Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 84.9%

Time: 24.3s

Alternatives: 14

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{+26}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_2 + t_3\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-168}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{t_1 + t_1}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_3 + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ (* l l) (* 2.0 (* t t))))
        (t_2 (/ (* l l) x))
        (t_3 (* 2.0 (+ (* t t) (/ (* t t) x)))))
   (if (<= t -2e+26)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t -5e-154)
       (* t (/ (sqrt 2.0) (sqrt (+ t_2 (+ t_2 t_3)))))
       (if (<= t 3e-301)
         (* (sqrt 2.0) (/ t (* (fabs l) (sqrt (/ 2.0 x)))))
         (if (<= t 4.5e-168)
           (*
            t
            (/
             (sqrt 2.0)
             (fma
              0.5
              (/ (+ t_1 t_1) (* (sqrt 2.0) (* t x)))
              (* t (sqrt 2.0)))))
           (if (<= t 8.2e+76)
             (*
              t
              (/
               (sqrt 2.0)
               (sqrt (+ t_2 (+ t_3 (/ (fma (* t 2.0) t (* l l)) x))))))
             (- 1.0 (/ 1.0 x)))))))))

C

double code(double x, double l, double t) {
	double t_1 = (l * l) + (2.0 * (t * t));
	double t_2 = (l * l) / x;
	double t_3 = 2.0 * ((t * t) + ((t * t) / x));
	double tmp;
	if (t <= -2e+26) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -5e-154) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + (t_2 + t_3))));
	} else if (t <= 3e-301) {
		tmp = sqrt(2.0) * (t / (fabs(l) * sqrt((2.0 / x))));
	} else if (t <= 4.5e-168) {
		tmp = t * (sqrt(2.0) / fma(0.5, ((t_1 + t_1) / (sqrt(2.0) * (t * x))), (t * sqrt(2.0))));
	} else if (t <= 8.2e+76) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + (t_3 + (fma((t * 2.0), t, (l * l)) / x)))));
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	tmp = 0.0
	if (t <= -2e+26)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= -5e-154)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(t_2 + t_3)))));
	elseif (t <= 3e-301)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(abs(l) * sqrt(Float64(2.0 / x)))));
	elseif (t <= 4.5e-168)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(t_1 + t_1) / Float64(sqrt(2.0) * Float64(t * x))), Float64(t * sqrt(2.0)))));
	elseif (t <= 8.2e+76)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(t_3 + Float64(fma(Float64(t * 2.0), t, Float64(l * l)) / x))))));
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+26], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -5e-154], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e-301], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e-168], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 + t$95$1), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.2e+76], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(t$95$3 + N[(N[(N[(t * 2.0), $MachinePrecision] * t + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -2.0000000000000001e26

   1. Initial program 31.7%

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

      1. associate-*r/ 31.6%

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

      2. fma-neg 31.6%

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

      3. sub-neg 31.6%

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

      4. metadata-eval 31.6%

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

      5. +-commutative 31.6%

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

      6. fma-def 31.6%

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

      7. distribute-rgt-neg-in 31.6%

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

   3. Simplified 31.6%

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

      1. associate-*r/ 31.7%

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

      2. metadata-eval 31.7%

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

      3. sub-neg 31.7%

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

      4. fma-udef 31.7%

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

      5. +-commutative 31.7%

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

      6. distribute-rgt-neg-out 31.7%

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

      7. fma-neg 31.7%

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

      8. *-commutative 31.7%

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

      9. fma-neg 31.7%

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

   5. Applied egg-rr 81.8%

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

   6. Taylor expanded in t around -inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. sub-neg 93.2%

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

      3. metadata-eval 93.2%

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

   8. Simplified 93.2%

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

   if -2.0000000000000001e26 < t < -5.0000000000000002e-154

   1. Initial program 43.3%

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

      1. associate-*l/ 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in x around inf 87.4%

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

      1. associate--l+ 87.4%

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

      2. unpow2 87.4%

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

      3. distribute-lft-out 87.4%

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

      4. unpow2 87.4%

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

      5. unpow2 87.4%

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

      6. associate-*r/ 87.4%

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

      7. mul-1-neg 87.4%

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

      8. +-commutative 87.4%

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

      9. unpow2 87.4%

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

      10. associate-*l* 87.4%

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

      11. unpow2 87.4%

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

      12. fma-udef 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in t around 0 87.4%

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

      1. unpow2 87.4%

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

      2. associate-*r/ 87.4%

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

      3. mul-1-neg 87.4%

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

      4. distribute-rgt-neg-in 87.4%

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

   9. Simplified 87.4%

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

   if -5.0000000000000002e-154 < t < 2.99999999999999999e-301

   1. Initial program 2.9%

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

      1. associate-*r/ 2.9%

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

      2. fma-neg 2.9%

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

      3. sub-neg 2.9%

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

      4. metadata-eval 2.9%

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

      5. +-commutative 2.9%

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

      6. fma-def 2.9%

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

      7. distribute-rgt-neg-in 2.9%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in l around inf 2.8%

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

      1. *-commutative 2.8%

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

      2. unpow2 2.8%

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

      3. associate--l+ 2.9%

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

      4. sub-neg 2.9%

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

      5. metadata-eval 2.9%

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

      6. +-commutative 2.9%

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

      7. sub-neg 2.9%

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

      8. metadata-eval 2.9%

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

      9. +-commutative 2.9%

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

   6. Simplified 2.9%

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

   7. Taylor expanded in x around inf 53.8%

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

      1. sqrt-prod 63.1%

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

   9. Applied egg-rr 63.1%

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

       1. rem-sqrt-square 76.4%

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

   11. Simplified 76.4%

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

   if 2.99999999999999999e-301 < t < 4.5000000000000001e-168

   1. Initial program 9.2%

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

      1. associate-*l/ 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in x around inf 50.5%

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

      1. associate--l+ 50.5%

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

      2. unpow2 50.5%

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

      3. distribute-lft-out 50.5%

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

      4. unpow2 50.5%

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

      5. unpow2 50.5%

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

      6. associate-*r/ 50.5%

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

      7. mul-1-neg 50.5%

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

      8. +-commutative 50.5%

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

      9. unpow2 50.5%

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

      10. associate-*l* 50.5%

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

      11. unpow2 50.5%

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

      12. fma-udef 50.5%

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

   6. Simplified 50.5%

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

   7. Taylor expanded in x around inf 72.3%

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

      1. fma-def 72.3%

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

      2. unpow2 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. mul-1-neg 72.3%

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

      6. unpow2 72.3%

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

      7. +-commutative 72.3%

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

      8. unpow2 72.3%

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

      9. *-commutative 72.3%

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

   9. Simplified 72.3%

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

   if 4.5000000000000001e-168 < t < 8.1999999999999997e76

   1. Initial program 57.4%

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

      1. associate-*l/ 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in x around inf 89.5%

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

      1. associate--l+ 89.5%

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

      2. unpow2 89.5%

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

      3. distribute-lft-out 89.5%

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

      4. unpow2 89.5%

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

      5. unpow2 89.5%

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

      6. associate-*r/ 89.5%

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

      7. mul-1-neg 89.5%

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

      8. +-commutative 89.5%

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

      9. unpow2 89.5%

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

      10. associate-*l* 89.5%

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

      11. unpow2 89.5%

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

      12. fma-udef 89.5%

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

   6. Simplified 89.5%

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

   if 8.1999999999999997e76 < t

   1. Initial program 30.1%

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

      1. associate-/l* 30.0%

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

      2. fma-neg 30.0%

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

      3. remove-double-neg 30.0%

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

      4. fma-neg 30.0%

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

      5. sub-neg 30.0%

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

      6. metadata-eval 30.0%

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

      7. remove-double-neg 30.0%

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

      8. fma-def 30.0%

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

   3. Simplified 30.0%

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

   4. Taylor expanded in l around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+26}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-168}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 2: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{+19}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x)))
   (if (<= t -8.2e+19)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t -5e-154)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt (+ t_1 (+ t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))))))
       (if (<= t 4.2e-107)
         (* (sqrt 2.0) (/ t (* (fabs l) (sqrt (/ 2.0 x)))))
         (+ (+ 1.0 (/ 0.5 (* x x))) (/ -1.0 x)))))))

C

double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double tmp;
	if (t <= -8.2e+19) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -5e-154) {
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else if (t <= 4.2e-107) {
		tmp = sqrt(2.0) * (t / (fabs(l) * sqrt((2.0 / x))));
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	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 = (l * l) / x
    if (t <= (-8.2d+19)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= (-5d-154)) then
        tmp = t * (sqrt(2.0d0) / sqrt((t_1 + (t_1 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    else if (t <= 4.2d-107) then
        tmp = sqrt(2.0d0) * (t / (abs(l) * sqrt((2.0d0 / x))))
    else
        tmp = (1.0d0 + (0.5d0 / (x * x))) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double tmp;
	if (t <= -8.2e+19) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -5e-154) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else if (t <= 4.2e-107) {
		tmp = Math.sqrt(2.0) * (t / (Math.abs(l) * Math.sqrt((2.0 / x))));
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (l * l) / x
	tmp = 0
	if t <= -8.2e+19:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= -5e-154:
		tmp = t * (math.sqrt(2.0) / math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))))
	elif t <= 4.2e-107:
		tmp = math.sqrt(2.0) * (t / (math.fabs(l) * math.sqrt((2.0 / x))))
	else:
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -8.2e+19)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= -5e-154)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_1 + Float64(t_1 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))));
	elseif (t <= 4.2e-107)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(abs(l) * sqrt(Float64(2.0 / x)))));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (l * l) / x;
	tmp = 0.0;
	if (t <= -8.2e+19)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= -5e-154)
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	elseif (t <= 4.2e-107)
		tmp = sqrt(2.0) * (t / (abs(l) * sqrt((2.0 / x))));
	else
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -8.2e+19], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -5e-154], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(t$95$1 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-107], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.2e19

   1. Initial program 31.7%

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

      1. associate-*r/ 31.6%

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

      2. fma-neg 31.6%

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

      3. sub-neg 31.6%

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

      4. metadata-eval 31.6%

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

      5. +-commutative 31.6%

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

      6. fma-def 31.6%

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

      7. distribute-rgt-neg-in 31.6%

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

   3. Simplified 31.6%

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

      1. associate-*r/ 31.7%

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

      2. metadata-eval 31.7%

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

      3. sub-neg 31.7%

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

      4. fma-udef 31.7%

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

      5. +-commutative 31.7%

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

      6. distribute-rgt-neg-out 31.7%

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

      7. fma-neg 31.7%

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

      8. *-commutative 31.7%

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

      9. fma-neg 31.7%

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

   5. Applied egg-rr 81.8%

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

   6. Taylor expanded in t around -inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. sub-neg 93.2%

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

      3. metadata-eval 93.2%

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

   8. Simplified 93.2%

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

   if -8.2e19 < t < -5.0000000000000002e-154

   1. Initial program 43.3%

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

      1. associate-*l/ 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in x around inf 87.4%

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

      1. associate--l+ 87.4%

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

      2. unpow2 87.4%

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

      3. distribute-lft-out 87.4%

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

      4. unpow2 87.4%

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

      5. unpow2 87.4%

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

      6. associate-*r/ 87.4%

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

      7. mul-1-neg 87.4%

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

      8. +-commutative 87.4%

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

      9. unpow2 87.4%

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

      10. associate-*l* 87.4%

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

      11. unpow2 87.4%

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

      12. fma-udef 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in t around 0 87.4%

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

      1. unpow2 87.4%

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

      2. associate-*r/ 87.4%

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

      3. mul-1-neg 87.4%

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

      4. distribute-rgt-neg-in 87.4%

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

   9. Simplified 87.4%

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

   if -5.0000000000000002e-154 < t < 4.1999999999999998e-107

   1. Initial program 5.6%

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

      1. associate-*r/ 5.6%

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

      2. fma-neg 5.4%

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

      3. sub-neg 5.4%

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

      4. metadata-eval 5.4%

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

      5. +-commutative 5.4%

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

      6. fma-def 5.4%

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

      7. distribute-rgt-neg-in 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in l around inf 6.8%

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

      1. *-commutative 6.8%

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

      2. unpow2 6.8%

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

      3. associate--l+ 7.0%

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

      4. sub-neg 7.0%

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

      5. metadata-eval 7.0%

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

      6. +-commutative 7.0%

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

      7. sub-neg 7.0%

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

      8. metadata-eval 7.0%

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

      9. +-commutative 7.0%

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

   6. Simplified 7.0%

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

   7. Taylor expanded in x around inf 54.2%

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

      1. sqrt-prod 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. rem-sqrt-square 67.0%

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

   11. Simplified 67.0%

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

   if 4.1999999999999998e-107 < t

   1. Initial program 42.7%

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

      1. associate-/l* 42.6%

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

      2. fma-neg 42.6%

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

      3. remove-double-neg 42.6%

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

      4. fma-neg 42.6%

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

      5. sub-neg 42.6%

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

      6. metadata-eval 42.6%

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

      7. remove-double-neg 42.6%

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

      8. fma-def 42.6%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in l around 0 93.2%

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

      1. +-commutative 93.2%

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

      2. sub-neg 93.2%

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

      3. metadata-eval 93.2%

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

      4. +-commutative 93.2%

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

   6. Simplified 93.2%

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

   7. Taylor expanded in x around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. metadata-eval 93.2%

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

      3. unpow2 93.2%

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

   9. Simplified 93.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+19}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+21}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+67}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x)))
   (if (<= t -8e+21)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t 1.7e+67)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt (+ t_1 (+ t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))))))
       (- 1.0 (/ 1.0 x))))))

C

double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double tmp;
	if (t <= -8e+21) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.7e+67) {
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	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 = (l * l) / x
    if (t <= (-8d+21)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 1.7d+67) then
        tmp = t * (sqrt(2.0d0) / sqrt((t_1 + (t_1 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    else
        tmp = 1.0d0 - (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double tmp;
	if (t <= -8e+21) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.7e+67) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (l * l) / x
	tmp = 0
	if t <= -8e+21:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 1.7e+67:
		tmp = t * (math.sqrt(2.0) / math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))))
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -8e+21)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 1.7e+67)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_1 + Float64(t_1 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))));
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (l * l) / x;
	tmp = 0.0;
	if (t <= -8e+21)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 1.7e+67)
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -8e+21], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 1.7e+67], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(t$95$1 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8e21

   1. Initial program 31.7%

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

      1. associate-*r/ 31.6%

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

      2. fma-neg 31.6%

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

      3. sub-neg 31.6%

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

      4. metadata-eval 31.6%

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

      5. +-commutative 31.6%

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

      6. fma-def 31.6%

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

      7. distribute-rgt-neg-in 31.6%

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

   3. Simplified 31.6%

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

      1. associate-*r/ 31.7%

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

      2. metadata-eval 31.7%

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

      3. sub-neg 31.7%

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

      4. fma-udef 31.7%

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

      5. +-commutative 31.7%

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

      6. distribute-rgt-neg-out 31.7%

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

      7. fma-neg 31.7%

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

      8. *-commutative 31.7%

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

      9. fma-neg 31.7%

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

   5. Applied egg-rr 81.8%

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

   6. Taylor expanded in t around -inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. sub-neg 93.2%

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

      3. metadata-eval 93.2%

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

   8. Simplified 93.2%

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

   if -8e21 < t < 1.7000000000000001e67

   1. Initial program 31.3%

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

      1. associate-*l/ 31.4%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in x around inf 73.3%

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

      1. associate--l+ 73.3%

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

      2. unpow2 73.3%

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

      3. distribute-lft-out 73.3%

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

      4. unpow2 73.3%

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

      5. unpow2 73.3%

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

      6. associate-*r/ 73.3%

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

      7. mul-1-neg 73.3%

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

      8. +-commutative 73.3%

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

      9. unpow2 73.3%

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

      10. associate-*l* 73.3%

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

      11. unpow2 73.3%

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

      12. fma-udef 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in t around 0 73.0%

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

      1. unpow2 73.0%

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

      2. associate-*r/ 73.0%

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

      3. mul-1-neg 73.0%

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

      4. distribute-rgt-neg-in 73.0%

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

   9. Simplified 73.0%

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

   if 1.7000000000000001e67 < t

   1. Initial program 30.1%

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

      1. associate-/l* 30.0%

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

      2. fma-neg 30.0%

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

      3. remove-double-neg 30.0%

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

      4. fma-neg 30.0%

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

      5. sub-neg 30.0%

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

      6. metadata-eval 30.0%

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

      7. remove-double-neg 30.0%

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

      8. fma-def 30.0%

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

   3. Simplified 30.0%

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

   4. Taylor expanded in l around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+21}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+67}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 4: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-140}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.8e-140)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 4e-107)
     (* (sqrt 2.0) (/ t (sqrt (* 2.0 (/ l (/ x l))))))
     (+ (+ 1.0 (/ 0.5 (* x x))) (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.8e-140) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 4e-107) {
		tmp = sqrt(2.0) * (t / sqrt((2.0 * (l / (x / l)))));
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	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 <= (-1.8d-140)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 4d-107) then
        tmp = sqrt(2.0d0) * (t / sqrt((2.0d0 * (l / (x / l)))))
    else
        tmp = (1.0d0 + (0.5d0 / (x * x))) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.8e-140) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 4e-107) {
		tmp = Math.sqrt(2.0) * (t / Math.sqrt((2.0 * (l / (x / l)))));
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.8e-140:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 4e-107:
		tmp = math.sqrt(2.0) * (t / math.sqrt((2.0 * (l / (x / l)))))
	else:
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.8e-140)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 4e-107)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(2.0 * Float64(l / Float64(x / l))))));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.8e-140)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 4e-107)
		tmp = sqrt(2.0) * (t / sqrt((2.0 * (l / (x / l)))));
	else
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.8e-140], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 4e-107], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(2.0 * N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.8e-140

   1. Initial program 38.3%

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

      1. associate-*r/ 38.2%

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

      2. fma-neg 38.2%

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

      3. sub-neg 38.2%

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

      4. metadata-eval 38.2%

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

      5. +-commutative 38.2%

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

      6. fma-def 38.2%

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

      7. distribute-rgt-neg-in 38.2%

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

   3. Simplified 38.2%

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

      1. associate-*r/ 38.3%

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

      2. metadata-eval 38.3%

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

      3. sub-neg 38.3%

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

      4. fma-udef 38.3%

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

      5. +-commutative 38.3%

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

      6. distribute-rgt-neg-out 38.3%

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

      7. fma-neg 38.3%

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

      8. *-commutative 38.3%

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

      9. fma-neg 38.3%

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

   5. Applied egg-rr 68.4%

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

   6. Taylor expanded in t around -inf 83.5%

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

      1. mul-1-neg 83.5%

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

      2. sub-neg 83.5%

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

      3. metadata-eval 83.5%

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

   8. Simplified 83.5%

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

   if -1.8e-140 < t < 4e-107

   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-*r/ 5.4%

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

      2. fma-neg 5.2%

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

      3. sub-neg 5.2%

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

      4. metadata-eval 5.2%

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

      5. +-commutative 5.2%

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

      6. fma-def 5.2%

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

      7. distribute-rgt-neg-in 5.2%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in l around inf 6.6%

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

      1. *-commutative 6.6%

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

      2. unpow2 6.6%

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

      3. associate--l+ 6.7%

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

      4. sub-neg 6.7%

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

      5. metadata-eval 6.7%

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

      6. +-commutative 6.7%

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

      7. sub-neg 6.7%

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

      8. metadata-eval 6.7%

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

      9. +-commutative 6.7%

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

   6. Simplified 6.7%

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

   7. Taylor expanded in x around inf 54.4%

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

   8. Taylor expanded in l around 0 54.4%

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

      1. unpow2 54.4%

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

      2. associate-/l* 57.1%

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

   10. Simplified 57.1%

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

   if 4e-107 < t

   1. Initial program 42.7%

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

      1. associate-/l* 42.6%

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

      2. fma-neg 42.6%

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

      3. remove-double-neg 42.6%

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

      4. fma-neg 42.6%

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

      5. sub-neg 42.6%

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

      6. metadata-eval 42.6%

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

      7. remove-double-neg 42.6%

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

      8. fma-def 42.6%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in l around 0 93.2%

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

      1. +-commutative 93.2%

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

      2. sub-neg 93.2%

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

      3. metadata-eval 93.2%

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

      4. +-commutative 93.2%

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

   6. Simplified 93.2%

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

   7. Taylor expanded in x around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. metadata-eval 93.2%

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

      3. unpow2 93.2%

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

   9. Simplified 93.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-140}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 5: 77.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-143}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-257}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -3.8e-143)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 1.1e-257)
     (* (/ t l) (sqrt x))
     (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.8e-143) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.1e-257) {
		tmp = (t / l) * sqrt(x);
	} else {
		tmp = 1.0 / 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) :: tmp
    if (t <= (-3.8d-143)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 1.1d-257) then
        tmp = (t / l) * sqrt(x)
    else
        tmp = 1.0d0 / 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 tmp;
	if (t <= -3.8e-143) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 1.1e-257) {
		tmp = (t / l) * Math.sqrt(x);
	} else {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -3.8e-143:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 1.1e-257:
		tmp = (t / l) * math.sqrt(x)
	else:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -3.8e-143)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 1.1e-257)
		tmp = Float64(Float64(t / l) * sqrt(x));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.8e-143)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 1.1e-257)
		tmp = (t / l) * sqrt(x);
	else
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -3.8e-143], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 1.1e-257], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.79999999999999981e-143

   1. Initial program 38.0%

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

      1. associate-*r/ 37.8%

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

      2. fma-neg 37.8%

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

      3. sub-neg 37.8%

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

      4. metadata-eval 37.8%

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

      5. +-commutative 37.8%

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

      6. fma-def 37.8%

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

      7. distribute-rgt-neg-in 37.8%

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

   3. Simplified 37.8%

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

      1. associate-*r/ 38.0%

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

      2. metadata-eval 38.0%

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

      3. sub-neg 38.0%

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

      4. fma-udef 38.0%

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

      5. +-commutative 38.0%

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

      6. distribute-rgt-neg-out 38.0%

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

      7. fma-neg 38.0%

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

      8. *-commutative 38.0%

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

      9. fma-neg 38.0%

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

   5. Applied egg-rr 67.7%

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

   6. Taylor expanded in t around -inf 82.7%

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

      1. mul-1-neg 82.7%

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

      2. sub-neg 82.7%

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

      3. metadata-eval 82.7%

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

   8. Simplified 82.7%

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

   if -3.79999999999999981e-143 < t < 1.09999999999999994e-257

   1. Initial program 4.2%

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

      1. associate-*r/ 4.2%

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

      2. fma-neg 4.2%

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

      3. sub-neg 4.2%

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

      4. metadata-eval 4.2%

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

      5. +-commutative 4.2%

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

      6. fma-def 4.2%

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

      7. distribute-rgt-neg-in 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in l around inf 4.1%

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

      1. *-commutative 4.1%

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

      2. unpow2 4.1%

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

      3. associate--l+ 4.2%

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

      4. sub-neg 4.2%

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

      5. metadata-eval 4.2%

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

      6. +-commutative 4.2%

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

      7. sub-neg 4.2%

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

      8. metadata-eval 4.2%

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

      9. +-commutative 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in x around inf 56.0%

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

   8. Taylor expanded in t around 0 51.5%

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

   if 1.09999999999999994e-257 < t

   1. Initial program 35.1%

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

      1. associate-/l* 35.0%

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

      2. fma-neg 34.9%

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

      3. remove-double-neg 34.9%

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

      4. fma-neg 35.0%

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

      5. sub-neg 35.0%

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

      6. metadata-eval 35.0%

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

      7. remove-double-neg 35.0%

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

      8. fma-def 35.0%

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

   3. Simplified 35.0%

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

   4. Taylor expanded in l around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. sub-neg 82.6%

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

      3. metadata-eval 82.6%

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

      4. +-commutative 82.6%

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

   6. Simplified 82.6%

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

      1. expm1-log1p-u 82.6%

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

      2. expm1-udef 82.6%

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

      3. associate-/r* 82.6%

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

      4. pow1/2 82.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{{2}^{0.5}}}{\sqrt{2}}}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1 \]

      5. pow1/2 82.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{{2}^{0.5}}{\color{blue}{{2}^{0.5}}}}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1 \]

      6. pow-div 82.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{2}^{\left(0.5 - 0.5\right)}}}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1 \]

      7. metadata-eval 82.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{2}^{\color{blue}{0}}}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1 \]

      8. metadata-eval 82.6%

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

      9. +-commutative 82.6%

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

      10. +-commutative 82.6%

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

   8. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{1 + x}{x + -1}}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 82.7%

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

      2. expm1-log1p 82.7%

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

   10. Simplified 82.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-143}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-257}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 6: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-257}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -7e-144)
   (- (/ 1.0 x) (+ 1.0 (/ 0.5 (* x x))))
   (if (<= t 7.2e-257) (* (/ t l) (sqrt x)) (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -7e-144) {
		tmp = (1.0 / x) - (1.0 + (0.5 / (x * x)));
	} else if (t <= 7.2e-257) {
		tmp = (t / l) * sqrt(x);
	} else {
		tmp = 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) :: tmp
    if (t <= (-7d-144)) then
        tmp = (1.0d0 / x) - (1.0d0 + (0.5d0 / (x * x)))
    else if (t <= 7.2d-257) then
        tmp = (t / l) * sqrt(x)
    else
        tmp = 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 tmp;
	if (t <= -7e-144) {
		tmp = (1.0 / x) - (1.0 + (0.5 / (x * x)));
	} else if (t <= 7.2e-257) {
		tmp = (t / l) * Math.sqrt(x);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -7e-144:
		tmp = (1.0 / x) - (1.0 + (0.5 / (x * x)))
	elif t <= 7.2e-257:
		tmp = (t / l) * math.sqrt(x)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -7e-144)
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 + Float64(0.5 / Float64(x * x))));
	elseif (t <= 7.2e-257)
		tmp = Float64(Float64(t / l) * sqrt(x));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -7e-144)
		tmp = (1.0 / x) - (1.0 + (0.5 / (x * x)));
	elseif (t <= 7.2e-257)
		tmp = (t / l) * sqrt(x);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -7e-144], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e-257], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.9999999999999997e-144

   1. Initial program 38.0%

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

      1. associate-*r/ 37.8%

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

      2. fma-neg 37.8%

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

      3. sub-neg 37.8%

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

      4. metadata-eval 37.8%

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

      5. +-commutative 37.8%

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

      6. fma-def 37.8%

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

      7. distribute-rgt-neg-in 37.8%

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

   3. Simplified 37.8%

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

   4. Taylor expanded in t around -inf 82.5%

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

      1. associate-*r* 82.5%

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

      2. *-commutative 82.5%

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

      3. neg-mul-1 82.5%

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

      4. distribute-rgt-neg-in 82.5%

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

      5. +-commutative 82.5%

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

      6. sub-neg 82.5%

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

      7. metadata-eval 82.5%

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

      8. +-commutative 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in x around inf 80.9%

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

      1. associate-*r/ 80.9%

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

      2. metadata-eval 80.9%

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

      3. unpow2 80.9%

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

   9. Simplified 80.9%

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

   if -6.9999999999999997e-144 < t < 7.20000000000000013e-257

   1. Initial program 4.2%

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

      1. associate-*r/ 4.2%

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

      2. fma-neg 4.2%

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

      3. sub-neg 4.2%

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

      4. metadata-eval 4.2%

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

      5. +-commutative 4.2%

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

      6. fma-def 4.2%

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

      7. distribute-rgt-neg-in 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in l around inf 4.1%

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

      1. *-commutative 4.1%

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

      2. unpow2 4.1%

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

      3. associate--l+ 4.2%

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

      4. sub-neg 4.2%

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

      5. metadata-eval 4.2%

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

      6. +-commutative 4.2%

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

      7. sub-neg 4.2%

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

      8. metadata-eval 4.2%

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

      9. +-commutative 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in x around inf 56.0%

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

   8. Taylor expanded in t around 0 51.5%

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

   if 7.20000000000000013e-257 < t

   1. Initial program 35.1%

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

      1. associate-*r/ 35.0%

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

      2. fma-neg 34.9%

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

      3. sub-neg 34.9%

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

      4. metadata-eval 34.9%

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

      5. +-commutative 34.9%

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

      6. fma-def 34.9%

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

      7. distribute-rgt-neg-in 34.9%

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

   3. Simplified 34.9%

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

      1. associate-*r/ 35.0%

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

      2. metadata-eval 35.0%

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

      3. sub-neg 35.0%

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

      4. fma-udef 35.0%

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

      5. +-commutative 35.0%

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

      6. distribute-rgt-neg-out 35.0%

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

      7. fma-neg 35.1%

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

      8. *-commutative 35.1%

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

      9. fma-neg 35.0%

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

   5. Applied egg-rr 73.1%

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

   6. Taylor expanded in t around inf 82.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-257}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7: 77.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -8.4 \cdot 10^{-144}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-257}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -8.4e-144) (- t_1) (if (<= t 3e-257) (* (/ t l) (sqrt x)) t_1))))

C

double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -8.4e-144) {
		tmp = -t_1;
	} else if (t <= 3e-257) {
		tmp = (t / l) * sqrt(x);
	} else {
		tmp = t_1;
	}
	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 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-8.4d-144)) then
        tmp = -t_1
    else if (t <= 3d-257) then
        tmp = (t / l) * sqrt(x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -8.4e-144) {
		tmp = -t_1;
	} else if (t <= 3e-257) {
		tmp = (t / l) * Math.sqrt(x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -8.4e-144:
		tmp = -t_1
	elif t <= 3e-257:
		tmp = (t / l) * math.sqrt(x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -8.4e-144)
		tmp = Float64(-t_1);
	elseif (t <= 3e-257)
		tmp = Float64(Float64(t / l) * sqrt(x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -8.4e-144)
		tmp = -t_1;
	elseif (t <= 3e-257)
		tmp = (t / l) * sqrt(x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -8.4e-144], (-t$95$1), If[LessEqual[t, 3e-257], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.4000000000000005e-144

   1. Initial program 38.0%

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

      1. associate-*r/ 37.8%

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

      2. fma-neg 37.8%

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

      3. sub-neg 37.8%

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

      4. metadata-eval 37.8%

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

      5. +-commutative 37.8%

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

      6. fma-def 37.8%

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

      7. distribute-rgt-neg-in 37.8%

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

   3. Simplified 37.8%

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

      1. associate-*r/ 38.0%

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

      2. metadata-eval 38.0%

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

      3. sub-neg 38.0%

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

      4. fma-udef 38.0%

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

      5. +-commutative 38.0%

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

      6. distribute-rgt-neg-out 38.0%

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

      7. fma-neg 38.0%

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

      8. *-commutative 38.0%

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

      9. fma-neg 38.0%

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

   5. Applied egg-rr 67.7%

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

   6. Taylor expanded in t around -inf 82.7%

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

      1. mul-1-neg 82.7%

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

      2. sub-neg 82.7%

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

      3. metadata-eval 82.7%

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

   8. Simplified 82.7%

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

   if -8.4000000000000005e-144 < t < 2.9999999999999999e-257

   1. Initial program 4.2%

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

      1. associate-*r/ 4.2%

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

      2. fma-neg 4.2%

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

      3. sub-neg 4.2%

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

      4. metadata-eval 4.2%

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

      5. +-commutative 4.2%

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

      6. fma-def 4.2%

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

      7. distribute-rgt-neg-in 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in l around inf 4.1%

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

      1. *-commutative 4.1%

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

      2. unpow2 4.1%

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

      3. associate--l+ 4.2%

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

      4. sub-neg 4.2%

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

      5. metadata-eval 4.2%

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

      6. +-commutative 4.2%

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

      7. sub-neg 4.2%

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

      8. metadata-eval 4.2%

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

      9. +-commutative 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in x around inf 56.0%

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

   8. Taylor expanded in t around 0 51.5%

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

   if 2.9999999999999999e-257 < t

   1. Initial program 35.1%

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

      1. associate-*r/ 35.0%

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

      2. fma-neg 34.9%

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

      3. sub-neg 34.9%

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

      4. metadata-eval 34.9%

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

      5. +-commutative 34.9%

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

      6. fma-def 34.9%

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

      7. distribute-rgt-neg-in 34.9%

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

   3. Simplified 34.9%

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

      1. associate-*r/ 35.0%

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

      2. metadata-eval 35.0%

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

      3. sub-neg 35.0%

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

      4. fma-udef 35.0%

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

      5. +-commutative 35.0%

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

      6. distribute-rgt-neg-out 35.0%

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

      7. fma-neg 35.1%

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

      8. *-commutative 35.1%

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

      9. fma-neg 35.0%

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

   5. Applied egg-rr 73.1%

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

   6. Taylor expanded in t around inf 82.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.4 \cdot 10^{-144}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-257}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 8: 76.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{x} - t_1\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-255}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ 0.5 (* x x)))))
   (if (<= t -4.8e-144)
     (- (/ 1.0 x) t_1)
     (if (<= t 2.25e-255) (* (/ t l) (sqrt x)) (+ t_1 (/ -1.0 x))))))

C

double code(double x, double l, double t) {
	double t_1 = 1.0 + (0.5 / (x * x));
	double tmp;
	if (t <= -4.8e-144) {
		tmp = (1.0 / x) - t_1;
	} else if (t <= 2.25e-255) {
		tmp = (t / l) * sqrt(x);
	} else {
		tmp = t_1 + (-1.0 / x);
	}
	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 = 1.0d0 + (0.5d0 / (x * x))
    if (t <= (-4.8d-144)) then
        tmp = (1.0d0 / x) - t_1
    else if (t <= 2.25d-255) then
        tmp = (t / l) * sqrt(x)
    else
        tmp = t_1 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = 1.0 + (0.5 / (x * x));
	double tmp;
	if (t <= -4.8e-144) {
		tmp = (1.0 / x) - t_1;
	} else if (t <= 2.25e-255) {
		tmp = (t / l) * Math.sqrt(x);
	} else {
		tmp = t_1 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = 1.0 + (0.5 / (x * x))
	tmp = 0
	if t <= -4.8e-144:
		tmp = (1.0 / x) - t_1
	elif t <= 2.25e-255:
		tmp = (t / l) * math.sqrt(x)
	else:
		tmp = t_1 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(1.0 + Float64(0.5 / Float64(x * x)))
	tmp = 0.0
	if (t <= -4.8e-144)
		tmp = Float64(Float64(1.0 / x) - t_1);
	elseif (t <= 2.25e-255)
		tmp = Float64(Float64(t / l) * sqrt(x));
	else
		tmp = Float64(t_1 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = 1.0 + (0.5 / (x * x));
	tmp = 0.0;
	if (t <= -4.8e-144)
		tmp = (1.0 / x) - t_1;
	elseif (t <= 2.25e-255)
		tmp = (t / l) * sqrt(x);
	else
		tmp = t_1 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.8e-144], N[(N[(1.0 / x), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t, 2.25e-255], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.79999999999999988e-144

   1. Initial program 38.0%

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

      1. associate-*r/ 37.8%

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

      2. fma-neg 37.8%

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

      3. sub-neg 37.8%

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

      4. metadata-eval 37.8%

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

      5. +-commutative 37.8%

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

      6. fma-def 37.8%

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

      7. distribute-rgt-neg-in 37.8%

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

   3. Simplified 37.8%

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

   4. Taylor expanded in t around -inf 82.5%

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

      1. associate-*r* 82.5%

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

      2. *-commutative 82.5%

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

      3. neg-mul-1 82.5%

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

      4. distribute-rgt-neg-in 82.5%

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

      5. +-commutative 82.5%

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

      6. sub-neg 82.5%

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

      7. metadata-eval 82.5%

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

      8. +-commutative 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in x around inf 80.9%

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

      1. associate-*r/ 80.9%

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

      2. metadata-eval 80.9%

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

      3. unpow2 80.9%

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

   9. Simplified 80.9%

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

   if -4.79999999999999988e-144 < t < 2.2499999999999999e-255

   1. Initial program 4.2%

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

      1. associate-*r/ 4.2%

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

      2. fma-neg 4.2%

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

      3. sub-neg 4.2%

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

      4. metadata-eval 4.2%

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

      5. +-commutative 4.2%

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

      6. fma-def 4.2%

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

      7. distribute-rgt-neg-in 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in l around inf 4.1%

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

      1. *-commutative 4.1%

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

      2. unpow2 4.1%

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

      3. associate--l+ 4.2%

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

      4. sub-neg 4.2%

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

      5. metadata-eval 4.2%

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

      6. +-commutative 4.2%

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

      7. sub-neg 4.2%

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

      8. metadata-eval 4.2%

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

      9. +-commutative 4.2%

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

   6. Simplified 4.2%

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

   7. Taylor expanded in x around inf 56.0%

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

   8. Taylor expanded in t around 0 51.5%

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

   if 2.2499999999999999e-255 < t

   1. Initial program 35.1%

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

      1. associate-/l* 35.0%

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

      2. fma-neg 34.9%

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

      3. remove-double-neg 34.9%

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

      4. fma-neg 35.0%

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

      5. sub-neg 35.0%

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

      6. metadata-eval 35.0%

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

      7. remove-double-neg 35.0%

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

      8. fma-def 35.0%

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

   3. Simplified 35.0%

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

   4. Taylor expanded in l around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. sub-neg 82.6%

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

      3. metadata-eval 82.6%

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

      4. +-commutative 82.6%

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

   6. Simplified 82.6%

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

   7. Taylor expanded in x around inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. metadata-eval 82.6%

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

      3. unpow2 82.6%

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

   9. Simplified 82.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-255}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 9: 76.8% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) (+ -1.0 (/ 1.0 x)) (+ (+ 1.0 (/ 0.5 (* x x))) (/ -1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	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-309)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = (1.0d0 + (0.5d0 / (x * x))) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = (1.0 + (0.5 / (x * x))) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-309], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 29.5%

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

      1. associate-*r/ 29.4%

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

      2. fma-neg 29.4%

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

      3. sub-neg 29.4%

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

      4. metadata-eval 29.4%

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

      5. +-commutative 29.4%

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

      6. fma-def 29.4%

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

      7. distribute-rgt-neg-in 29.4%

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

   3. Simplified 29.4%

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

   4. Taylor expanded in t around -inf 69.8%

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

      1. associate-*r* 69.8%

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

      2. *-commutative 69.8%

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

      3. neg-mul-1 69.8%

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

      4. distribute-rgt-neg-in 69.8%

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

      5. +-commutative 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. +-commutative 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in x around inf 68.1%

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

   if -1.000000000000002e-309 < t

   1. Initial program 32.7%

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

      1. associate-/l* 32.7%

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

      2. fma-neg 32.6%

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

      3. remove-double-neg 32.6%

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

      4. fma-neg 32.7%

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

      5. sub-neg 32.7%

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

      6. metadata-eval 32.7%

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

      7. remove-double-neg 32.7%

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

      8. fma-def 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in l around 0 76.8%

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

      1. +-commutative 76.8%

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

      2. sub-neg 76.8%

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

      3. metadata-eval 76.8%

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

      4. +-commutative 76.8%

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

   6. Simplified 76.8%

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

   7. Taylor expanded in x around inf 76.7%

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

      1. associate-*r/ 76.7%

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

      2. metadata-eval 76.7%

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

      3. unpow2 76.7%

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

   9. Simplified 76.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 10: 76.8% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ 0.5 (* x x)))))
   (if (<= t -1e-309) (- (/ 1.0 x) t_1) (+ t_1 (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double t_1 = 1.0 + (0.5 / (x * x));
	double tmp;
	if (t <= -1e-309) {
		tmp = (1.0 / x) - t_1;
	} else {
		tmp = t_1 + (-1.0 / x);
	}
	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 = 1.0d0 + (0.5d0 / (x * x))
    if (t <= (-1d-309)) then
        tmp = (1.0d0 / x) - t_1
    else
        tmp = t_1 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = 1.0 + (0.5 / (x * x));
	double tmp;
	if (t <= -1e-309) {
		tmp = (1.0 / x) - t_1;
	} else {
		tmp = t_1 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = 1.0 + (0.5 / (x * x))
	tmp = 0
	if t <= -1e-309:
		tmp = (1.0 / x) - t_1
	else:
		tmp = t_1 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(1.0 + Float64(0.5 / Float64(x * x)))
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(Float64(1.0 / x) - t_1);
	else
		tmp = Float64(t_1 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = 1.0 + (0.5 / (x * x));
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = (1.0 / x) - t_1;
	else
		tmp = t_1 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e-309], N[(N[(1.0 / x), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 29.5%

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

      1. associate-*r/ 29.4%

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

      2. fma-neg 29.4%

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

      3. sub-neg 29.4%

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

      4. metadata-eval 29.4%

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

      5. +-commutative 29.4%

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

      6. fma-def 29.4%

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

      7. distribute-rgt-neg-in 29.4%

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

   3. Simplified 29.4%

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

   4. Taylor expanded in t around -inf 69.8%

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

      1. associate-*r* 69.8%

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

      2. *-commutative 69.8%

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

      3. neg-mul-1 69.8%

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

      4. distribute-rgt-neg-in 69.8%

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

      5. +-commutative 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. +-commutative 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in x around inf 68.5%

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

      1. associate-*r/ 68.5%

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

      2. metadata-eval 68.5%

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

      3. unpow2 68.5%

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

   9. Simplified 68.5%

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

   if -1.000000000000002e-309 < t

   1. Initial program 32.7%

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

      1. associate-/l* 32.7%

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

      2. fma-neg 32.6%

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

      3. remove-double-neg 32.6%

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

      4. fma-neg 32.7%

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

      5. sub-neg 32.7%

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

      6. metadata-eval 32.7%

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

      7. remove-double-neg 32.7%

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

      8. fma-def 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in l around 0 76.8%

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

      1. +-commutative 76.8%

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

      2. sub-neg 76.8%

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

      3. metadata-eval 76.8%

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

      4. +-commutative 76.8%

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

   6. Simplified 76.8%

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

   7. Taylor expanded in x around inf 76.7%

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

      1. associate-*r/ 76.7%

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

      2. metadata-eval 76.7%

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

      3. unpow2 76.7%

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

   9. Simplified 76.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 11: 76.4% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) -1.0 (+ 1.0 (/ -1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	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-309)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-309], -1.0, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 29.5%

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

      1. associate-*r/ 29.4%

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

      2. fma-neg 29.4%

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

      3. sub-neg 29.4%

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

      4. metadata-eval 29.4%

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

      5. +-commutative 29.4%

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

      6. fma-def 29.4%

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

      7. distribute-rgt-neg-in 29.4%

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

   3. Simplified 29.4%

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

      1. associate-*r/ 29.5%

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

      2. metadata-eval 29.5%

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

      3. sub-neg 29.5%

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

      4. fma-udef 29.5%

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

      5. +-commutative 29.5%

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

      6. distribute-rgt-neg-out 29.5%

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

      7. fma-neg 29.5%

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

      8. *-commutative 29.5%

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

      9. fma-neg 29.5%

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

   5. Applied egg-rr 58.2%

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

   6. Taylor expanded in t around inf 1.8%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 67.3%

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

   9. Simplified 67.3%

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

   if -1.000000000000002e-309 < t

   1. Initial program 32.7%

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

      1. associate-/l* 32.7%

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

      2. fma-neg 32.6%

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

      3. remove-double-neg 32.6%

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

      4. fma-neg 32.7%

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

      5. sub-neg 32.7%

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

      6. metadata-eval 32.7%

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

      7. remove-double-neg 32.7%

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

      8. fma-def 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in l around 0 76.8%

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

      1. +-commutative 76.8%

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

      2. sub-neg 76.8%

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

      3. metadata-eval 76.8%

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

      4. +-commutative 76.8%

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

   6. Simplified 76.8%

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

   7. Taylor expanded in x around inf 76.3%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12: 76.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) (+ -1.0 (/ 1.0 x)) (- 1.0 (/ 1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	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-309)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 - (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-309], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 29.5%

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

      1. associate-*r/ 29.4%

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

      2. fma-neg 29.4%

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

      3. sub-neg 29.4%

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

      4. metadata-eval 29.4%

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

      5. +-commutative 29.4%

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

      6. fma-def 29.4%

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

      7. distribute-rgt-neg-in 29.4%

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

   3. Simplified 29.4%

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

   4. Taylor expanded in t around -inf 69.8%

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

      1. associate-*r* 69.8%

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

      2. *-commutative 69.8%

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

      3. neg-mul-1 69.8%

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

      4. distribute-rgt-neg-in 69.8%

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

      5. +-commutative 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. +-commutative 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in x around inf 68.1%

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

   if -1.000000000000002e-309 < t

   1. Initial program 32.7%

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

      1. associate-/l* 32.7%

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

      2. fma-neg 32.6%

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

      3. remove-double-neg 32.6%

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

      4. fma-neg 32.7%

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

      5. sub-neg 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 32.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around 0 76.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 76.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 76.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval 76.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 76.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 76.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 76.3%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 13: 76.1% accurate, 73.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (if (<= t -1e-309) -1.0 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		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-309)) 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-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		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-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1e-309], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 29.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 29.4%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 29.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. sub-neg 29.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

      4. metadata-eval 29.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

      5. +-commutative 29.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]

      6. fma-def 29.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]

      7. distribute-rgt-neg-in 29.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]
   4. Step-by-step derivation

      1. associate-*r/ 29.5%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

      2. metadata-eval 29.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{\left(-1\right)}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]

      3. sub-neg 29.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x - 1}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]

      4. fma-udef 29.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, \ell \cdot \left(-\ell\right)\right)}} \]

      5. +-commutative 29.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}, \ell \cdot \left(-\ell\right)\right)}} \]

      6. distribute-rgt-neg-out 29.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), \color{blue}{-\ell \cdot \ell}\right)}} \]

      7. fma-neg 29.5%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      8. *-commutative 29.5%

         \[\leadsto \frac{\color{blue}{t \cdot \sqrt{2}}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

      9. fma-neg 29.5%

         \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

   5. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}} \]

   6. Taylor expanded in t around inf 1.8%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 0.0%

         \[\leadsto \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

      2. rem-square-sqrt 67.3%

         \[\leadsto \color{blue}{-1} \]

   9. Simplified 67.3%

      \[\leadsto \color{blue}{-1} \]

   if -1.000000000000002e-309 < t

   1. Initial program 32.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 32.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 32.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 32.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 32.7%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 32.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around 0 76.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 76.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 76.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval 76.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 76.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 76.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 75.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 38.6% 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 31.1%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*r/ 31.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

   2. fma-neg 31.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

   3. sub-neg 31.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

   4. metadata-eval 31.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

   5. +-commutative 31.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]

   6. fma-def 31.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]

   7. distribute-rgt-neg-in 31.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

3. Simplified 31.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]
4. Step-by-step derivation

   1. associate-*r/ 31.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   2. metadata-eval 31.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{\left(-1\right)}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]

   3. sub-neg 31.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x - 1}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]

   4. fma-udef 31.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, \ell \cdot \left(-\ell\right)\right)}} \]

   5. +-commutative 31.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}, \ell \cdot \left(-\ell\right)\right)}} \]

   6. distribute-rgt-neg-out 31.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), \color{blue}{-\ell \cdot \ell}\right)}} \]

   7. fma-neg 31.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

   8. *-commutative 31.1%

      \[\leadsto \frac{\color{blue}{t \cdot \sqrt{2}}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   9. fma-neg 31.1%

      \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

5. Applied egg-rr 64.4%

   \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}} \]

6. Taylor expanded in t around inf 39.3%

   \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]

7. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

   2. rem-square-sqrt 34.6%

      \[\leadsto \color{blue}{-1} \]

9. Simplified 34.6%

   \[\leadsto \color{blue}{-1} \]

10. Final simplification 34.6%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023178 
(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)))))