Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 86.2%

Time: 26.1s

Alternatives: 16

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: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-233}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t_1\right)}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2
         (*
          t
          (/
           (sqrt 2.0)
           (pow
            (+
             (/ l (/ x l))
             (- (* 2.0 (+ (* t t) (/ (* t t) x))) (/ l (/ x (- l)))))
            0.5)))))
   (if (<= t -2.8e+135)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t 7.4e-233)
       t_2
       (if (<= t 3.4e-161)
         (*
          t
          (/
           (sqrt 2.0)
           (fma
            0.5
            (/ (* 2.0 (fma 2.0 (* t t) (* l l))) (* (sqrt 2.0) (* t x)))
            t_1)))
         (if (<= t 1.32e+73)
           t_2
           (* t (/ (sqrt 2.0) (* t_1 (sqrt (/ (+ x 1.0) (+ x -1.0))))))))))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = t * (sqrt(2.0) / pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double tmp;
	if (t <= -2.8e+135) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 7.4e-233) {
		tmp = t_2;
	} else if (t <= 3.4e-161) {
		tmp = t * (sqrt(2.0) / fma(0.5, ((2.0 * fma(2.0, (t * t), (l * l))) / (sqrt(2.0) * (t * x))), t_1));
	} else if (t <= 1.32e+73) {
		tmp = t_2;
	} else {
		tmp = t * (sqrt(2.0) / (t_1 * sqrt(((x + 1.0) / (x + -1.0)))));
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(t * Float64(sqrt(2.0) / (Float64(Float64(l / Float64(x / l)) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) - Float64(l / Float64(x / Float64(-l))))) ^ 0.5)))
	tmp = 0.0
	if (t <= -2.8e+135)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 7.4e-233)
		tmp = t_2;
	elseif (t <= 3.4e-161)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(2.0 * fma(2.0, Float64(t * t), Float64(l * l))) / Float64(sqrt(2.0) * Float64(t * x))), t_1)));
	elseif (t <= 1.32e+73)
		tmp = t_2;
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_1 * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l / N[(x / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e+135], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 7.4e-233], t$95$2, If[LessEqual[t, 3.4e-161], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(2.0 * N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.32e+73], t$95$2, N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$1 * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.80000000000000002e135

   1. Initial program 5.4%

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

      1. associate-*l/ 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. +-commutative 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-rgt-neg-out 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -2.80000000000000002e135 < t < 7.3999999999999996e-233 or 3.39999999999999982e-161 < t < 1.32e73

   1. Initial program 52.6%

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

      1. associate-*l/ 52.7%

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

   3. Simplified 52.7%

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

   4. Taylor expanded in x around inf 78.8%

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

      1. associate--l+ 78.8%

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

      2. unpow2 78.8%

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

      3. distribute-lft-out 78.8%

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

      4. unpow2 78.8%

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

      5. unpow2 78.8%

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

      6. associate-*r/ 78.8%

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

      7. mul-1-neg 78.8%

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

      8. +-commutative 78.8%

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

      9. unpow2 78.8%

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

      10. unpow2 78.8%

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

      11. fma-udef 78.8%

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

   6. Simplified 78.8%

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

   7. Taylor expanded in t around 0 77.8%

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

      1. associate-*r/ 77.8%

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

      2. neg-mul-1 77.8%

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

      3. unpow2 77.8%

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

      4. distribute-rgt-neg-in 77.8%

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

   9. Simplified 77.8%

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

       1. pow1/2 77.9%

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

       2. associate-/l* 77.8%

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

       3. +-commutative 77.8%

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

       4. associate-/l* 85.9%

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

   11. Applied egg-rr 85.9%

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

   if 7.3999999999999996e-233 < t < 3.39999999999999982e-161

   1. Initial program 2.4%

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

      1. associate-*l/ 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in x around inf 79.7%

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

      1. fma-def 79.7%

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

      2. cancel-sign-sub-inv 79.7%

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

      3. metadata-eval 79.7%

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

      4. distribute-rgt1-in 79.7%

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

      5. metadata-eval 79.7%

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

      6. +-commutative 79.7%

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

      7. unpow2 79.7%

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

      8. unpow2 79.7%

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

      9. fma-udef 79.7%

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

      10. *-commutative 79.7%

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

   6. Simplified 79.7%

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

   if 1.32e73 < t

   1. Initial program 17.2%

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

      1. associate-*l/ 17.2%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in t around inf 98.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-233}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+73}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 2: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{if}\;t \leq -2.85 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{3}}\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (pow
            (+
             (/ l (/ x l))
             (- (* 2.0 (+ (* t t) (/ (* t t) x))) (/ l (/ x (- l)))))
            0.5)))))
   (if (<= t -2.85e+135)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t -1.75e-239)
       t_1
       (if (<= t 7e-188)
         (*
          t
          (/
           (sqrt 2.0)
           (* l (sqrt (+ (/ 2.0 x) (+ (/ 2.0 (* x x)) (/ 2.0 (pow x 3.0))))))))
         (if (<= t 3.1e+73)
           t_1
           (*
            t
            (/
             (sqrt 2.0)
             (* (* t (sqrt 2.0)) (sqrt (/ (+ x 1.0) (+ x -1.0))))))))))))

C

double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double tmp;
	if (t <= -2.85e+135) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -1.75e-239) {
		tmp = t_1;
	} else if (t <= 7e-188) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + ((2.0 / (x * x)) + (2.0 / pow(x, 3.0)))))));
	} else if (t <= 3.1e+73) {
		tmp = t_1;
	} else {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * sqrt(((x + 1.0) / (x + -1.0)))));
	}
	return tmp;
}

Fortran

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 / (x / l)) + ((2.0d0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ** 0.5d0))
    if (t <= (-2.85d+135)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= (-1.75d-239)) then
        tmp = t_1
    else if (t <= 7d-188) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((2.0d0 / x) + ((2.0d0 / (x * x)) + (2.0d0 / (x ** 3.0d0)))))))
    else if (t <= 3.1d+73) then
        tmp = t_1
    else
        tmp = t * (sqrt(2.0d0) / ((t * sqrt(2.0d0)) * sqrt(((x + 1.0d0) / (x + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(2.0) / Math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double tmp;
	if (t <= -2.85e+135) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -1.75e-239) {
		tmp = t_1;
	} else if (t <= 7e-188) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 / x) + ((2.0 / (x * x)) + (2.0 / Math.pow(x, 3.0)))))));
	} else if (t <= 3.1e+73) {
		tmp = t_1;
	} else {
		tmp = t * (Math.sqrt(2.0) / ((t * Math.sqrt(2.0)) * Math.sqrt(((x + 1.0) / (x + -1.0)))));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * (math.sqrt(2.0) / math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5))
	tmp = 0
	if t <= -2.85e+135:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= -1.75e-239:
		tmp = t_1
	elif t <= 7e-188:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((2.0 / x) + ((2.0 / (x * x)) + (2.0 / math.pow(x, 3.0)))))))
	elif t <= 3.1e+73:
		tmp = t_1
	else:
		tmp = t * (math.sqrt(2.0) / ((t * math.sqrt(2.0)) * math.sqrt(((x + 1.0) / (x + -1.0)))))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / (Float64(Float64(l / Float64(x / l)) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) - Float64(l / Float64(x / Float64(-l))))) ^ 0.5)))
	tmp = 0.0
	if (t <= -2.85e+135)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= -1.75e-239)
		tmp = t_1;
	elseif (t <= 7e-188)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / (x ^ 3.0))))))));
	elseif (t <= 3.1e+73)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(2.0) / (((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ^ 0.5));
	tmp = 0.0;
	if (t <= -2.85e+135)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= -1.75e-239)
		tmp = t_1;
	elseif (t <= 7e-188)
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + ((2.0 / (x * x)) + (2.0 / (x ^ 3.0)))))));
	elseif (t <= 3.1e+73)
		tmp = t_1;
	else
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * sqrt(((x + 1.0) / (x + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l / N[(x / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.85e+135], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -1.75e-239], t$95$1, If[LessEqual[t, 7e-188], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.1e+73], t$95$1, N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.8500000000000001e135

   1. Initial program 5.4%

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

      1. associate-*l/ 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. +-commutative 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-rgt-neg-out 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -2.8500000000000001e135 < t < -1.75000000000000003e-239 or 7.000000000000001e-188 < t < 3.1e73

   1. Initial program 58.2%

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

      1. associate-*l/ 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in x around inf 78.1%

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

      1. associate--l+ 78.1%

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

      2. unpow2 78.1%

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

      3. distribute-lft-out 78.1%

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

      4. unpow2 78.1%

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

      5. unpow2 78.1%

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

      6. associate-*r/ 78.1%

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

      7. mul-1-neg 78.1%

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

      8. +-commutative 78.1%

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

      9. unpow2 78.1%

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

      10. unpow2 78.1%

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

      11. fma-udef 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in t around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

      3. unpow2 77.0%

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

      4. distribute-rgt-neg-in 77.0%

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

   9. Simplified 77.0%

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

       1. pow1/2 77.1%

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

       2. associate-/l* 77.1%

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

       3. +-commutative 77.1%

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

       4. associate-/l* 86.1%

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

   11. Applied egg-rr 86.1%

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

   if -1.75000000000000003e-239 < t < 7.000000000000001e-188

   1. Initial program 3.7%

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

      1. associate-*l/ 3.7%

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

   3. Simplified 3.7%

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

   4. Taylor expanded in l around inf 3.4%

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

   5. Taylor expanded in x around inf 59.5%

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

      1. associate-*r/ 59.5%

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

      2. metadata-eval 59.5%

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

      3. +-commutative 59.5%

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

      4. associate-*r/ 59.5%

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

      5. metadata-eval 59.5%

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

      6. unpow2 59.5%

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

      7. associate-*r/ 59.5%

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

      8. metadata-eval 59.5%

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

   7. Simplified 59.5%

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

   if 3.1e73 < t

   1. Initial program 17.2%

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

      1. associate-*l/ 17.2%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in t around inf 98.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-239}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{3}}\right)}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+73}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 3: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.85 \cdot 10^{+135}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -1.78 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t_2 \cdot \frac{\sqrt{2} \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (pow
            (+
             (/ l (/ x l))
             (- (* 2.0 (+ (* t t) (/ (* t t) x))) (/ l (/ x (- l)))))
            0.5))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -2.85e+135)
     (- t_2)
     (if (<= t -1.78e-239)
       t_1
       (if (<= t 2.05e-188)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 7.5e+151)
           t_1
           (* t (* t_2 (/ (* (sqrt 2.0) (sqrt 0.5)) t)))))))))

C

double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -2.85e+135) {
		tmp = -t_2;
	} else if (t <= -1.78e-239) {
		tmp = t_1;
	} else if (t <= 2.05e-188) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 7.5e+151) {
		tmp = t_1;
	} else {
		tmp = t * (t_2 * ((sqrt(2.0) * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (sqrt(2.0d0) / (((l / (x / l)) + ((2.0d0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ** 0.5d0))
    t_2 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-2.85d+135)) then
        tmp = -t_2
    else if (t <= (-1.78d-239)) then
        tmp = t_1
    else if (t <= 2.05d-188) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((2.0d0 / x) + (2.0d0 / (x * x))))))
    else if (t <= 7.5d+151) then
        tmp = t_1
    else
        tmp = t * (t_2 * ((sqrt(2.0d0) * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(2.0) / Math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double t_2 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -2.85e+135) {
		tmp = -t_2;
	} else if (t <= -1.78e-239) {
		tmp = t_1;
	} else if (t <= 2.05e-188) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 7.5e+151) {
		tmp = t_1;
	} else {
		tmp = t * (t_2 * ((Math.sqrt(2.0) * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * (math.sqrt(2.0) / math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5))
	t_2 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -2.85e+135:
		tmp = -t_2
	elif t <= -1.78e-239:
		tmp = t_1
	elif t <= 2.05e-188:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((2.0 / x) + (2.0 / (x * x))))))
	elif t <= 7.5e+151:
		tmp = t_1
	else:
		tmp = t * (t_2 * ((math.sqrt(2.0) * math.sqrt(0.5)) / t))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / (Float64(Float64(l / Float64(x / l)) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) - Float64(l / Float64(x / Float64(-l))))) ^ 0.5)))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -2.85e+135)
		tmp = Float64(-t_2);
	elseif (t <= -1.78e-239)
		tmp = t_1;
	elseif (t <= 2.05e-188)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 7.5e+151)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(t_2 * Float64(Float64(sqrt(2.0) * sqrt(0.5)) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(2.0) / (((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ^ 0.5));
	t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -2.85e+135)
		tmp = -t_2;
	elseif (t <= -1.78e-239)
		tmp = t_1;
	elseif (t <= 2.05e-188)
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	elseif (t <= 7.5e+151)
		tmp = t_1;
	else
		tmp = t * (t_2 * ((sqrt(2.0) * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l / N[(x / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.85e+135], (-t$95$2), If[LessEqual[t, -1.78e-239], t$95$1, If[LessEqual[t, 2.05e-188], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e+151], t$95$1, N[(t * N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.8500000000000001e135

   1. Initial program 5.4%

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

      1. associate-*l/ 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. +-commutative 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-rgt-neg-out 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -2.8500000000000001e135 < t < -1.78e-239 or 2.04999999999999991e-188 < t < 7.49999999999999977e151

   1. Initial program 58.9%

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

      1. associate-*l/ 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in x around inf 78.0%

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

      1. associate--l+ 78.0%

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

      2. unpow2 78.0%

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

      3. distribute-lft-out 78.0%

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

      4. unpow2 78.0%

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

      5. unpow2 78.0%

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

      6. associate-*r/ 78.0%

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

      7. mul-1-neg 78.0%

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

      8. +-commutative 78.0%

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

      9. unpow2 78.0%

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

      10. unpow2 78.0%

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

      11. fma-udef 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in t around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

      3. unpow2 77.0%

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

      4. distribute-rgt-neg-in 77.0%

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

   9. Simplified 77.0%

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

       1. pow1/2 77.1%

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

       2. associate-/l* 77.1%

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

       3. +-commutative 77.1%

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

       4. associate-/l* 87.2%

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

   11. Applied egg-rr 87.2%

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

   if -1.78e-239 < t < 2.04999999999999991e-188

   1. Initial program 3.7%

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

      1. associate-*l/ 3.7%

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

   3. Simplified 3.7%

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

   4. Taylor expanded in l around inf 3.4%

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

   5. Taylor expanded in x around inf 59.5%

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

      1. +-commutative 59.5%

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

      2. associate-*r/ 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*r/ 59.5%

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

      5. metadata-eval 59.5%

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

      6. unpow2 59.5%

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

   7. Simplified 59.5%

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

   if 7.49999999999999977e151 < t

   1. Initial program 2.4%

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

      1. associate-*l/ 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in t around inf 96.3%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.78 \cdot 10^{-239}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \frac{\sqrt{2} \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.78 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-189}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (pow
            (+
             (/ l (/ x l))
             (- (* 2.0 (+ (* t t) (/ (* t t) x))) (/ l (/ x (- l)))))
            0.5)))))
   (if (<= t -2.8e+135)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t -1.78e-239)
       t_1
       (if (<= t 7e-189)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 2.9e+73)
           t_1
           (*
            t
            (/
             (sqrt 2.0)
             (* (* t (sqrt 2.0)) (sqrt (/ (+ x 1.0) (+ x -1.0))))))))))))

C

double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double tmp;
	if (t <= -2.8e+135) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -1.78e-239) {
		tmp = t_1;
	} else if (t <= 7e-189) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 2.9e+73) {
		tmp = t_1;
	} else {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * sqrt(((x + 1.0) / (x + -1.0)))));
	}
	return tmp;
}

Fortran

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 / (x / l)) + ((2.0d0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ** 0.5d0))
    if (t <= (-2.8d+135)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= (-1.78d-239)) then
        tmp = t_1
    else if (t <= 7d-189) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((2.0d0 / x) + (2.0d0 / (x * x))))))
    else if (t <= 2.9d+73) then
        tmp = t_1
    else
        tmp = t * (sqrt(2.0d0) / ((t * sqrt(2.0d0)) * sqrt(((x + 1.0d0) / (x + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(2.0) / Math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double tmp;
	if (t <= -2.8e+135) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -1.78e-239) {
		tmp = t_1;
	} else if (t <= 7e-189) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 2.9e+73) {
		tmp = t_1;
	} else {
		tmp = t * (Math.sqrt(2.0) / ((t * Math.sqrt(2.0)) * Math.sqrt(((x + 1.0) / (x + -1.0)))));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * (math.sqrt(2.0) / math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5))
	tmp = 0
	if t <= -2.8e+135:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= -1.78e-239:
		tmp = t_1
	elif t <= 7e-189:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((2.0 / x) + (2.0 / (x * x))))))
	elif t <= 2.9e+73:
		tmp = t_1
	else:
		tmp = t * (math.sqrt(2.0) / ((t * math.sqrt(2.0)) * math.sqrt(((x + 1.0) / (x + -1.0)))))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / (Float64(Float64(l / Float64(x / l)) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) - Float64(l / Float64(x / Float64(-l))))) ^ 0.5)))
	tmp = 0.0
	if (t <= -2.8e+135)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= -1.78e-239)
		tmp = t_1;
	elseif (t <= 7e-189)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 2.9e+73)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(2.0) / (((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ^ 0.5));
	tmp = 0.0;
	if (t <= -2.8e+135)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= -1.78e-239)
		tmp = t_1;
	elseif (t <= 7e-189)
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	elseif (t <= 2.9e+73)
		tmp = t_1;
	else
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * sqrt(((x + 1.0) / (x + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l / N[(x / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e+135], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -1.78e-239], t$95$1, If[LessEqual[t, 7e-189], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.9e+73], t$95$1, N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.80000000000000002e135

   1. Initial program 5.4%

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

      1. associate-*l/ 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. +-commutative 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-rgt-neg-out 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -2.80000000000000002e135 < t < -1.78e-239 or 7.0000000000000003e-189 < t < 2.9000000000000002e73

   1. Initial program 58.2%

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

      1. associate-*l/ 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in x around inf 78.1%

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

      1. associate--l+ 78.1%

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

      2. unpow2 78.1%

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

      3. distribute-lft-out 78.1%

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

      4. unpow2 78.1%

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

      5. unpow2 78.1%

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

      6. associate-*r/ 78.1%

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

      7. mul-1-neg 78.1%

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

      8. +-commutative 78.1%

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

      9. unpow2 78.1%

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

      10. unpow2 78.1%

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

      11. fma-udef 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in t around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

      3. unpow2 77.0%

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

      4. distribute-rgt-neg-in 77.0%

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

   9. Simplified 77.0%

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

       1. pow1/2 77.1%

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

       2. associate-/l* 77.1%

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

       3. +-commutative 77.1%

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

       4. associate-/l* 86.1%

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

   11. Applied egg-rr 86.1%

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

   if -1.78e-239 < t < 7.0000000000000003e-189

   1. Initial program 3.7%

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

      1. associate-*l/ 3.7%

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

   3. Simplified 3.7%

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

   4. Taylor expanded in l around inf 3.4%

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

   5. Taylor expanded in x around inf 59.5%

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

      1. +-commutative 59.5%

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

      2. associate-*r/ 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*r/ 59.5%

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

      5. metadata-eval 59.5%

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

      6. unpow2 59.5%

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

   7. Simplified 59.5%

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

   if 2.9000000000000002e73 < t

   1. Initial program 17.2%

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

      1. associate-*l/ 17.2%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in t around inf 98.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.78 \cdot 10^{-239}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-189}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+73}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 5: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -1.34 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(t_2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (pow
            (+
             (/ l (/ x l))
             (- (* 2.0 (+ (* t t) (/ (* t t) x))) (/ l (/ x (- l)))))
            0.5))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -2.8e+135)
     (- t_2)
     (if (<= t -1.34e-244)
       t_1
       (if (<= t 7e-188)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 1.45e+146) t_1 (* (sqrt 2.0) (* t_2 (sqrt 0.5)))))))))

C

double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -2.8e+135) {
		tmp = -t_2;
	} else if (t <= -1.34e-244) {
		tmp = t_1;
	} else if (t <= 7e-188) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 1.45e+146) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) * (t_2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (sqrt(2.0d0) / (((l / (x / l)) + ((2.0d0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ** 0.5d0))
    t_2 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-2.8d+135)) then
        tmp = -t_2
    else if (t <= (-1.34d-244)) then
        tmp = t_1
    else if (t <= 7d-188) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((2.0d0 / x) + (2.0d0 / (x * x))))))
    else if (t <= 1.45d+146) then
        tmp = t_1
    else
        tmp = sqrt(2.0d0) * (t_2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(2.0) / Math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double t_2 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -2.8e+135) {
		tmp = -t_2;
	} else if (t <= -1.34e-244) {
		tmp = t_1;
	} else if (t <= 7e-188) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 1.45e+146) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(2.0) * (t_2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * (math.sqrt(2.0) / math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5))
	t_2 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -2.8e+135:
		tmp = -t_2
	elif t <= -1.34e-244:
		tmp = t_1
	elif t <= 7e-188:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((2.0 / x) + (2.0 / (x * x))))))
	elif t <= 1.45e+146:
		tmp = t_1
	else:
		tmp = math.sqrt(2.0) * (t_2 * math.sqrt(0.5))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / (Float64(Float64(l / Float64(x / l)) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) - Float64(l / Float64(x / Float64(-l))))) ^ 0.5)))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -2.8e+135)
		tmp = Float64(-t_2);
	elseif (t <= -1.34e-244)
		tmp = t_1;
	elseif (t <= 7e-188)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 1.45e+146)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) * Float64(t_2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(2.0) / (((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ^ 0.5));
	t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -2.8e+135)
		tmp = -t_2;
	elseif (t <= -1.34e-244)
		tmp = t_1;
	elseif (t <= 7e-188)
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	elseif (t <= 1.45e+146)
		tmp = t_1;
	else
		tmp = sqrt(2.0) * (t_2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l / N[(x / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.8e+135], (-t$95$2), If[LessEqual[t, -1.34e-244], t$95$1, If[LessEqual[t, 7e-188], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+146], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.80000000000000002e135

   1. Initial program 5.4%

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

      1. associate-*l/ 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. +-commutative 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-rgt-neg-out 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -2.80000000000000002e135 < t < -1.34e-244 or 7.000000000000001e-188 < t < 1.4499999999999999e146

   1. Initial program 58.9%

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

      1. associate-*l/ 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in x around inf 78.0%

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

      1. associate--l+ 78.0%

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

      2. unpow2 78.0%

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

      3. distribute-lft-out 78.0%

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

      4. unpow2 78.0%

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

      5. unpow2 78.0%

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

      6. associate-*r/ 78.0%

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

      7. mul-1-neg 78.0%

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

      8. +-commutative 78.0%

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

      9. unpow2 78.0%

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

      10. unpow2 78.0%

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

      11. fma-udef 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in t around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

      3. unpow2 77.0%

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

      4. distribute-rgt-neg-in 77.0%

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

   9. Simplified 77.0%

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

       1. pow1/2 77.1%

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

       2. associate-/l* 77.1%

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

       3. +-commutative 77.1%

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

       4. associate-/l* 87.2%

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

   11. Applied egg-rr 87.2%

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

   if -1.34e-244 < t < 7.000000000000001e-188

   1. Initial program 3.7%

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

      1. associate-*l/ 3.7%

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

   3. Simplified 3.7%

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

   4. Taylor expanded in l around inf 3.4%

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

   5. Taylor expanded in x around inf 59.5%

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

      1. +-commutative 59.5%

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

      2. associate-*r/ 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*r/ 59.5%

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

      5. metadata-eval 59.5%

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

      6. unpow2 59.5%

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

   7. Simplified 59.5%

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

   if 1.4499999999999999e146 < t

   1. Initial program 2.4%

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

      1. associate-*l/ 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in t around inf 96.3%

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

      1. associate-*l* 96.3%

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

      2. sub-neg 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

   6. Simplified 96.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.34 \cdot 10^{-244}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+146}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 6: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.78 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-189}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (pow
            (+
             (/ l (/ x l))
             (- (* 2.0 (+ (* t t) (/ (* t t) x))) (/ l (/ x (- l)))))
            0.5)))))
   (if (<= t -2.8e+135)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t -1.78e-239)
       t_1
       (if (<= t 4.2e-189)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 2.25e+70) t_1 (- (/ -1.0 x) -1.0)))))))

C

double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double tmp;
	if (t <= -2.8e+135) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -1.78e-239) {
		tmp = t_1;
	} else if (t <= 4.2e-189) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 2.25e+70) {
		tmp = t_1;
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Fortran

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 / (x / l)) + ((2.0d0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ** 0.5d0))
    if (t <= (-2.8d+135)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= (-1.78d-239)) then
        tmp = t_1
    else if (t <= 4.2d-189) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((2.0d0 / x) + (2.0d0 / (x * x))))))
    else if (t <= 2.25d+70) then
        tmp = t_1
    else
        tmp = ((-1.0d0) / x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(2.0) / Math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5));
	double tmp;
	if (t <= -2.8e+135) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -1.78e-239) {
		tmp = t_1;
	} else if (t <= 4.2e-189) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 2.25e+70) {
		tmp = t_1;
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * (math.sqrt(2.0) / math.pow(((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))), 0.5))
	tmp = 0
	if t <= -2.8e+135:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= -1.78e-239:
		tmp = t_1
	elif t <= 4.2e-189:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((2.0 / x) + (2.0 / (x * x))))))
	elif t <= 2.25e+70:
		tmp = t_1
	else:
		tmp = (-1.0 / x) - -1.0
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / (Float64(Float64(l / Float64(x / l)) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) - Float64(l / Float64(x / Float64(-l))))) ^ 0.5)))
	tmp = 0.0
	if (t <= -2.8e+135)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= -1.78e-239)
		tmp = t_1;
	elseif (t <= 4.2e-189)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 2.25e+70)
		tmp = t_1;
	else
		tmp = Float64(Float64(-1.0 / x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(2.0) / (((l / (x / l)) + ((2.0 * ((t * t) + ((t * t) / x))) - (l / (x / -l)))) ^ 0.5));
	tmp = 0.0;
	if (t <= -2.8e+135)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= -1.78e-239)
		tmp = t_1;
	elseif (t <= 4.2e-189)
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	elseif (t <= 2.25e+70)
		tmp = t_1;
	else
		tmp = (-1.0 / x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l / N[(x / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e+135], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -1.78e-239], t$95$1, If[LessEqual[t, 4.2e-189], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.25e+70], t$95$1, N[(N[(-1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.80000000000000002e135

   1. Initial program 5.4%

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

      1. associate-*l/ 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. +-commutative 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-rgt-neg-out 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -2.80000000000000002e135 < t < -1.78e-239 or 4.20000000000000033e-189 < t < 2.25e70

   1. Initial program 58.0%

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

      1. associate-*l/ 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in x around inf 77.7%

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

      1. associate--l+ 77.7%

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

      2. unpow2 77.7%

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

      3. distribute-lft-out 77.7%

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

      4. unpow2 77.7%

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

      5. unpow2 77.7%

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

      6. associate-*r/ 77.7%

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

      7. mul-1-neg 77.7%

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

      8. +-commutative 77.7%

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

      9. unpow2 77.7%

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

      10. unpow2 77.7%

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

      11. fma-udef 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in t around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. neg-mul-1 76.5%

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

      3. unpow2 76.5%

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

      4. distribute-rgt-neg-in 76.5%

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

   9. Simplified 76.5%

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

       1. pow1/2 76.6%

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

       2. associate-/l* 76.6%

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

       3. +-commutative 76.6%

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

       4. associate-/l* 85.8%

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

   11. Applied egg-rr 85.8%

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

   if -1.78e-239 < t < 4.20000000000000033e-189

   1. Initial program 3.7%

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

      1. associate-*l/ 3.7%

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

   3. Simplified 3.7%

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

   4. Taylor expanded in l around inf 3.4%

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

   5. Taylor expanded in x around inf 59.5%

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

      1. +-commutative 59.5%

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

      2. associate-*r/ 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*r/ 59.5%

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

      5. metadata-eval 59.5%

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

      6. unpow2 59.5%

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

   7. Simplified 59.5%

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

   if 2.25e70 < t

   1. Initial program 19.7%

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

      1. associate-*l/ 19.7%

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

   3. Simplified 19.7%

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

   4. Taylor expanded in t around -inf 1.6%

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

      1. mul-1-neg 1.6%

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

      2. *-commutative 1.6%

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

      3. distribute-rgt-neg-in 1.6%

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

      4. +-commutative 1.6%

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

      5. sub-neg 1.6%

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

      6. metadata-eval 1.6%

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

      7. +-commutative 1.6%

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

      8. distribute-rgt-neg-out 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in t around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. +-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 97.0%

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

   12. Simplified 97.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.78 \cdot 10^{-239}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-189}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+70}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \]

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-140}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-223}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -7.5e-140)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 4.4e-223)
     (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
     (- (/ -1.0 x) -1.0))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -7.5e-140) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 4.4e-223) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Fortran

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 <= (-7.5d-140)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 4.4d-223) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((2.0d0 / x) + (2.0d0 / (x * x))))))
    else
        tmp = ((-1.0d0) / x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -7.5e-140) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 4.4e-223) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -7.5e-140:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 4.4e-223:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((2.0 / x) + (2.0 / (x * x))))))
	else:
		tmp = (-1.0 / x) - -1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -7.5e-140)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 4.4e-223)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	else
		tmp = Float64(Float64(-1.0 / x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -7.5e-140)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 4.4e-223)
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	else
		tmp = (-1.0 / x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -7.5e-140], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 4.4e-223], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.4999999999999998e-140

   1. Initial program 49.1%

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

      1. associate-*l/ 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in t around -inf 91.3%

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

      1. mul-1-neg 91.3%

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

      2. *-commutative 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

      4. +-commutative 91.3%

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

      5. sub-neg 91.3%

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

      6. metadata-eval 91.3%

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

      7. +-commutative 91.3%

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

      8. distribute-rgt-neg-out 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in t around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. sub-neg 91.6%

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

      3. metadata-eval 91.6%

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

      4. +-commutative 91.6%

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

   9. Simplified 91.6%

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

   if -7.4999999999999998e-140 < t < 4.40000000000000018e-223

   1. Initial program 4.4%

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

      1. associate-*l/ 4.4%

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

   3. Simplified 4.4%

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

   4. Taylor expanded in l around inf 3.0%

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

   5. Taylor expanded in x around inf 55.2%

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

      1. +-commutative 55.2%

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

      2. associate-*r/ 55.2%

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

      3. metadata-eval 55.2%

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

      4. associate-*r/ 55.2%

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

      5. metadata-eval 55.2%

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

      6. unpow2 55.2%

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

   7. Simplified 55.2%

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

   if 4.40000000000000018e-223 < t

   1. Initial program 36.1%

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

      1. associate-*l/ 36.2%

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

   3. Simplified 36.2%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. distribute-rgt-neg-in 1.7%

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

      4. +-commutative 1.7%

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

      5. sub-neg 1.7%

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

      6. metadata-eval 1.7%

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

      7. +-commutative 1.7%

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

      8. distribute-rgt-neg-out 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-140}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-223}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \]

Alternative 8: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-222}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.06e-141)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 1.9e-222) (* t (/ (sqrt x) l)) (- (/ -1.0 x) -1.0))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.06e-141) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.9e-222) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.06d-141)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 1.9d-222) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = ((-1.0d0) / x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.06e-141) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.9e-222) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.06e-141:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 1.9e-222:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = (-1.0 / x) - -1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.06e-141)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 1.9e-222)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = Float64(Float64(-1.0 / x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.06e-141)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 1.9e-222)
		tmp = t * (sqrt(x) / l);
	else
		tmp = (-1.0 / x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.06e-141], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 1.9e-222], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.06e-141

   1. Initial program 49.1%

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

      1. associate-*l/ 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in t around -inf 91.3%

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

      1. mul-1-neg 91.3%

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

      2. *-commutative 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

      4. +-commutative 91.3%

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

      5. sub-neg 91.3%

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

      6. metadata-eval 91.3%

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

      7. +-commutative 91.3%

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

      8. distribute-rgt-neg-out 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in t around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. sub-neg 91.6%

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

      3. metadata-eval 91.6%

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

      4. +-commutative 91.6%

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

   9. Simplified 91.6%

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

   if -1.06e-141 < t < 1.89999999999999998e-222

   1. Initial program 4.4%

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

      1. associate-*l/ 4.4%

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

   3. Simplified 4.4%

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

   4. Taylor expanded in x around inf 69.6%

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

      1. associate--l+ 69.6%

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

      2. unpow2 69.6%

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

      3. distribute-lft-out 69.6%

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

      4. unpow2 69.6%

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

      5. unpow2 69.6%

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

      6. associate-*r/ 69.6%

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

      7. mul-1-neg 69.6%

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

      8. +-commutative 69.6%

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

      9. unpow2 69.6%

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

      10. unpow2 69.6%

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

      11. fma-udef 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in l around inf 54.4%

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

      1. associate-*l* 54.4%

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

   9. Simplified 54.4%

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

   10. Taylor expanded in l around 0 54.5%

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

       1. associate-*l/ 54.4%

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

       2. *-lft-identity 54.4%

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

   12. Simplified 54.4%

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

   if 1.89999999999999998e-222 < t

   1. Initial program 36.1%

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

      1. associate-*l/ 36.2%

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

   3. Simplified 36.2%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. distribute-rgt-neg-in 1.7%

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

      4. +-commutative 1.7%

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

      5. sub-neg 1.7%

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

      6. metadata-eval 1.7%

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

      7. +-commutative 1.7%

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

      8. distribute-rgt-neg-out 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-222}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \]

Alternative 9: 76.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-213}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -3.6e-213)
   (+ -1.0 (- (/ 1.0 x) (/ 0.5 (* x x))))
   (if (<= t 3.2e-223) (* (sqrt x) (/ t l)) (- (/ -1.0 x) -1.0))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.6e-213) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else if (t <= 3.2e-223) {
		tmp = sqrt(x) * (t / l);
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Fortran

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 <= (-3.6d-213)) then
        tmp = (-1.0d0) + ((1.0d0 / x) - (0.5d0 / (x * x)))
    else if (t <= 3.2d-223) then
        tmp = sqrt(x) * (t / l)
    else
        tmp = ((-1.0d0) / x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.6e-213) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else if (t <= 3.2e-223) {
		tmp = Math.sqrt(x) * (t / l);
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -3.6e-213:
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)))
	elif t <= 3.2e-223:
		tmp = math.sqrt(x) * (t / l)
	else:
		tmp = (-1.0 / x) - -1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -3.6e-213)
		tmp = Float64(-1.0 + Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))));
	elseif (t <= 3.2e-223)
		tmp = Float64(sqrt(x) * Float64(t / l));
	else
		tmp = Float64(Float64(-1.0 / x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.6e-213)
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	elseif (t <= 3.2e-223)
		tmp = sqrt(x) * (t / l);
	else
		tmp = (-1.0 / x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -3.6e-213], N[(-1.0 + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e-223], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.6000000000000001e-213

   1. Initial program 44.9%

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

      1. associate-*l/ 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in t around -inf 86.4%

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

      1. mul-1-neg 86.4%

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

      2. *-commutative 86.4%

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

      3. distribute-rgt-neg-in 86.4%

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

      4. +-commutative 86.4%

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

      5. sub-neg 86.4%

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

      6. metadata-eval 86.4%

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

      7. +-commutative 86.4%

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

      8. distribute-rgt-neg-out 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in t around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. sub-neg 86.8%

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

      3. metadata-eval 86.8%

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

      4. +-commutative 86.8%

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

   9. Simplified 86.8%

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

   10. Taylor expanded in x around inf 86.1%

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

       1. associate--l+ 86.1%

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

       2. associate-*r/ 86.1%

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

       3. metadata-eval 86.1%

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

       4. unpow2 86.1%

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

   12. Simplified 86.1%

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

   if -3.6000000000000001e-213 < t < 3.2000000000000001e-223

   1. Initial program 3.6%

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

      1. associate-*l/ 3.6%

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

   3. Simplified 3.6%

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

   4. Taylor expanded in x around inf 74.4%

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

      1. associate--l+ 74.4%

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

      2. unpow2 74.4%

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

      3. distribute-lft-out 74.4%

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

      4. unpow2 74.4%

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

      5. unpow2 74.4%

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

      6. associate-*r/ 74.4%

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

      7. mul-1-neg 74.4%

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

      8. +-commutative 74.4%

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

      9. unpow2 74.4%

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

      10. unpow2 74.4%

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

      11. fma-udef 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in l around inf 62.8%

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

      1. associate-*l* 62.8%

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

   9. Simplified 62.8%

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

   10. Taylor expanded in l around 0 53.2%

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

   if 3.2000000000000001e-223 < t

   1. Initial program 36.1%

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

      1. associate-*l/ 36.2%

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

   3. Simplified 36.2%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. distribute-rgt-neg-in 1.7%

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

      4. +-commutative 1.7%

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

      5. sub-neg 1.7%

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

      6. metadata-eval 1.7%

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

      7. +-commutative 1.7%

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

      8. distribute-rgt-neg-out 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-213}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \]

Alternative 10: 76.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-137}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-222}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.1e-137)
   (+ -1.0 (- (/ 1.0 x) (/ 0.5 (* x x))))
   (if (<= t 1.9e-222) (* t (/ (sqrt x) l)) (- (/ -1.0 x) -1.0))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.1e-137) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else if (t <= 1.9e-222) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.1d-137)) then
        tmp = (-1.0d0) + ((1.0d0 / x) - (0.5d0 / (x * x)))
    else if (t <= 1.9d-222) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = ((-1.0d0) / x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.1e-137) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else if (t <= 1.9e-222) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.1e-137:
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)))
	elif t <= 1.9e-222:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = (-1.0 / x) - -1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.1e-137)
		tmp = Float64(-1.0 + Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))));
	elseif (t <= 1.9e-222)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = Float64(Float64(-1.0 / x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.1e-137)
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	elseif (t <= 1.9e-222)
		tmp = t * (sqrt(x) / l);
	else
		tmp = (-1.0 / x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.1e-137], N[(-1.0 + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-222], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1000000000000001e-137

   1. Initial program 49.1%

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

      1. associate-*l/ 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in t around -inf 91.3%

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

      1. mul-1-neg 91.3%

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

      2. *-commutative 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

      4. +-commutative 91.3%

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

      5. sub-neg 91.3%

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

      6. metadata-eval 91.3%

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

      7. +-commutative 91.3%

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

      8. distribute-rgt-neg-out 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in t around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. sub-neg 91.6%

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

      3. metadata-eval 91.6%

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

      4. +-commutative 91.6%

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

   9. Simplified 91.6%

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

   10. Taylor expanded in x around inf 90.8%

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

       1. associate--l+ 90.9%

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

       2. associate-*r/ 90.9%

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

       3. metadata-eval 90.9%

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

       4. unpow2 90.9%

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

   12. Simplified 90.9%

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

   if -1.1000000000000001e-137 < t < 1.89999999999999998e-222

   1. Initial program 4.4%

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

      1. associate-*l/ 4.4%

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

   3. Simplified 4.4%

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

   4. Taylor expanded in x around inf 69.6%

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

      1. associate--l+ 69.6%

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

      2. unpow2 69.6%

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

      3. distribute-lft-out 69.6%

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

      4. unpow2 69.6%

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

      5. unpow2 69.6%

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

      6. associate-*r/ 69.6%

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

      7. mul-1-neg 69.6%

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

      8. +-commutative 69.6%

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

      9. unpow2 69.6%

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

      10. unpow2 69.6%

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

      11. fma-udef 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in l around inf 54.4%

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

      1. associate-*l* 54.4%

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

   9. Simplified 54.4%

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

   10. Taylor expanded in l around 0 54.5%

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

       1. associate-*l/ 54.4%

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

       2. *-lft-identity 54.4%

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

   12. Simplified 54.4%

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

   if 1.89999999999999998e-222 < t

   1. Initial program 36.1%

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

      1. associate-*l/ 36.2%

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

   3. Simplified 36.2%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. distribute-rgt-neg-in 1.7%

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

      4. +-commutative 1.7%

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

      5. sub-neg 1.7%

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

      6. metadata-eval 1.7%

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

      7. +-commutative 1.7%

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

      8. distribute-rgt-neg-out 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-137}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-222}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \]

Alternative 11: 76.6% accurate, 17.2× speedup

Math

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

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-310)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (- (/ -1.0 x) -1.0)))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Fortran

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 <= (-1d-310)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else
        tmp = ((-1.0d0) / x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-310:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	else:
		tmp = (-1.0 / x) - -1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-310)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(Float64(-1.0 / x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-310)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	else
		tmp = (-1.0 / x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-310], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.999999999999969e-311

   1. Initial program 41.1%

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

      1. associate-*l/ 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in t around -inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. distribute-rgt-neg-in 79.2%

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

      4. +-commutative 79.2%

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

      5. sub-neg 79.2%

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

      6. metadata-eval 79.2%

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

      7. +-commutative 79.2%

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

      8. distribute-rgt-neg-out 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in x around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. metadata-eval 78.9%

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

      3. unpow2 78.9%

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

   9. Simplified 78.9%

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

   if -9.999999999999969e-311 < t

   1. Initial program 32.1%

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

      1. associate-*l/ 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in t around -inf 1.8%

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

      1. mul-1-neg 1.8%

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

      2. *-commutative 1.8%

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

      3. distribute-rgt-neg-in 1.8%

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

      4. +-commutative 1.8%

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

      5. sub-neg 1.8%

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

      6. metadata-eval 1.8%

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

      7. +-commutative 1.8%

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

      8. distribute-rgt-neg-out 1.8%

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

   6. Simplified 1.8%

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

   7. Taylor expanded in t around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

      4. +-commutative 1.8%

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

   9. Simplified 1.8%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 74.0%

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

   12. Simplified 74.0%

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

4. Final simplification 76.1%

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

Alternative 12: 76.6% accurate, 17.2× speedup

Math

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

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-310)
   (+ -1.0 (- (/ 1.0 x) (/ 0.5 (* x x))))
   (- (/ -1.0 x) -1.0)))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Fortran

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 <= (-1d-310)) then
        tmp = (-1.0d0) + ((1.0d0 / x) - (0.5d0 / (x * x)))
    else
        tmp = ((-1.0d0) / x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-310:
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)))
	else:
		tmp = (-1.0 / x) - -1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-310)
		tmp = Float64(-1.0 + Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(Float64(-1.0 / x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-310)
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	else
		tmp = (-1.0 / x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-310], N[(-1.0 + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.999999999999969e-311

   1. Initial program 41.1%

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

      1. associate-*l/ 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in t around -inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. distribute-rgt-neg-in 79.2%

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

      4. +-commutative 79.2%

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

      5. sub-neg 79.2%

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

      6. metadata-eval 79.2%

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

      7. +-commutative 79.2%

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

      8. distribute-rgt-neg-out 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in t around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. sub-neg 79.5%

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

      3. metadata-eval 79.5%

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

      4. +-commutative 79.5%

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

   9. Simplified 79.5%

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

   10. Taylor expanded in x around inf 78.9%

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

       1. associate--l+ 78.9%

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

       2. associate-*r/ 78.9%

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

       3. metadata-eval 78.9%

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

       4. unpow2 78.9%

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

   12. Simplified 78.9%

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

   if -9.999999999999969e-311 < t

   1. Initial program 32.1%

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

      1. associate-*l/ 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in t around -inf 1.8%

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

      1. mul-1-neg 1.8%

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

      2. *-commutative 1.8%

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

      3. distribute-rgt-neg-in 1.8%

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

      4. +-commutative 1.8%

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

      5. sub-neg 1.8%

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

      6. metadata-eval 1.8%

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

      7. +-commutative 1.8%

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

      8. distribute-rgt-neg-out 1.8%

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

   6. Simplified 1.8%

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

   7. Taylor expanded in t around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

      4. +-commutative 1.8%

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

   9. Simplified 1.8%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 74.0%

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

   12. Simplified 74.0%

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

4. Final simplification 76.1%

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

Alternative 13: 76.1% accurate, 31.9× speedup

Math

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

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-310) (- -1.0 (/ -1.0 x)) 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = -1.0 - (-1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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 <= (-1d-310)) then
        tmp = (-1.0d0) - ((-1.0d0) / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = -1.0 - (-1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-310:
		tmp = -1.0 - (-1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-310)
		tmp = Float64(-1.0 - Float64(-1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-310)
		tmp = -1.0 - (-1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-310], N[(-1.0 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -9.999999999999969e-311

   1. Initial program 41.1%

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

      1. associate-*l/ 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in t around -inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. distribute-rgt-neg-in 79.2%

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

      4. +-commutative 79.2%

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

      5. sub-neg 79.2%

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

      6. metadata-eval 79.2%

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

      7. +-commutative 79.2%

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

      8. distribute-rgt-neg-out 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in x around inf 78.5%

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

   if -9.999999999999969e-311 < t

   1. Initial program 32.1%

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

      1. associate-*l/ 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in x around inf 71.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 72.6%

         \[\leadsto \color{blue}{\sqrt{2 \cdot 0.5}} \]

      2. metadata-eval 72.6%

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

      3. metadata-eval 72.6%

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

   6. Applied egg-rr 72.6%

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

4. Final simplification 75.2%

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

Alternative 14: 76.5% accurate, 31.9× speedup

Math

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

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-310) (- -1.0 (/ -1.0 x)) (- (/ -1.0 x) -1.0)))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = -1.0 - (-1.0 / x);
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Fortran

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 <= (-1d-310)) then
        tmp = (-1.0d0) - ((-1.0d0) / x)
    else
        tmp = ((-1.0d0) / x) - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = -1.0 - (-1.0 / x);
	} else {
		tmp = (-1.0 / x) - -1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-310:
		tmp = -1.0 - (-1.0 / x)
	else:
		tmp = (-1.0 / x) - -1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-310)
		tmp = Float64(-1.0 - Float64(-1.0 / x));
	else
		tmp = Float64(Float64(-1.0 / x) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-310)
		tmp = -1.0 - (-1.0 / x);
	else
		tmp = (-1.0 / x) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-310], N[(-1.0 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.999999999999969e-311

   1. Initial program 41.1%

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

      1. associate-*l/ 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in t around -inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. distribute-rgt-neg-in 79.2%

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

      4. +-commutative 79.2%

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

      5. sub-neg 79.2%

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

      6. metadata-eval 79.2%

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

      7. +-commutative 79.2%

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

      8. distribute-rgt-neg-out 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in x around inf 78.5%

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

   if -9.999999999999969e-311 < t

   1. Initial program 32.1%

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

      1. associate-*l/ 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in t around -inf 1.8%

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

      1. mul-1-neg 1.8%

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

      2. *-commutative 1.8%

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

      3. distribute-rgt-neg-in 1.8%

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

      4. +-commutative 1.8%

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

      5. sub-neg 1.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      6. metadata-eval 1.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      7. +-commutative 1.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      8. distribute-rgt-neg-out 1.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)}} \cdot t \]

   6. Simplified 1.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}} \cdot t \]

   7. Taylor expanded in t around 0 1.8%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 1.8%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.8%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 1.8%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 1.8%

         \[\leadsto -\sqrt{\frac{x + -1}{\color{blue}{x + 1}}} \]

   9. Simplified 1.8%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{x + 1}}} \]

   10. Taylor expanded in x around -inf 0.0%

       \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 0.0%

          \[\leadsto -\color{blue}{\left(\frac{1}{x} + {\left(\sqrt{-1}\right)}^{2}\right)} \]

       2. unpow2 0.0%

          \[\leadsto -\left(\frac{1}{x} + \color{blue}{\sqrt{-1} \cdot \sqrt{-1}}\right) \]

       3. rem-square-sqrt 74.0%

          \[\leadsto -\left(\frac{1}{x} + \color{blue}{-1}\right) \]

   12. Simplified 74.0%

       \[\leadsto -\color{blue}{\left(\frac{1}{x} + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} - -1\\ \end{array} \]

Alternative 15: 75.8% accurate, 73.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (if (<= t -1e-310) -1.0 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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 <= (-1d-310)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -9.999999999999969e-311

   1. Initial program 41.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 41.1%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 41.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around -inf 79.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \cdot t \]
   5. Step-by-step derivation

      1. mul-1-neg 79.2%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

      2. *-commutative 79.2%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \cdot t \]

      3. distribute-rgt-neg-in 79.2%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)}} \cdot t \]

      4. +-commutative 79.2%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      5. sub-neg 79.2%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      6. metadata-eval 79.2%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      7. +-commutative 79.2%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

      8. distribute-rgt-neg-out 79.2%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)}} \cdot t \]

   6. Simplified 79.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}} \cdot t \]

   7. Taylor expanded in x around inf 77.9%

      \[\leadsto \color{blue}{-1} \]

   if -9.999999999999969e-311 < t

   1. Initial program 32.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 32.1%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 71.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 72.6%

         \[\leadsto \color{blue}{\sqrt{2 \cdot 0.5}} \]

      2. metadata-eval 72.6%

         \[\leadsto \sqrt{\color{blue}{1}} \]

      3. metadata-eval 72.6%

         \[\leadsto \color{blue}{1} \]

   6. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16: 39.2% accurate, 225.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 -1.0)

C

double code(double x, double l, double t) {
	return -1.0;
}

Fortran

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

public static double code(double x, double l, double t) {
	return -1.0;
}

Python

def code(x, l, t):
	return -1.0

Julia

function code(x, l, t)
	return -1.0
end

MATLAB

function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

code[x_, l_, t_] := -1.0

Derivation

1. Initial program 36.0%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 36.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 36.1%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

4. Taylor expanded in t around -inf 35.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \cdot t \]
5. Step-by-step derivation

   1. mul-1-neg 35.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   2. *-commutative 35.6%

      \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \cdot t \]

   3. distribute-rgt-neg-in 35.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)}} \cdot t \]

   4. +-commutative 35.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

   5. sub-neg 35.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

   6. metadata-eval 35.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

   7. +-commutative 35.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-\sqrt{2} \cdot t\right)} \cdot t \]

   8. distribute-rgt-neg-out 35.6%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)}} \cdot t \]

6. Simplified 35.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}} \cdot t \]

7. Taylor expanded in x around inf 35.1%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 35.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023275 
(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)))))