}}{x - 1}}} \cdot t \]

      6. sub-neg 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      7. metadata-eval 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \cdot t \]

   10. Simplified 78.6%

       \[\leadsto \color{blue}{\frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{x + -1}}}} \cdot t \]

   11. Taylor expanded in x around inf 79.3%

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

       1. sub-neg 79.3%

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

       2. associate-*r/ 79.3%

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

       3. metadata-eval 79.3%

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

       4. unpow2 79.3%

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

       5. distribute-neg-frac 79.3%

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

       6. metadata-eval 79.3%

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

   13. Simplified 79.3%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 9: 76.5% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ (/ 1.0 x) -1.0) (- 1.0 (/ 1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = (1.0 / x) + -1.0;
	} 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-310)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    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-310) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = (1.0 / x) + -1.0
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	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-310)
		tmp = (1.0 / x) + -1.0;
	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-310], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 32.0%

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

      1. associate-*r/ 31.9%

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

      2. fma-neg 31.9%

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

      3. remove-double-neg 31.9%

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

      4. fma-neg 31.9%

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

      5. sub-neg 31.9%

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

      6. metadata-eval 31.9%

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

      7. remove-double-neg 31.9%

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

      8. +-commutative 31.9%

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

      9. fma-def 31.9%

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

   3. Simplified 31.9%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in t around -inf 75.4%

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

      1. mul-1-neg 75.4%

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

      2. sub-neg 75.4%

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

      3. metadata-eval 75.4%

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

   8. Simplified 75.4%

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

   9. Taylor expanded in x around inf 75.3%

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

   if -4.999999999999985e-310 < t

   1. Initial program 33.9%

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

      1. associate-*l/ 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in t around inf 32.2%

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

      1. associate-/l* 36.2%

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

      2. +-commutative 36.2%

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

      3. sub-neg 36.2%

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

      4. metadata-eval 36.2%

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

      5. +-commutative 36.2%

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

      6. unpow2 36.2%

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

   6. Simplified 36.2%

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

      1. add-sqr-sqrt 35.9%

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

      2. sqrt-prod 35.7%

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

      3. times-frac 35.9%

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

      4. pow1/2 35.9%

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

      5. sqrt-pow1 35.9%

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

      6. metadata-eval 35.9%

         \[\leadsto \left(\frac{{2}^{\color{blue}{0.25}}}{\sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      7. pow1/2 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{\sqrt{\color{blue}{{2}^{0.5}}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      8. sqrt-pow1 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      9. metadata-eval 35.9%

         \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{\color{blue}{0.25}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

      10. associate-/r/ 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\color{blue}{\frac{x + 1}{-1 + x} \cdot \left(t \cdot t\right)}}}\right) \cdot t \]

      11. +-commutative 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}} \cdot \left(t \cdot t\right)}}\right) \cdot t \]

      12. sqrt-prod 47.0%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \sqrt{t \cdot t}}}\right) \cdot t \]

      13. sqrt-prod 78.7%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}}\right) \cdot t \]

      14. add-sqr-sqrt 78.9%

          \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \color{blue}{t}}\right) \cdot t \]

   8. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot t}\right)} \cdot t \]
   9. Step-by-step derivation

      1. associate-*r/ 78.6%

         \[\leadsto \color{blue}{\frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot t}} \cdot t \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{\color{blue}{t \cdot \sqrt{\frac{x + 1}{x + -1}}}} \cdot t \]

      3. metadata-eval 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{x + 1}{x + \color{blue}{\left(-1\right)}}}} \cdot t \]

      4. sub-neg 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{x + 1}{\color{blue}{x - 1}}}} \cdot t \]

      5. +-commutative 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{\color{blue}{1 + x}}{x - 1}}} \cdot t \]

      6. sub-neg 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

      7. metadata-eval 78.6%

         \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \cdot t \]

   10. Simplified 78.6%

       \[\leadsto \color{blue}{\frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{x + -1}}}} \cdot t \]

   11. Taylor expanded in x around inf 79.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 10: 39.5% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

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 - (1.0d0 / x)
end function

Java

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

Python

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

Julia

l = abs(l)
function code(x, l, t)
	return Float64(1.0 - Float64(1.0 / x))
end

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.9%

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

   1. associate-*l/ 32.9%

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

3. Simplified 32.9%

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

4. Taylor expanded in t around inf 30.9%

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

   1. associate-/l* 35.3%

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

   2. +-commutative 35.3%

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

   3. sub-neg 35.3%

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

   4. metadata-eval 35.3%

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

   5. +-commutative 35.3%

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

   6. unpow2 35.3%

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

6. Simplified 35.3%

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

   1. add-sqr-sqrt 35.2%

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

   2. sqrt-prod 35.0%

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

   3. times-frac 35.2%

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

   4. pow1/2 35.2%

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

   5. sqrt-pow1 35.2%

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

   6. metadata-eval 35.2%

      \[\leadsto \left(\frac{{2}^{\color{blue}{0.25}}}{\sqrt{2}} \cdot \frac{\sqrt{\sqrt{2}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

   7. pow1/2 35.2%

      \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{\sqrt{\color{blue}{{2}^{0.5}}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

   8. sqrt-pow1 35.2%

      \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

   9. metadata-eval 35.2%

      \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{\color{blue}{0.25}}}{\sqrt{\frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}\right) \cdot t \]

   10. associate-/r/ 44.3%

       \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\color{blue}{\frac{x + 1}{-1 + x} \cdot \left(t \cdot t\right)}}}\right) \cdot t \]

   11. +-commutative 44.3%

       \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}} \cdot \left(t \cdot t\right)}}\right) \cdot t \]

   12. sqrt-prod 44.3%

       \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \sqrt{t \cdot t}}}\right) \cdot t \]

   13. sqrt-prod 37.2%

       \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}}\right) \cdot t \]

   14. add-sqr-sqrt 38.2%

       \[\leadsto \left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \color{blue}{t}}\right) \cdot t \]

8. Applied egg-rr 38.2%

   \[\leadsto \color{blue}{\left(\frac{{2}^{0.25}}{\sqrt{2}} \cdot \frac{{2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot t}\right)} \cdot t \]
9. Step-by-step derivation

   1. associate-*r/ 38.1%

      \[\leadsto \color{blue}{\frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{\sqrt{\frac{x + 1}{x + -1}} \cdot t}} \cdot t \]

   2. *-commutative 38.1%

      \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{\color{blue}{t \cdot \sqrt{\frac{x + 1}{x + -1}}}} \cdot t \]

   3. metadata-eval 38.1%

      \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{x + 1}{x + \color{blue}{\left(-1\right)}}}} \cdot t \]

   4. sub-neg 38.1%

      \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{x + 1}{\color{blue}{x - 1}}}} \cdot t \]

   5. +-commutative 38.1%

      \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{\color{blue}{1 + x}}{x - 1}}} \cdot t \]

   6. sub-neg 38.1%

      \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]

   7. metadata-eval 38.1%

      \[\leadsto \frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \cdot t \]

10. Simplified 38.1%

    \[\leadsto \color{blue}{\frac{\frac{{2}^{0.25}}{\sqrt{2}} \cdot {2}^{0.25}}{t \cdot \sqrt{\frac{1 + x}{x + -1}}}} \cdot t \]

11. Taylor expanded in x around inf 38.3%

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

12. Final simplification 38.3%

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

Alternative 11: 39.2% 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 32.9%

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

   1. associate-*l/ 32.9%

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

3. Simplified 32.9%

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

4. Taylor expanded in t around inf 30.9%

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

   1. associate-/l* 35.3%

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

   2. +-commutative 35.3%

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

   3. sub-neg 35.3%

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

   4. metadata-eval 35.3%

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

   5. +-commutative 35.3%

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

   6. unpow2 35.3%

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

6. Simplified 35.3%

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

7. Taylor expanded in x around inf 38.0%

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

8. Taylor expanded in t around 0 38.1%

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

9. Final simplification 38.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023224 
(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)))))