Toniolo and Linder, Equation (7)

Percentage Accurate: 32.7% → 85.7%

Time: 20.9s

Alternatives: 12

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 32.7% 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: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+106}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\ell \cdot \frac{\ell}{x} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -2.25e+14)
   (/ (sqrt 2.0) (* (sqrt (/ (+ x 1.0) (+ x -1.0))) (- (sqrt 2.0))))
   (if (<= t 1.8e+106)
     (*
      t
      (/
       (sqrt 2.0)
       (sqrt
        (+
         (* l (/ l x))
         (- (* 2.0 (fma t t (/ t (/ x t)))) (/ l (/ x (- l))))))))
     (exp (+ (/ -1.0 x) (/ -0.3333333333333333 (pow x 3.0)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.25e+14) {
		tmp = sqrt(2.0) / (sqrt(((x + 1.0) / (x + -1.0))) * -sqrt(2.0));
	} else if (t <= 1.8e+106) {
		tmp = t * (sqrt(2.0) / sqrt(((l * (l / x)) + ((2.0 * fma(t, t, (t / (x / t)))) - (l / (x / -l))))));
	} else {
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / pow(x, 3.0))));
	}
	return tmp;
}

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2.25e+14)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))) * Float64(-sqrt(2.0))));
	elseif (t <= 1.8e+106)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(l * Float64(l / x)) + Float64(Float64(2.0 * fma(t, t, Float64(t / Float64(x / t)))) - Float64(l / Float64(x / Float64(-l))))))));
	else
		tmp = exp(Float64(Float64(-1.0 / x) + Float64(-0.3333333333333333 / (x ^ 3.0))));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2.25e+14], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8e+106], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l / N[(x / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(-1.0 / x), $MachinePrecision] + N[(-0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.25e14

   1. Initial program 39.0%

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

      1. associate-/l* 39.0%

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

      2. fma-neg 39.0%

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

      3. remove-double-neg 39.0%

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

      4. fma-neg 39.0%

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

      5. sub-neg 39.0%

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

      6. metadata-eval 39.0%

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

      7. remove-double-neg 39.0%

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

      8. fma-def 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around -inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. distribute-rgt-neg-in 96.5%

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

      3. +-commutative 96.5%

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

      4. sub-neg 96.5%

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

      5. metadata-eval 96.5%

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

      6. +-commutative 96.5%

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

   6. Simplified 96.5%

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

   if -2.25e14 < t < 1.8e106

   1. Initial program 38.5%

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

      1. associate-*l/ 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in x around inf 75.9%

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

      1. associate--l+ 75.9%

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

      2. unpow2 75.9%

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

      3. distribute-lft-out 75.9%

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

      4. unpow2 75.9%

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

      5. unpow2 75.9%

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

      6. associate-*r/ 75.9%

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

      7. mul-1-neg 75.9%

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

      8. unpow2 75.9%

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

      9. +-commutative 75.9%

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

      10. unpow2 75.9%

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

      11. fma-udef 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in t around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. neg-mul-1 75.4%

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

      3. unpow2 75.4%

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

      4. distribute-rgt-neg-in 75.4%

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

   9. Simplified 75.4%

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

       1. *-un-lft-identity 75.4%

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

       2. associate-/l* 75.4%

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

       3. +-commutative 75.4%

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

   11. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-/r/ 75.4%

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

       3. fma-def 75.4%

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

       4. associate-/l* 75.4%

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

       5. associate-/l* 80.6%

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

   13. Simplified 80.6%

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

   if 1.8e106 < t

   1. Initial program 21.7%

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

      1. associate-*l/ 21.7%

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

   3. Simplified 21.7%

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

   5. Applied egg-rr 5.9%

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

   6. Taylor expanded in t around inf 98.4%

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

   7. Taylor expanded in x around inf 98.4%

      \[\leadsto e^{\color{blue}{-\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto e^{-\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      2. distribute-neg-in 98.4%

         \[\leadsto e^{\color{blue}{\left(-\frac{1}{x}\right) + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      3. distribute-neg-frac 98.4%

         \[\leadsto e^{\color{blue}{\frac{-1}{x}} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. metadata-eval 98.4%

         \[\leadsto e^{\frac{\color{blue}{-1}}{x} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      5. associate-*r/ 98.4%

         \[\leadsto e^{\frac{-1}{x} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right)} \]

      6. metadata-eval 98.4%

         \[\leadsto e^{\frac{-1}{x} + \left(-\frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right)} \]

      7. distribute-neg-frac 98.4%

         \[\leadsto e^{\frac{-1}{x} + \color{blue}{\frac{-0.3333333333333333}{{x}^{3}}}} \]

      8. metadata-eval 98.4%

         \[\leadsto e^{\frac{-1}{x} + \frac{\color{blue}{-0.3333333333333333}}{{x}^{3}}} \]

   9. Simplified 98.4%

      \[\leadsto e^{\color{blue}{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+106}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\ell \cdot \frac{\ell}{x} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{\ell}{\frac{x}{-\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \]

Alternative 2: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -660000000000:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 10^{+106}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, \mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -660000000000.0)
   (/ (sqrt 2.0) (* (sqrt (/ (+ x 1.0) (+ x -1.0))) (- (sqrt 2.0))))
   (if (<= t 1e+106)
     (*
      t
      (sqrt
       (/ 2.0 (fma (/ l x) l (fma 2.0 (* t (+ t (/ t x))) (* l (/ l x)))))))
     (exp (+ (/ -1.0 x) (/ -0.3333333333333333 (pow x 3.0)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -660000000000.0) {
		tmp = sqrt(2.0) / (sqrt(((x + 1.0) / (x + -1.0))) * -sqrt(2.0));
	} else if (t <= 1e+106) {
		tmp = t * sqrt((2.0 / fma((l / x), l, fma(2.0, (t * (t + (t / x))), (l * (l / x))))));
	} else {
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / pow(x, 3.0))));
	}
	return tmp;
}

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -660000000000.0)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))) * Float64(-sqrt(2.0))));
	elseif (t <= 1e+106)
		tmp = Float64(t * sqrt(Float64(2.0 / fma(Float64(l / x), l, fma(2.0, Float64(t * Float64(t + Float64(t / x))), Float64(l * Float64(l / x)))))));
	else
		tmp = exp(Float64(Float64(-1.0 / x) + Float64(-0.3333333333333333 / (x ^ 3.0))));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -660000000000.0], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+106], N[(t * N[Sqrt[N[(2.0 / N[(N[(l / x), $MachinePrecision] * l + N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(-1.0 / x), $MachinePrecision] + N[(-0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.6e11

   1. Initial program 39.0%

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

      1. associate-/l* 39.0%

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

      2. fma-neg 39.0%

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

      3. remove-double-neg 39.0%

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

      4. fma-neg 39.0%

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

      5. sub-neg 39.0%

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

      6. metadata-eval 39.0%

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

      7. remove-double-neg 39.0%

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

      8. fma-def 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around -inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. distribute-rgt-neg-in 96.5%

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

      3. +-commutative 96.5%

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

      4. sub-neg 96.5%

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

      5. metadata-eval 96.5%

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

      6. +-commutative 96.5%

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

   6. Simplified 96.5%

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

   if -6.6e11 < t < 1.00000000000000009e106

   1. Initial program 38.5%

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

      1. associate-*l/ 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in x around inf 75.9%

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

      1. associate--l+ 75.9%

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

      2. unpow2 75.9%

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

      3. distribute-lft-out 75.9%

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

      4. unpow2 75.9%

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

      5. unpow2 75.9%

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

      6. associate-*r/ 75.9%

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

      7. mul-1-neg 75.9%

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

      8. unpow2 75.9%

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

      9. +-commutative 75.9%

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

      10. unpow2 75.9%

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

      11. fma-udef 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in t around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. neg-mul-1 75.4%

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

      3. unpow2 75.4%

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

      4. distribute-rgt-neg-in 75.4%

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

   9. Simplified 75.4%

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

       1. *-un-lft-identity 75.4%

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

       2. associate-/l* 75.4%

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

       3. +-commutative 75.4%

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

   11. Applied egg-rr 75.4%

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

       1. *-lft-identity 75.4%

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

       2. associate-/r/ 75.4%

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

       3. fma-def 75.4%

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

       4. associate-/l* 75.4%

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

       5. associate-/l* 80.6%

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

   13. Simplified 80.6%

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

       1. *-un-lft-identity 80.6%

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

       2. sqrt-undiv 79.2%

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

       3. fma-def 79.1%

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

       4. associate-/r/ 79.1%

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

       5. associate-/r/ 79.1%

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

   15. Applied egg-rr 79.1%

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

       1. *-lft-identity 79.1%

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

       2. fma-neg 79.1%

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

       3. fma-udef 79.1%

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

       4. distribute-rgt-out 79.1%

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

       5. distribute-rgt-neg-out 79.1%

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

       6. remove-double-neg 79.1%

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

   17. Simplified 79.1%

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

   if 1.00000000000000009e106 < t

   1. Initial program 21.7%

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

      1. associate-*l/ 21.7%

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

   3. Simplified 21.7%

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

   5. Applied egg-rr 5.9%

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

   6. Taylor expanded in t around inf 98.4%

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

   7. Taylor expanded in x around inf 98.4%

      \[\leadsto e^{\color{blue}{-\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto e^{-\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      2. distribute-neg-in 98.4%

         \[\leadsto e^{\color{blue}{\left(-\frac{1}{x}\right) + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      3. distribute-neg-frac 98.4%

         \[\leadsto e^{\color{blue}{\frac{-1}{x}} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. metadata-eval 98.4%

         \[\leadsto e^{\frac{\color{blue}{-1}}{x} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      5. associate-*r/ 98.4%

         \[\leadsto e^{\frac{-1}{x} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right)} \]

      6. metadata-eval 98.4%

         \[\leadsto e^{\frac{-1}{x} + \left(-\frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right)} \]

      7. distribute-neg-frac 98.4%

         \[\leadsto e^{\frac{-1}{x} + \color{blue}{\frac{-0.3333333333333333}{{x}^{3}}}} \]

      8. metadata-eval 98.4%

         \[\leadsto e^{\frac{-1}{x} + \frac{\color{blue}{-0.3333333333333333}}{{x}^{3}}} \]

   9. Simplified 98.4%

      \[\leadsto e^{\color{blue}{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -660000000000:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 10^{+106}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, \mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \]

Alternative 3: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-195}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-176}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x)))
   (if (<= t -1e-10)
     (/ (sqrt 2.0) (* (sqrt (/ (+ x 1.0) (+ x -1.0))) (- (sqrt 2.0))))
     (if (<= t -3.9e-156)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt (+ t_1 (+ t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))))))
       (if (<= t -2.6e-195)
         (* t (/ (sqrt 2.0) (* t (- (sqrt (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x))))))))
         (if (<= t 9e-176)
           (* t (/ (sqrt x) l))
           (exp (+ (/ -1.0 x) (/ -0.3333333333333333 (pow x 3.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double tmp;
	if (t <= -1e-10) {
		tmp = sqrt(2.0) / (sqrt(((x + 1.0) / (x + -1.0))) * -sqrt(2.0));
	} else if (t <= -3.9e-156) {
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else if (t <= -2.6e-195) {
		tmp = t * (sqrt(2.0) / (t * -sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	} else if (t <= 9e-176) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / pow(x, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * l) / x
    if (t <= (-1d-10)) then
        tmp = sqrt(2.0d0) / (sqrt(((x + 1.0d0) / (x + (-1.0d0)))) * -sqrt(2.0d0))
    else if (t <= (-3.9d-156)) then
        tmp = t * (sqrt(2.0d0) / sqrt((t_1 + (t_1 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    else if (t <= (-2.6d-195)) then
        tmp = t * (sqrt(2.0d0) / (t * -sqrt(((2.0d0 / x) + (2.0d0 + (2.0d0 / x))))))
    else if (t <= 9d-176) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = exp((((-1.0d0) / x) + ((-0.3333333333333333d0) / (x ** 3.0d0))))
    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 <= -1e-10) {
		tmp = Math.sqrt(2.0) / (Math.sqrt(((x + 1.0) / (x + -1.0))) * -Math.sqrt(2.0));
	} else if (t <= -3.9e-156) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else if (t <= -2.6e-195) {
		tmp = t * (Math.sqrt(2.0) / (t * -Math.sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	} else if (t <= 9e-176) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.exp(((-1.0 / x) + (-0.3333333333333333 / Math.pow(x, 3.0))));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (l * l) / x
	tmp = 0
	if t <= -1e-10:
		tmp = math.sqrt(2.0) / (math.sqrt(((x + 1.0) / (x + -1.0))) * -math.sqrt(2.0))
	elif t <= -3.9e-156:
		tmp = t * (math.sqrt(2.0) / math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))))
	elif t <= -2.6e-195:
		tmp = t * (math.sqrt(2.0) / (t * -math.sqrt(((2.0 / x) + (2.0 + (2.0 / x))))))
	elif t <= 9e-176:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.exp(((-1.0 / x) + (-0.3333333333333333 / math.pow(x, 3.0))))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -1e-10)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))) * Float64(-sqrt(2.0))));
	elseif (t <= -3.9e-156)
		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 <= -2.6e-195)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * Float64(-sqrt(Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x))))))));
	elseif (t <= 9e-176)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = exp(Float64(Float64(-1.0 / x) + Float64(-0.3333333333333333 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (l * l) / x;
	tmp = 0.0;
	if (t <= -1e-10)
		tmp = sqrt(2.0) / (sqrt(((x + 1.0) / (x + -1.0))) * -sqrt(2.0));
	elseif (t <= -3.9e-156)
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	elseif (t <= -2.6e-195)
		tmp = t * (sqrt(2.0) / (t * -sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	elseif (t <= 9e-176)
		tmp = t * (sqrt(x) / l);
	else
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / (x ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -1e-10], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.9e-156], 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, -2.6e-195], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * (-N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e-176], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(-1.0 / x), $MachinePrecision] + N[(-0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.00000000000000004e-10

   1. Initial program 41.3%

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

      1. associate-/l* 41.3%

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

      2. fma-neg 41.3%

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

      3. remove-double-neg 41.3%

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

      4. fma-neg 41.3%

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

      5. sub-neg 41.3%

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

      6. metadata-eval 41.3%

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

      7. remove-double-neg 41.3%

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

      8. fma-def 41.3%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in t around -inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. distribute-rgt-neg-in 95.6%

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

      3. +-commutative 95.6%

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

      4. sub-neg 95.6%

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

      5. metadata-eval 95.6%

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

      6. +-commutative 95.6%

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

   6. Simplified 95.6%

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

   if -1.00000000000000004e-10 < t < -3.9000000000000001e-156

   1. Initial program 46.8%

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

      1. associate-*l/ 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in x around inf 84.5%

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

      1. associate--l+ 84.5%

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

      2. unpow2 84.5%

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

      3. distribute-lft-out 84.5%

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

      4. unpow2 84.5%

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

      5. unpow2 84.5%

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

      6. associate-*r/ 84.5%

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

      7. mul-1-neg 84.5%

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

      8. unpow2 84.5%

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

      9. +-commutative 84.5%

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

      10. unpow2 84.5%

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

      11. fma-udef 84.5%

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

   6. Simplified 84.5%

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

   7. Taylor expanded in t around 0 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

      3. unpow2 84.5%

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

      4. distribute-rgt-neg-in 84.5%

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

   9. Simplified 84.5%

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

   if -3.9000000000000001e-156 < t < -2.6000000000000002e-195

   1. Initial program 12.0%

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

      1. associate-*l/ 12.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in x around inf 33.8%

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

      1. associate--l+ 33.8%

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

      2. unpow2 33.8%

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

      3. distribute-lft-out 33.8%

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

      4. unpow2 33.8%

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

      5. unpow2 33.8%

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

      6. associate-*r/ 33.8%

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

      7. mul-1-neg 33.8%

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

      8. unpow2 33.8%

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

      9. +-commutative 33.8%

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

      10. unpow2 33.8%

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

      11. fma-udef 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in t around -inf 65.1%

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

      1. mul-1-neg 65.1%

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

      2. *-commutative 65.1%

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

      3. distribute-rgt-neg-in 65.1%

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

      4. distribute-lft-in 65.1%

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

      5. metadata-eval 65.1%

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

      6. associate-*r/ 65.1%

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

      7. metadata-eval 65.1%

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

      8. associate-*r/ 65.1%

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

      9. metadata-eval 65.1%

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

   9. Simplified 65.1%

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

   if -2.6000000000000002e-195 < t < 9e-176

   1. Initial program 1.6%

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

      1. associate-*l/ 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

      1. associate--l+ 63.8%

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

      2. unpow2 63.8%

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

      3. distribute-lft-out 63.8%

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

      4. unpow2 63.8%

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

      5. unpow2 63.8%

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

      6. associate-*r/ 63.8%

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

      7. mul-1-neg 63.8%

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

      8. unpow2 63.8%

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

      9. +-commutative 63.8%

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

      10. unpow2 63.8%

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

      11. fma-udef 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in l around inf 60.3%

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

   8. Taylor expanded in l around 0 60.3%

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

      1. associate-*l/ 60.3%

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

      2. *-lft-identity 60.3%

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

   10. Simplified 60.3%

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

   if 9e-176 < t

   1. Initial program 40.5%

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

      1. associate-*l/ 40.5%

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

   3. Simplified 40.5%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in t around inf 91.5%

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

   7. Taylor expanded in x around inf 91.5%

      \[\leadsto e^{\color{blue}{-\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto e^{-\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      2. distribute-neg-in 91.5%

         \[\leadsto e^{\color{blue}{\left(-\frac{1}{x}\right) + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      3. distribute-neg-frac 91.5%

         \[\leadsto e^{\color{blue}{\frac{-1}{x}} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. metadata-eval 91.5%

         \[\leadsto e^{\frac{\color{blue}{-1}}{x} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      5. associate-*r/ 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right)} \]

      6. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right)} \]

      7. distribute-neg-frac 91.5%

         \[\leadsto e^{\frac{-1}{x} + \color{blue}{\frac{-0.3333333333333333}{{x}^{3}}}} \]

      8. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \frac{\color{blue}{-0.3333333333333333}}{{x}^{3}}} \]

   9. Simplified 91.5%

      \[\leadsto e^{\color{blue}{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-195}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-176}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \]

Alternative 4: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -1350000000:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{x + -1}{x + 1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-194}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-176}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x)))
   (if (<= t -1350000000.0)
     (* (sqrt (* 0.5 (/ (+ x -1.0) (+ x 1.0)))) (- (sqrt 2.0)))
     (if (<= t -3.9e-156)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt (+ t_1 (+ t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))))))
       (if (<= t -9.2e-194)
         (* t (/ (sqrt 2.0) (* t (- (sqrt (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x))))))))
         (if (<= t 1.2e-176)
           (* t (/ (sqrt x) l))
           (exp (+ (/ -1.0 x) (/ -0.3333333333333333 (pow x 3.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double tmp;
	if (t <= -1350000000.0) {
		tmp = sqrt((0.5 * ((x + -1.0) / (x + 1.0)))) * -sqrt(2.0);
	} else if (t <= -3.9e-156) {
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else if (t <= -9.2e-194) {
		tmp = t * (sqrt(2.0) / (t * -sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	} else if (t <= 1.2e-176) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / pow(x, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * l) / x
    if (t <= (-1350000000.0d0)) then
        tmp = sqrt((0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))) * -sqrt(2.0d0)
    else if (t <= (-3.9d-156)) then
        tmp = t * (sqrt(2.0d0) / sqrt((t_1 + (t_1 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    else if (t <= (-9.2d-194)) then
        tmp = t * (sqrt(2.0d0) / (t * -sqrt(((2.0d0 / x) + (2.0d0 + (2.0d0 / x))))))
    else if (t <= 1.2d-176) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = exp((((-1.0d0) / x) + ((-0.3333333333333333d0) / (x ** 3.0d0))))
    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 <= -1350000000.0) {
		tmp = Math.sqrt((0.5 * ((x + -1.0) / (x + 1.0)))) * -Math.sqrt(2.0);
	} else if (t <= -3.9e-156) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else if (t <= -9.2e-194) {
		tmp = t * (Math.sqrt(2.0) / (t * -Math.sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	} else if (t <= 1.2e-176) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.exp(((-1.0 / x) + (-0.3333333333333333 / Math.pow(x, 3.0))));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = (l * l) / x
	tmp = 0
	if t <= -1350000000.0:
		tmp = math.sqrt((0.5 * ((x + -1.0) / (x + 1.0)))) * -math.sqrt(2.0)
	elif t <= -3.9e-156:
		tmp = t * (math.sqrt(2.0) / math.sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))))
	elif t <= -9.2e-194:
		tmp = t * (math.sqrt(2.0) / (t * -math.sqrt(((2.0 / x) + (2.0 + (2.0 / x))))))
	elif t <= 1.2e-176:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.exp(((-1.0 / x) + (-0.3333333333333333 / math.pow(x, 3.0))))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -1350000000.0)
		tmp = Float64(sqrt(Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))) * Float64(-sqrt(2.0)));
	elseif (t <= -3.9e-156)
		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 <= -9.2e-194)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * Float64(-sqrt(Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x))))))));
	elseif (t <= 1.2e-176)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = exp(Float64(Float64(-1.0 / x) + Float64(-0.3333333333333333 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = (l * l) / x;
	tmp = 0.0;
	if (t <= -1350000000.0)
		tmp = sqrt((0.5 * ((x + -1.0) / (x + 1.0)))) * -sqrt(2.0);
	elseif (t <= -3.9e-156)
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x)))))));
	elseif (t <= -9.2e-194)
		tmp = t * (sqrt(2.0) / (t * -sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	elseif (t <= 1.2e-176)
		tmp = t * (sqrt(x) / l);
	else
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / (x ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -1350000000.0], N[(N[Sqrt[N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -3.9e-156], 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, -9.2e-194], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * (-N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e-176], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(-1.0 / x), $MachinePrecision] + N[(-0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.35e9

   1. Initial program 38.5%

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

      1. associate-*l/ 38.6%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in t around -inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. associate-*l* 93.9%

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

      3. distribute-rgt-neg-in 93.9%

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

      4. sub-neg 93.9%

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

      5. metadata-eval 93.9%

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

      6. +-commutative 93.9%

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

      7. +-commutative 93.9%

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

   6. Simplified 93.9%

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

      1. pow1 93.9%

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

      2. sqrt-unprod 94.0%

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

      3. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. unpow1 94.0%

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

   10. Simplified 94.0%

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

   if -1.35e9 < t < -3.9000000000000001e-156

   1. Initial program 54.1%

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

      1. associate-*l/ 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in x around inf 86.6%

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

      1. associate--l+ 86.6%

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

      2. unpow2 86.6%

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

      3. distribute-lft-out 86.6%

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

      4. unpow2 86.6%

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

      5. unpow2 86.6%

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

      6. associate-*r/ 86.6%

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

      7. mul-1-neg 86.6%

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

      8. unpow2 86.6%

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

      9. +-commutative 86.6%

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

      10. unpow2 86.6%

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

      11. fma-udef 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in t around 0 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

      3. unpow2 86.6%

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

      4. distribute-rgt-neg-in 86.6%

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

   9. Simplified 86.6%

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

   if -3.9000000000000001e-156 < t < -9.2000000000000001e-194

   1. Initial program 12.0%

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

      1. associate-*l/ 12.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in x around inf 33.8%

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

      1. associate--l+ 33.8%

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

      2. unpow2 33.8%

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

      3. distribute-lft-out 33.8%

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

      4. unpow2 33.8%

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

      5. unpow2 33.8%

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

      6. associate-*r/ 33.8%

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

      7. mul-1-neg 33.8%

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

      8. unpow2 33.8%

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

      9. +-commutative 33.8%

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

      10. unpow2 33.8%

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

      11. fma-udef 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in t around -inf 65.1%

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

      1. mul-1-neg 65.1%

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

      2. *-commutative 65.1%

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

      3. distribute-rgt-neg-in 65.1%

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

      4. distribute-lft-in 65.1%

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

      5. metadata-eval 65.1%

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

      6. associate-*r/ 65.1%

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

      7. metadata-eval 65.1%

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

      8. associate-*r/ 65.1%

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

      9. metadata-eval 65.1%

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

   9. Simplified 65.1%

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

   if -9.2000000000000001e-194 < t < 1.20000000000000003e-176

   1. Initial program 1.6%

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

      1. associate-*l/ 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

      1. associate--l+ 63.8%

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

      2. unpow2 63.8%

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

      3. distribute-lft-out 63.8%

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

      4. unpow2 63.8%

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

      5. unpow2 63.8%

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

      6. associate-*r/ 63.8%

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

      7. mul-1-neg 63.8%

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

      8. unpow2 63.8%

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

      9. +-commutative 63.8%

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

      10. unpow2 63.8%

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

      11. fma-udef 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in l around inf 60.3%

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

   8. Taylor expanded in l around 0 60.3%

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

      1. associate-*l/ 60.3%

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

      2. *-lft-identity 60.3%

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

   10. Simplified 60.3%

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

   if 1.20000000000000003e-176 < t

   1. Initial program 40.5%

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

      1. associate-*l/ 40.5%

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

   3. Simplified 40.5%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in t around inf 91.5%

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

   7. Taylor expanded in x around inf 91.5%

      \[\leadsto e^{\color{blue}{-\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto e^{-\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      2. distribute-neg-in 91.5%

         \[\leadsto e^{\color{blue}{\left(-\frac{1}{x}\right) + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      3. distribute-neg-frac 91.5%

         \[\leadsto e^{\color{blue}{\frac{-1}{x}} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. metadata-eval 91.5%

         \[\leadsto e^{\frac{\color{blue}{-1}}{x} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      5. associate-*r/ 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right)} \]

      6. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right)} \]

      7. distribute-neg-frac 91.5%

         \[\leadsto e^{\frac{-1}{x} + \color{blue}{\frac{-0.3333333333333333}{{x}^{3}}}} \]

      8. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \frac{\color{blue}{-0.3333333333333333}}{{x}^{3}}} \]

   9. Simplified 91.5%

      \[\leadsto e^{\color{blue}{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1350000000:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{x + -1}{x + 1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-194}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-176}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \]

Alternative 5: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-198}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-176}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -4.2e-198)
   (* t (/ (sqrt 2.0) (* t (- (sqrt (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x))))))))
   (if (<= t 9e-176)
     (* t (/ (sqrt x) l))
     (exp (+ (/ -1.0 x) (/ -0.3333333333333333 (pow x 3.0)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.2e-198) {
		tmp = t * (sqrt(2.0) / (t * -sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	} else if (t <= 9e-176) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / pow(x, 3.0))));
	}
	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 <= (-4.2d-198)) then
        tmp = t * (sqrt(2.0d0) / (t * -sqrt(((2.0d0 / x) + (2.0d0 + (2.0d0 / x))))))
    else if (t <= 9d-176) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = exp((((-1.0d0) / x) + ((-0.3333333333333333d0) / (x ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.2e-198) {
		tmp = t * (Math.sqrt(2.0) / (t * -Math.sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	} else if (t <= 9e-176) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.exp(((-1.0 / x) + (-0.3333333333333333 / Math.pow(x, 3.0))));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -4.2e-198:
		tmp = t * (math.sqrt(2.0) / (t * -math.sqrt(((2.0 / x) + (2.0 + (2.0 / x))))))
	elif t <= 9e-176:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.exp(((-1.0 / x) + (-0.3333333333333333 / math.pow(x, 3.0))))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -4.2e-198)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * Float64(-sqrt(Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x))))))));
	elseif (t <= 9e-176)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = exp(Float64(Float64(-1.0 / x) + Float64(-0.3333333333333333 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -4.2e-198)
		tmp = t * (sqrt(2.0) / (t * -sqrt(((2.0 / x) + (2.0 + (2.0 / x))))));
	elseif (t <= 9e-176)
		tmp = t * (sqrt(x) / l);
	else
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / (x ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -4.2e-198], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * (-N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e-176], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(-1.0 / x), $MachinePrecision] + N[(-0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.19999999999999986e-198

   1. Initial program 40.2%

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

      1. associate-*l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in x around inf 49.2%

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

      1. associate--l+ 49.2%

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

      2. unpow2 49.2%

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

      3. distribute-lft-out 49.2%

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

      4. unpow2 49.2%

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

      5. unpow2 49.2%

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

      6. associate-*r/ 49.2%

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

      7. mul-1-neg 49.2%

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

      8. unpow2 49.2%

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

      9. +-commutative 49.2%

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

      10. unpow2 49.2%

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

      11. fma-udef 49.2%

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

   6. Simplified 49.2%

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

   7. Taylor expanded in t around -inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. *-commutative 85.2%

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

      3. distribute-rgt-neg-in 85.2%

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

      4. distribute-lft-in 85.2%

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

      5. metadata-eval 85.2%

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

      6. associate-*r/ 85.2%

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

      7. metadata-eval 85.2%

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

      8. associate-*r/ 85.2%

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

      9. metadata-eval 85.2%

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

   9. Simplified 85.2%

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

   if -4.19999999999999986e-198 < t < 9e-176

   1. Initial program 1.6%

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

      1. associate-*l/ 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

      1. associate--l+ 63.8%

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

      2. unpow2 63.8%

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

      3. distribute-lft-out 63.8%

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

      4. unpow2 63.8%

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

      5. unpow2 63.8%

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

      6. associate-*r/ 63.8%

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

      7. mul-1-neg 63.8%

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

      8. unpow2 63.8%

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

      9. +-commutative 63.8%

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

      10. unpow2 63.8%

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

      11. fma-udef 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in l around inf 60.3%

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

   8. Taylor expanded in l around 0 60.3%

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

      1. associate-*l/ 60.3%

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

      2. *-lft-identity 60.3%

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

   10. Simplified 60.3%

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

   if 9e-176 < t

   1. Initial program 40.5%

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

      1. associate-*l/ 40.5%

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

   3. Simplified 40.5%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in t around inf 91.5%

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

   7. Taylor expanded in x around inf 91.5%

      \[\leadsto e^{\color{blue}{-\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto e^{-\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      2. distribute-neg-in 91.5%

         \[\leadsto e^{\color{blue}{\left(-\frac{1}{x}\right) + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      3. distribute-neg-frac 91.5%

         \[\leadsto e^{\color{blue}{\frac{-1}{x}} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. metadata-eval 91.5%

         \[\leadsto e^{\frac{\color{blue}{-1}}{x} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      5. associate-*r/ 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right)} \]

      6. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right)} \]

      7. distribute-neg-frac 91.5%

         \[\leadsto e^{\frac{-1}{x} + \color{blue}{\frac{-0.3333333333333333}{{x}^{3}}}} \]

      8. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \frac{\color{blue}{-0.3333333333333333}}{{x}^{3}}} \]

   9. Simplified 91.5%

      \[\leadsto e^{\color{blue}{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-198}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-176}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \]

Alternative 6: 76.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{x + -1}{x + 1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-176}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -6e-199)
   (* (sqrt (* 0.5 (/ (+ x -1.0) (+ x 1.0)))) (- (sqrt 2.0)))
   (if (<= t 5.4e-176)
     (* t (/ (sqrt x) l))
     (exp (+ (/ -1.0 x) (/ -0.3333333333333333 (pow x 3.0)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -6e-199) {
		tmp = sqrt((0.5 * ((x + -1.0) / (x + 1.0)))) * -sqrt(2.0);
	} else if (t <= 5.4e-176) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / pow(x, 3.0))));
	}
	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 <= (-6d-199)) then
        tmp = sqrt((0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))) * -sqrt(2.0d0)
    else if (t <= 5.4d-176) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = exp((((-1.0d0) / x) + ((-0.3333333333333333d0) / (x ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -6e-199) {
		tmp = Math.sqrt((0.5 * ((x + -1.0) / (x + 1.0)))) * -Math.sqrt(2.0);
	} else if (t <= 5.4e-176) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.exp(((-1.0 / x) + (-0.3333333333333333 / Math.pow(x, 3.0))));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -6e-199:
		tmp = math.sqrt((0.5 * ((x + -1.0) / (x + 1.0)))) * -math.sqrt(2.0)
	elif t <= 5.4e-176:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.exp(((-1.0 / x) + (-0.3333333333333333 / math.pow(x, 3.0))))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -6e-199)
		tmp = Float64(sqrt(Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))) * Float64(-sqrt(2.0)));
	elseif (t <= 5.4e-176)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = exp(Float64(Float64(-1.0 / x) + Float64(-0.3333333333333333 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -6e-199)
		tmp = sqrt((0.5 * ((x + -1.0) / (x + 1.0)))) * -sqrt(2.0);
	elseif (t <= 5.4e-176)
		tmp = t * (sqrt(x) / l);
	else
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / (x ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -6e-199], N[(N[Sqrt[N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 5.4e-176], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(-1.0 / x), $MachinePrecision] + N[(-0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.99999999999999966e-199

   1. Initial program 40.2%

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

      1. associate-*l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around -inf 85.0%

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

      1. mul-1-neg 85.0%

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

      2. associate-*l* 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

      4. sub-neg 85.0%

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

      5. metadata-eval 85.0%

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

      6. +-commutative 85.0%

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

      7. +-commutative 85.0%

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

   6. Simplified 85.0%

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

      1. pow1 85.0%

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

      2. sqrt-unprod 85.1%

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

      3. +-commutative 85.1%

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

   8. Applied egg-rr 85.1%

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

      1. unpow1 85.1%

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

   10. Simplified 85.1%

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

   if -5.99999999999999966e-199 < t < 5.3999999999999997e-176

   1. Initial program 1.6%

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

      1. associate-*l/ 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

      1. associate--l+ 63.8%

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

      2. unpow2 63.8%

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

      3. distribute-lft-out 63.8%

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

      4. unpow2 63.8%

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

      5. unpow2 63.8%

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

      6. associate-*r/ 63.8%

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

      7. mul-1-neg 63.8%

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

      8. unpow2 63.8%

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

      9. +-commutative 63.8%

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

      10. unpow2 63.8%

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

      11. fma-udef 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in l around inf 60.3%

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

   8. Taylor expanded in l around 0 60.3%

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

      1. associate-*l/ 60.3%

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

      2. *-lft-identity 60.3%

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

   10. Simplified 60.3%

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

   if 5.3999999999999997e-176 < t

   1. Initial program 40.5%

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

      1. associate-*l/ 40.5%

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

   3. Simplified 40.5%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in t around inf 91.5%

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

   7. Taylor expanded in x around inf 91.5%

      \[\leadsto e^{\color{blue}{-\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto e^{-\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      2. distribute-neg-in 91.5%

         \[\leadsto e^{\color{blue}{\left(-\frac{1}{x}\right) + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      3. distribute-neg-frac 91.5%

         \[\leadsto e^{\color{blue}{\frac{-1}{x}} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. metadata-eval 91.5%

         \[\leadsto e^{\frac{\color{blue}{-1}}{x} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      5. associate-*r/ 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right)} \]

      6. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right)} \]

      7. distribute-neg-frac 91.5%

         \[\leadsto e^{\frac{-1}{x} + \color{blue}{\frac{-0.3333333333333333}{{x}^{3}}}} \]

      8. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \frac{\color{blue}{-0.3333333333333333}}{{x}^{3}}} \]

   9. Simplified 91.5%

      \[\leadsto e^{\color{blue}{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{x + -1}{x + 1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-176}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \]

Alternative 7: 75.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-194}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{0.5}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-194)
   (- (* (sqrt 2.0) (sqrt 0.5)))
   (if (<= t 2.4e-175)
     (* t (/ (sqrt x) l))
     (exp (+ (/ -1.0 x) (/ -0.3333333333333333 (pow x 3.0)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-194) {
		tmp = -(sqrt(2.0) * sqrt(0.5));
	} else if (t <= 2.4e-175) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / pow(x, 3.0))));
	}
	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 <= (-5d-194)) then
        tmp = -(sqrt(2.0d0) * sqrt(0.5d0))
    else if (t <= 2.4d-175) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = exp((((-1.0d0) / x) + ((-0.3333333333333333d0) / (x ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-194) {
		tmp = -(Math.sqrt(2.0) * Math.sqrt(0.5));
	} else if (t <= 2.4e-175) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.exp(((-1.0 / x) + (-0.3333333333333333 / Math.pow(x, 3.0))));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -5e-194:
		tmp = -(math.sqrt(2.0) * math.sqrt(0.5))
	elif t <= 2.4e-175:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.exp(((-1.0 / x) + (-0.3333333333333333 / math.pow(x, 3.0))))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-194)
		tmp = Float64(-Float64(sqrt(2.0) * sqrt(0.5)));
	elseif (t <= 2.4e-175)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = exp(Float64(Float64(-1.0 / x) + Float64(-0.3333333333333333 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-194)
		tmp = -(sqrt(2.0) * sqrt(0.5));
	elseif (t <= 2.4e-175)
		tmp = t * (sqrt(x) / l);
	else
		tmp = exp(((-1.0 / x) + (-0.3333333333333333 / (x ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -5e-194], (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), If[LessEqual[t, 2.4e-175], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(-1.0 / x), $MachinePrecision] + N[(-0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.0000000000000002e-194

   1. Initial program 40.2%

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

      1. associate-*l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around -inf 85.0%

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

      1. mul-1-neg 85.0%

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

      2. associate-*l* 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

      4. sub-neg 85.0%

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

      5. metadata-eval 85.0%

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

      6. +-commutative 85.0%

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

      7. +-commutative 85.0%

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

   6. Simplified 85.0%

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

   7. Taylor expanded in x around inf 82.4%

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

      1. mul-1-neg 82.4%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{0.5}} \]

      2. distribute-rgt-neg-in 82.4%

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

   9. Simplified 82.4%

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

   if -5.0000000000000002e-194 < t < 2.4e-175

   1. Initial program 1.6%

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

      1. associate-*l/ 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

      1. associate--l+ 63.8%

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

      2. unpow2 63.8%

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

      3. distribute-lft-out 63.8%

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

      4. unpow2 63.8%

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

      5. unpow2 63.8%

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

      6. associate-*r/ 63.8%

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

      7. mul-1-neg 63.8%

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

      8. unpow2 63.8%

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

      9. +-commutative 63.8%

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

      10. unpow2 63.8%

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

      11. fma-udef 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in l around inf 60.3%

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

   8. Taylor expanded in l around 0 60.3%

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

      1. associate-*l/ 60.3%

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

      2. *-lft-identity 60.3%

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

   10. Simplified 60.3%

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

   if 2.4e-175 < t

   1. Initial program 40.5%

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

      1. associate-*l/ 40.5%

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

   3. Simplified 40.5%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in t around inf 91.5%

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

   7. Taylor expanded in x around inf 91.5%

      \[\leadsto e^{\color{blue}{-\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto e^{-\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      2. distribute-neg-in 91.5%

         \[\leadsto e^{\color{blue}{\left(-\frac{1}{x}\right) + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)}} \]

      3. distribute-neg-frac 91.5%

         \[\leadsto e^{\color{blue}{\frac{-1}{x}} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. metadata-eval 91.5%

         \[\leadsto e^{\frac{\color{blue}{-1}}{x} + \left(-0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} \]

      5. associate-*r/ 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right)} \]

      6. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \left(-\frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right)} \]

      7. distribute-neg-frac 91.5%

         \[\leadsto e^{\frac{-1}{x} + \color{blue}{\frac{-0.3333333333333333}{{x}^{3}}}} \]

      8. metadata-eval 91.5%

         \[\leadsto e^{\frac{-1}{x} + \frac{\color{blue}{-0.3333333333333333}}{{x}^{3}}} \]

   9. Simplified 91.5%

      \[\leadsto e^{\color{blue}{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-194}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{0.5}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x} + \frac{-0.3333333333333333}{{x}^{3}}}\\ \end{array} \]

Alternative 8: 75.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-194}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{0.5}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -2.2e-194)
   (- (* (sqrt 2.0) (sqrt 0.5)))
   (if (<= t 4.8e-175) (* t (/ (sqrt x) l)) (exp (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.2e-194) {
		tmp = -(sqrt(2.0) * sqrt(0.5));
	} else if (t <= 4.8e-175) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = exp((-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 <= (-2.2d-194)) then
        tmp = -(sqrt(2.0d0) * sqrt(0.5d0))
    else if (t <= 4.8d-175) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = exp(((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.2e-194) {
		tmp = -(Math.sqrt(2.0) * Math.sqrt(0.5));
	} else if (t <= 4.8e-175) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.exp((-1.0 / x));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2.2e-194:
		tmp = -(math.sqrt(2.0) * math.sqrt(0.5))
	elif t <= 4.8e-175:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.exp((-1.0 / x))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2.2e-194)
		tmp = Float64(-Float64(sqrt(2.0) * sqrt(0.5)));
	elseif (t <= 4.8e-175)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = exp(Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2.2e-194)
		tmp = -(sqrt(2.0) * sqrt(0.5));
	elseif (t <= 4.8e-175)
		tmp = t * (sqrt(x) / l);
	else
		tmp = exp((-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2.2e-194], (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), If[LessEqual[t, 4.8e-175], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Exp[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.2000000000000001e-194

   1. Initial program 40.2%

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

      1. associate-*l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around -inf 85.0%

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

      1. mul-1-neg 85.0%

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

      2. associate-*l* 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

      4. sub-neg 85.0%

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

      5. metadata-eval 85.0%

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

      6. +-commutative 85.0%

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

      7. +-commutative 85.0%

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

   6. Simplified 85.0%

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

   7. Taylor expanded in x around inf 82.4%

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

      1. mul-1-neg 82.4%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{0.5}} \]

      2. distribute-rgt-neg-in 82.4%

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

   9. Simplified 82.4%

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

   if -2.2000000000000001e-194 < t < 4.8e-175

   1. Initial program 1.6%

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

      1. associate-*l/ 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in x around inf 63.8%

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

      1. associate--l+ 63.8%

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

      2. unpow2 63.8%

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

      3. distribute-lft-out 63.8%

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

      4. unpow2 63.8%

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

      5. unpow2 63.8%

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

      6. associate-*r/ 63.8%

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

      7. mul-1-neg 63.8%

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

      8. unpow2 63.8%

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

      9. +-commutative 63.8%

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

      10. unpow2 63.8%

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

      11. fma-udef 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in l around inf 60.3%

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

   8. Taylor expanded in l around 0 60.3%

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

      1. associate-*l/ 60.3%

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

      2. *-lft-identity 60.3%

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

   10. Simplified 60.3%

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

   if 4.8e-175 < t

   1. Initial program 40.5%

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

      1. associate-*l/ 40.5%

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

   3. Simplified 40.5%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in t around inf 91.5%

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

   7. Taylor expanded in x around inf 91.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-194}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{0.5}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x}}\\ \end{array} \]

Alternative 9: 45.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 4.2e-177) (* (sqrt x) (/ t l)) (exp (/ -1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 4.2e-177) {
		tmp = sqrt(x) * (t / l);
	} else {
		tmp = exp((-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 <= 4.2d-177) then
        tmp = sqrt(x) * (t / l)
    else
        tmp = exp(((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 4.2e-177) {
		tmp = Math.sqrt(x) * (t / l);
	} else {
		tmp = Math.exp((-1.0 / x));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 4.2e-177:
		tmp = math.sqrt(x) * (t / l)
	else:
		tmp = math.exp((-1.0 / x))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 4.2e-177)
		tmp = Float64(sqrt(x) * Float64(t / l));
	else
		tmp = exp(Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 4.2e-177)
		tmp = sqrt(x) * (t / l);
	else
		tmp = exp((-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 4.2e-177], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision], N[Exp[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.20000000000000002e-177

   1. Initial program 31.2%

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

      1. associate-*l/ 31.2%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in x around inf 52.6%

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

      1. associate--l+ 52.6%

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

      2. unpow2 52.6%

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

      3. distribute-lft-out 52.6%

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

      4. unpow2 52.6%

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

      5. unpow2 52.6%

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

      6. associate-*r/ 52.6%

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

      7. mul-1-neg 52.6%

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

      8. unpow2 52.6%

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

      9. +-commutative 52.6%

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

      10. unpow2 52.6%

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

      11. fma-udef 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in l around inf 21.6%

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

   8. Taylor expanded in l around 0 19.4%

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

   if 4.20000000000000002e-177 < t

   1. Initial program 40.5%

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

      1. associate-*l/ 40.5%

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

   3. Simplified 40.5%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in t around inf 91.5%

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

   7. Taylor expanded in x around inf 91.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x}}\\ \end{array} \]

Alternative 10: 46.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-177}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 4e-177) (* t (/ (sqrt x) l)) (exp (/ -1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 4e-177) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = exp((-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 <= 4d-177) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = exp(((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 4e-177) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.exp((-1.0 / x));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 4e-177:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.exp((-1.0 / x))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 4e-177)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = exp(Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 4e-177)
		tmp = t * (sqrt(x) / l);
	else
		tmp = exp((-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 4e-177], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Exp[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.99999999999999981e-177

   1. Initial program 31.2%

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

      1. associate-*l/ 31.2%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in x around inf 52.6%

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

      1. associate--l+ 52.6%

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

      2. unpow2 52.6%

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

      3. distribute-lft-out 52.6%

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

      4. unpow2 52.6%

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

      5. unpow2 52.6%

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

      6. associate-*r/ 52.6%

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

      7. mul-1-neg 52.6%

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

      8. unpow2 52.6%

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

      9. +-commutative 52.6%

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

      10. unpow2 52.6%

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

      11. fma-udef 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in l around inf 21.6%

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

   8. Taylor expanded in l around 0 21.6%

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

      1. associate-*l/ 21.6%

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

      2. *-lft-identity 21.6%

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

   10. Simplified 21.6%

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

   if 3.99999999999999981e-177 < t

   1. Initial program 40.5%

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

      1. associate-*l/ 40.5%

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

   3. Simplified 40.5%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in t around inf 91.5%

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

   7. Taylor expanded in x around inf 91.2%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-177}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-1}{x}}\\ \end{array} \]

Alternative 11: 39.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ e^{\frac{-1}{x}} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (exp (/ -1.0 x)))

C

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

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = exp(((-1.0d0) / x))
end function

Java

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

Python

def code(x, l, t):
	return math.exp((-1.0 / x))

Julia

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

MATLAB

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

Wolfram

code[x_, l_, t_] := N[Exp[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 34.9%

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

   1. associate-*l/ 34.9%

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

3. Simplified 34.9%

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

5. Applied egg-rr 11.2%

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

6. Taylor expanded in t around inf 40.0%

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

7. Taylor expanded in x around inf 39.9%

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

8. Final simplification 39.9%

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

Alternative 12: 38.9% 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 34.9%

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

   1. associate-*l/ 34.9%

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

3. Simplified 34.9%

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

4. Taylor expanded in x around inf 38.5%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
5. Step-by-step derivation

   1. sqrt-unprod 39.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot 0.5}} \]

   2. metadata-eval 39.1%

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

   3. metadata-eval 39.1%

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

6. Applied egg-rr 39.1%

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

7. Final simplification 39.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023217 
(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)))))