Toniolo and Linder, Equation (7)

Percentage Accurate: 32.8% → 83.9%

Time: 23.7s

Alternatives: 16

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 32.8% 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: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := 2 \cdot \left(t \cdot t\right)\\ t_4 := \mathsf{fma}\left(\ell, \ell, t_3\right)\\ t_5 := \frac{t_4}{x}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(t_1 + \frac{\ell \cdot \ell}{{x}^{3}}\right) + \frac{t_4 + t_4}{x \cdot x}\right) + \left(\frac{t_4}{{x}^{3}} + t_5\right)\right)}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \frac{1}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(\ell \cdot \ell + t_3\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+60}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_1 + t_5\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))
        (t_2 (/ (* l l) x))
        (t_3 (* 2.0 (* t t)))
        (t_4 (fma l l t_3))
        (t_5 (/ t_4 x)))
   (if (<= t -5.6e+136)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (<= t -4e-155)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt
          (+
           t_2
           (+
            (fma
             2.0
             (/ (* t t) (pow x 3.0))
             (+ (+ t_1 (/ (* l l) (pow x 3.0))) (/ (+ t_4 t_4) (* x x))))
            (+ (/ t_4 (pow x 3.0)) t_5))))))
       (if (<= t 1.2e-278)
         (*
          (sqrt 2.0)
          (* t (/ 1.0 (* l (sqrt (+ (/ 2.0 (* x x)) (/ 2.0 x)))))))
         (if (<= t 2.45e-183)
           (*
            t
            (/
             (sqrt 2.0)
             (fma
              0.5
              (/ (+ (* l l) (+ (* l l) t_3)) (* (sqrt 2.0) (* t x)))
              (* t (sqrt 2.0)))))
           (if (<= t 1.85e+60)
             (* t (/ (sqrt 2.0) (sqrt (+ t_2 (+ t_1 t_5)))))
             1.0)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 2.0 * ((t * t) + ((t * t) / x));
	double t_2 = (l * l) / x;
	double t_3 = 2.0 * (t * t);
	double t_4 = fma(l, l, t_3);
	double t_5 = t_4 / x;
	double tmp;
	if (t <= -5.6e+136) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -4e-155) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + (fma(2.0, ((t * t) / pow(x, 3.0)), ((t_1 + ((l * l) / pow(x, 3.0))) + ((t_4 + t_4) / (x * x)))) + ((t_4 / pow(x, 3.0)) + t_5)))));
	} else if (t <= 1.2e-278) {
		tmp = sqrt(2.0) * (t * (1.0 / (l * sqrt(((2.0 / (x * x)) + (2.0 / x))))));
	} else if (t <= 2.45e-183) {
		tmp = t * (sqrt(2.0) / fma(0.5, (((l * l) + ((l * l) + t_3)) / (sqrt(2.0) * (t * x))), (t * sqrt(2.0))));
	} else if (t <= 1.85e+60) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + (t_1 + t_5))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(2.0 * Float64(t * t))
	t_4 = fma(l, l, t_3)
	t_5 = Float64(t_4 / x)
	tmp = 0.0
	if (t <= -5.6e+136)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= -4e-155)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(fma(2.0, Float64(Float64(t * t) / (x ^ 3.0)), Float64(Float64(t_1 + Float64(Float64(l * l) / (x ^ 3.0))) + Float64(Float64(t_4 + t_4) / Float64(x * x)))) + Float64(Float64(t_4 / (x ^ 3.0)) + t_5))))));
	elseif (t <= 1.2e-278)
		tmp = Float64(sqrt(2.0) * Float64(t * Float64(1.0 / Float64(l * sqrt(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x)))))));
	elseif (t <= 2.45e-183)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(Float64(l * l) + Float64(Float64(l * l) + t_3)) / Float64(sqrt(2.0) * Float64(t * x))), Float64(t * sqrt(2.0)))));
	elseif (t <= 1.85e+60)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(t_1 + t_5)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(l * l + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / x), $MachinePrecision]}, If[LessEqual[t, -5.6e+136], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -4e-155], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[(N[(l * l), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 + t$95$4), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e-278], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * N[(1.0 / N[(l * N[Sqrt[N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.45e-183], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(N[(l * l), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.85e+60], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(t$95$1 + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -5.6000000000000004e136

   1. Initial program 3.9%

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

      1. associate-*r/ 3.9%

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

      2. fma-neg 3.9%

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

      3. sub-neg 3.9%

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

      4. metadata-eval 3.9%

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

      5. +-commutative 3.9%

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

      6. fma-def 3.9%

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

      7. distribute-rgt-neg-in 3.9%

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

   3. Simplified 3.9%

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

   5. Applied egg-rr 83.5%

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

   6. Taylor expanded in t around -inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

      4. +-commutative 94.5%

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

   8. Simplified 94.5%

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

   if -5.6000000000000004e136 < t < -4.00000000000000006e-155

   1. Initial program 62.5%

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

      1. associate-*l/ 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in x around -inf 86.4%

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

   6. Simplified 86.4%

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

   if -4.00000000000000006e-155 < t < 1.2e-278

   1. Initial program 1.7%

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

      1. associate-*r/ 1.7%

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

      2. fma-neg 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. fma-def 1.7%

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

      7. distribute-rgt-neg-in 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in l around inf 8.5%

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

   5. Taylor expanded in x around inf 49.7%

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

      1. associate-*r/ 49.7%

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

      2. metadata-eval 49.7%

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

      3. unpow2 49.7%

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

      4. associate-*r/ 49.7%

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

      5. metadata-eval 49.7%

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

   7. Simplified 49.7%

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

      1. div-inv 49.7%

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

      2. *-commutative 49.7%

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

   9. Applied egg-rr 49.7%

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

   if 1.2e-278 < t < 2.45e-183

   1. Initial program 2.4%

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

      1. associate-*l/ 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in x around inf 34.8%

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

      1. associate--l+ 34.8%

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

      2. unpow2 34.8%

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

      3. distribute-lft-out 34.8%

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

      4. unpow2 34.8%

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

      5. unpow2 34.8%

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

      6. associate-*r/ 34.8%

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

      7. mul-1-neg 34.8%

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

      8. unpow2 34.8%

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

      9. fma-udef 34.8%

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

      10. unpow2 34.8%

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

   6. Simplified 34.8%

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

   7. Taylor expanded in l around inf 34.8%

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

      1. associate-*r/ 34.8%

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

      2. mul-1-neg 34.8%

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

      3. unpow2 34.8%

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

      4. distribute-rgt-neg-in 34.8%

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

   9. Simplified 34.8%

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

   10. Taylor expanded in x around inf 75.9%

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

       1. fma-def 75.9%

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

       2. unpow2 75.9%

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

       3. unpow2 75.9%

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

       4. mul-1-neg 75.9%

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

       5. unpow2 75.9%

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

       6. distribute-rgt-neg-out 75.9%

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

   12. Simplified 75.9%

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

   if 2.45e-183 < t < 1.84999999999999994e60

   1. Initial program 56.8%

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

      1. associate-*l/ 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in x around inf 91.5%

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

      1. associate--l+ 91.5%

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

      2. unpow2 91.5%

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

      3. distribute-lft-out 91.5%

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

      4. unpow2 91.5%

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

      5. unpow2 91.5%

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

      6. associate-*r/ 91.5%

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

      7. mul-1-neg 91.5%

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

      8. unpow2 91.5%

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

      9. fma-udef 91.5%

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

      10. unpow2 91.5%

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

   6. Simplified 91.5%

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

   if 1.84999999999999994e60 < t

   1. Initial program 39.7%

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

      1. associate-*l/ 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. sqrt-unprod 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell}{{x}^{3}}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) + \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x \cdot x}\right) + \left(\frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{{x}^{3}} + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \frac{1}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+60}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(t \cdot t\right)\\ t_2 := \mathsf{fma}\left(\ell, \ell, t_1\right)\\ t_3 := \frac{t_2}{x}\\ t_4 := \frac{\ell \cdot \ell}{x}\\ t_5 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_4 + \left(\left(t_5 + \frac{t_2 + t_2}{x \cdot x}\right) + t_3\right)}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \frac{1}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(\ell \cdot \ell + t_1\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+59}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_4 + \left(t_5 + t_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (* t t)))
        (t_2 (fma l l t_1))
        (t_3 (/ t_2 x))
        (t_4 (/ (* l l) x))
        (t_5 (* 2.0 (+ (* t t) (/ (* t t) x)))))
   (if (<= t -5.6e+136)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (<= t -4e-155)
       (*
        t
        (/ (sqrt 2.0) (sqrt (+ t_4 (+ (+ t_5 (/ (+ t_2 t_2) (* x x))) t_3)))))
       (if (<= t 2e-280)
         (*
          (sqrt 2.0)
          (* t (/ 1.0 (* l (sqrt (+ (/ 2.0 (* x x)) (/ 2.0 x)))))))
         (if (<= t 2.6e-183)
           (*
            t
            (/
             (sqrt 2.0)
             (fma
              0.5
              (/ (+ (* l l) (+ (* l l) t_1)) (* (sqrt 2.0) (* t x)))
              (* t (sqrt 2.0)))))
           (if (<= t 3.9e+59)
             (* t (/ (sqrt 2.0) (sqrt (+ t_4 (+ t_5 t_3)))))
             1.0)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 2.0 * (t * t);
	double t_2 = fma(l, l, t_1);
	double t_3 = t_2 / x;
	double t_4 = (l * l) / x;
	double t_5 = 2.0 * ((t * t) + ((t * t) / x));
	double tmp;
	if (t <= -5.6e+136) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -4e-155) {
		tmp = t * (sqrt(2.0) / sqrt((t_4 + ((t_5 + ((t_2 + t_2) / (x * x))) + t_3))));
	} else if (t <= 2e-280) {
		tmp = sqrt(2.0) * (t * (1.0 / (l * sqrt(((2.0 / (x * x)) + (2.0 / x))))));
	} else if (t <= 2.6e-183) {
		tmp = t * (sqrt(2.0) / fma(0.5, (((l * l) + ((l * l) + t_1)) / (sqrt(2.0) * (t * x))), (t * sqrt(2.0))));
	} else if (t <= 3.9e+59) {
		tmp = t * (sqrt(2.0) / sqrt((t_4 + (t_5 + t_3))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(2.0 * Float64(t * t))
	t_2 = fma(l, l, t_1)
	t_3 = Float64(t_2 / x)
	t_4 = Float64(Float64(l * l) / x)
	t_5 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	tmp = 0.0
	if (t <= -5.6e+136)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= -4e-155)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_4 + Float64(Float64(t_5 + Float64(Float64(t_2 + t_2) / Float64(x * x))) + t_3)))));
	elseif (t <= 2e-280)
		tmp = Float64(sqrt(2.0) * Float64(t * Float64(1.0 / Float64(l * sqrt(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x)))))));
	elseif (t <= 2.6e-183)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(Float64(l * l) + Float64(Float64(l * l) + t_1)) / Float64(sqrt(2.0) * Float64(t * x))), Float64(t * sqrt(2.0)))));
	elseif (t <= 3.9e+59)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_4 + Float64(t_5 + t_3)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * l + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e+136], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -4e-155], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$4 + N[(N[(t$95$5 + N[(N[(t$95$2 + t$95$2), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e-280], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * N[(1.0 / N[(l * N[Sqrt[N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e-183], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(N[(l * l), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.9e+59], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$4 + N[(t$95$5 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -5.6000000000000004e136

   1. Initial program 3.9%

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

      1. associate-*r/ 3.9%

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

      2. fma-neg 3.9%

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

      3. sub-neg 3.9%

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

      4. metadata-eval 3.9%

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

      5. +-commutative 3.9%

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

      6. fma-def 3.9%

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

      7. distribute-rgt-neg-in 3.9%

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

   3. Simplified 3.9%

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

   5. Applied egg-rr 83.5%

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

   6. Taylor expanded in t around -inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

      4. +-commutative 94.5%

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

   8. Simplified 94.5%

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

   if -5.6000000000000004e136 < t < -4.00000000000000006e-155

   1. Initial program 62.5%

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

      1. associate-*l/ 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in x around -inf 86.3%

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

      1. associate--l+ 86.3%

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

      2. unpow2 86.3%

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

      3. sub-neg 86.3%

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

   6. Simplified 86.3%

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

   if -4.00000000000000006e-155 < t < 1.9999999999999999e-280

   1. Initial program 1.7%

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

      1. associate-*r/ 1.7%

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

      2. fma-neg 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. fma-def 1.7%

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

      7. distribute-rgt-neg-in 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in l around inf 8.5%

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

   5. Taylor expanded in x around inf 49.7%

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

      1. associate-*r/ 49.7%

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

      2. metadata-eval 49.7%

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

      3. unpow2 49.7%

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

      4. associate-*r/ 49.7%

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

      5. metadata-eval 49.7%

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

   7. Simplified 49.7%

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

      1. div-inv 49.7%

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

      2. *-commutative 49.7%

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

   9. Applied egg-rr 49.7%

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

   if 1.9999999999999999e-280 < t < 2.5999999999999999e-183

   1. Initial program 2.4%

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

      1. associate-*l/ 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in x around inf 34.8%

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

      1. associate--l+ 34.8%

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

      2. unpow2 34.8%

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

      3. distribute-lft-out 34.8%

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

      4. unpow2 34.8%

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

      5. unpow2 34.8%

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

      6. associate-*r/ 34.8%

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

      7. mul-1-neg 34.8%

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

      8. unpow2 34.8%

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

      9. fma-udef 34.8%

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

      10. unpow2 34.8%

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

   6. Simplified 34.8%

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

   7. Taylor expanded in l around inf 34.8%

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

      1. associate-*r/ 34.8%

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

      2. mul-1-neg 34.8%

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

      3. unpow2 34.8%

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

      4. distribute-rgt-neg-in 34.8%

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

   9. Simplified 34.8%

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

   10. Taylor expanded in x around inf 75.9%

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

       1. fma-def 75.9%

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

       2. unpow2 75.9%

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

       3. unpow2 75.9%

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

       4. mul-1-neg 75.9%

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

       5. unpow2 75.9%

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

       6. distribute-rgt-neg-out 75.9%

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

   12. Simplified 75.9%

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

   if 2.5999999999999999e-183 < t < 3.90000000000000021e59

   1. Initial program 56.8%

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

      1. associate-*l/ 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in x around inf 91.5%

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

      1. associate--l+ 91.5%

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

      2. unpow2 91.5%

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

      3. distribute-lft-out 91.5%

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

      4. unpow2 91.5%

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

      5. unpow2 91.5%

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

      6. associate-*r/ 91.5%

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

      7. mul-1-neg 91.5%

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

      8. unpow2 91.5%

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

      9. fma-udef 91.5%

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

      10. unpow2 91.5%

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

   6. Simplified 91.5%

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

   if 3.90000000000000021e59 < t

   1. Initial program 39.7%

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

      1. associate-*l/ 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. sqrt-unprod 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) + \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x \cdot x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \frac{1}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\right)\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+59}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(t \cdot t\right)\\ t_2 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, t_1\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-281}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(\ell \cdot \ell + t_1\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (* t t)))
        (t_2
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             (/ (* l l) x)
             (+ (* 2.0 (+ (* t t) (/ (* t t) x))) (/ (fma l l t_1) x))))))))
   (if (<= t -5.6e+136)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (<= t 4e-281)
       t_2
       (if (<= t 1.65e-183)
         (*
          t
          (/
           (sqrt 2.0)
           (fma
            0.5
            (/ (+ (* l l) (+ (* l l) t_1)) (* (sqrt 2.0) (* t x)))
            (* t (sqrt 2.0)))))
         (if (<= t 9.2e+60) t_2 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 2.0 * (t * t);
	double t_2 = t * (sqrt(2.0) / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(l, l, t_1) / x)))));
	double tmp;
	if (t <= -5.6e+136) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= 4e-281) {
		tmp = t_2;
	} else if (t <= 1.65e-183) {
		tmp = t * (sqrt(2.0) / fma(0.5, (((l * l) + ((l * l) + t_1)) / (sqrt(2.0) * (t * x))), (t * sqrt(2.0))));
	} else if (t <= 9.2e+60) {
		tmp = t_2;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(2.0 * Float64(t * t))
	t_2 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(l, l, t_1) / x))))))
	tmp = 0.0
	if (t <= -5.6e+136)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= 4e-281)
		tmp = t_2;
	elseif (t <= 1.65e-183)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(Float64(l * l) + Float64(Float64(l * l) + t_1)) / Float64(sqrt(2.0) * Float64(t * x))), Float64(t * sqrt(2.0)))));
	elseif (t <= 9.2e+60)
		tmp = t_2;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l + t$95$1), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e+136], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 4e-281], t$95$2, If[LessEqual[t, 1.65e-183], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(N[(l * l), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e+60], t$95$2, 1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.6000000000000004e136

   1. Initial program 3.9%

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

      1. associate-*r/ 3.9%

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

      2. fma-neg 3.9%

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

      3. sub-neg 3.9%

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

      4. metadata-eval 3.9%

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

      5. +-commutative 3.9%

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

      6. fma-def 3.9%

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

      7. distribute-rgt-neg-in 3.9%

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

   3. Simplified 3.9%

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

   5. Applied egg-rr 83.5%

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

   6. Taylor expanded in t around -inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

      4. +-commutative 94.5%

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

   8. Simplified 94.5%

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

   if -5.6000000000000004e136 < t < 4.0000000000000001e-281 or 1.65e-183 < t < 9.20000000000000068e60

   1. Initial program 46.6%

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

      1. associate-*l/ 46.6%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in x around inf 82.9%

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

      1. associate--l+ 82.9%

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

      2. unpow2 82.9%

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

      3. distribute-lft-out 82.9%

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

      4. unpow2 82.9%

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

      5. unpow2 82.9%

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

      6. associate-*r/ 82.9%

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

      7. mul-1-neg 82.9%

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

      8. unpow2 82.9%

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

      9. fma-udef 82.9%

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

      10. unpow2 82.9%

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

   6. Simplified 82.9%

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

   if 4.0000000000000001e-281 < t < 1.65e-183

   1. Initial program 2.4%

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

      1. associate-*l/ 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in x around inf 34.8%

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

      1. associate--l+ 34.8%

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

      2. unpow2 34.8%

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

      3. distribute-lft-out 34.8%

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

      4. unpow2 34.8%

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

      5. unpow2 34.8%

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

      6. associate-*r/ 34.8%

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

      7. mul-1-neg 34.8%

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

      8. unpow2 34.8%

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

      9. fma-udef 34.8%

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

      10. unpow2 34.8%

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

   6. Simplified 34.8%

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

   7. Taylor expanded in l around inf 34.8%

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

      1. associate-*r/ 34.8%

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

      2. mul-1-neg 34.8%

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

      3. unpow2 34.8%

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

      4. distribute-rgt-neg-in 34.8%

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

   9. Simplified 34.8%

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

   10. Taylor expanded in x around inf 75.9%

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

       1. fma-def 75.9%

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

       2. unpow2 75.9%

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

       3. unpow2 75.9%

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

       4. mul-1-neg 75.9%

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

       5. unpow2 75.9%

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

       6. distribute-rgt-neg-out 75.9%

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

   12. Simplified 75.9%

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

   if 9.20000000000000068e60 < t

   1. Initial program 39.7%

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

      1. associate-*l/ 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. sqrt-unprod 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-281}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+60}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ t_2 := \sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             (/ (* l l) x)
             (+
              (* 2.0 (+ (* t t) (/ (* t t) x)))
              (/ (fma l l (* 2.0 (* t t))) x)))))))
        (t_2 (* (sqrt 2.0) (/ t (* l (sqrt (/ 2.0 x)))))))
   (if (<= t -5.6e+136)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (<= t -4.4e-155)
       t_1
       (if (<= t 6e-217)
         t_2
         (if (<= t 6.8e-189)
           1.0
           (if (<= t 3.2e-164) t_2 (if (<= t 9e+60) t_1 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(l, l, (2.0 * (t * t))) / x)))));
	double t_2 = sqrt(2.0) * (t / (l * sqrt((2.0 / x))));
	double tmp;
	if (t <= -5.6e+136) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -4.4e-155) {
		tmp = t_1;
	} else if (t <= 6e-217) {
		tmp = t_2;
	} else if (t <= 6.8e-189) {
		tmp = 1.0;
	} else if (t <= 3.2e-164) {
		tmp = t_2;
	} else if (t <= 9e+60) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(l, l, Float64(2.0 * Float64(t * t))) / x))))))
	t_2 = Float64(sqrt(2.0) * Float64(t / Float64(l * sqrt(Float64(2.0 / x)))))
	tmp = 0.0
	if (t <= -5.6e+136)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= -4.4e-155)
		tmp = t_1;
	elseif (t <= 6e-217)
		tmp = t_2;
	elseif (t <= 6.8e-189)
		tmp = 1.0;
	elseif (t <= 3.2e-164)
		tmp = t_2;
	elseif (t <= 9e+60)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e+136], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -4.4e-155], t$95$1, If[LessEqual[t, 6e-217], t$95$2, If[LessEqual[t, 6.8e-189], 1.0, If[LessEqual[t, 3.2e-164], t$95$2, If[LessEqual[t, 9e+60], t$95$1, 1.0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.6000000000000004e136

   1. Initial program 3.9%

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

      1. associate-*r/ 3.9%

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

      2. fma-neg 3.9%

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

      3. sub-neg 3.9%

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

      4. metadata-eval 3.9%

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

      5. +-commutative 3.9%

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

      6. fma-def 3.9%

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

      7. distribute-rgt-neg-in 3.9%

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

   3. Simplified 3.9%

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

   5. Applied egg-rr 83.5%

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

   6. Taylor expanded in t around -inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

      4. +-commutative 94.5%

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

   8. Simplified 94.5%

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

   if -5.6000000000000004e136 < t < -4.3999999999999998e-155 or 3.2e-164 < t < 9.00000000000000026e60

   1. Initial program 62.2%

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

      1. associate-*l/ 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. associate--l+ 88.2%

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

      2. unpow2 88.2%

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

      3. distribute-lft-out 88.2%

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

      4. unpow2 88.2%

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

      5. unpow2 88.2%

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

      6. associate-*r/ 88.2%

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

      7. mul-1-neg 88.2%

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

      8. unpow2 88.2%

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

      9. fma-udef 88.2%

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

      10. unpow2 88.2%

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

   6. Simplified 88.2%

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

   if -4.3999999999999998e-155 < t < 6.00000000000000009e-217 or 6.8000000000000002e-189 < t < 3.2e-164

   1. Initial program 1.7%

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

      1. associate-*r/ 1.7%

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

      2. fma-neg 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. fma-def 1.7%

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

      7. distribute-rgt-neg-in 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in l around inf 8.8%

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

   5. Taylor expanded in x around inf 52.4%

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

   if 6.00000000000000009e-217 < t < 6.8000000000000002e-189 or 9.00000000000000026e60 < t

   1. Initial program 34.5%

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

      1. associate-*l/ 34.5%

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

   3. Simplified 34.5%

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

   4. Taylor expanded in x around inf 94.6%

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

      1. sqrt-unprod 96.1%

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

      2. metadata-eval 96.1%

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

      3. metadata-eval 96.1%

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

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+60}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-155}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+59}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) (/ t (* l (sqrt (/ 2.0 x))))))
        (t_2 (/ (* l l) x))
        (t_3
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt (+ t_2 (+ t_2 (* 2.0 (+ (* t t) (/ (* t t) x))))))))))
   (if (<= t -5.6e+136)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (<= t -4e-155)
       t_3
       (if (<= t 1.25e-216)
         t_1
         (if (<= t 2.3e-190)
           1.0
           (if (<= t 2.45e-163) t_1 (if (<= t 5.9e+59) t_3 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) * (t / (l * sqrt((2.0 / x))));
	double t_2 = (l * l) / x;
	double t_3 = t * (sqrt(2.0) / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -5.6e+136) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -4e-155) {
		tmp = t_3;
	} else if (t <= 1.25e-216) {
		tmp = t_1;
	} else if (t <= 2.3e-190) {
		tmp = 1.0;
	} else if (t <= 2.45e-163) {
		tmp = t_1;
	} else if (t <= 5.9e+59) {
		tmp = t_3;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(2.0d0) * (t / (l * sqrt((2.0d0 / x))))
    t_2 = (l * l) / x
    t_3 = t * (sqrt(2.0d0) / sqrt((t_2 + (t_2 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    if (t <= (-5.6d+136)) then
        tmp = -sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else if (t <= (-4d-155)) then
        tmp = t_3
    else if (t <= 1.25d-216) then
        tmp = t_1
    else if (t <= 2.3d-190) then
        tmp = 1.0d0
    else if (t <= 2.45d-163) then
        tmp = t_1
    else if (t <= 5.9d+59) then
        tmp = t_3
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(2.0) * (t / (l * Math.sqrt((2.0 / x))));
	double t_2 = (l * l) / x;
	double t_3 = t * (Math.sqrt(2.0) / Math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -5.6e+136) {
		tmp = -Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -4e-155) {
		tmp = t_3;
	} else if (t <= 1.25e-216) {
		tmp = t_1;
	} else if (t <= 2.3e-190) {
		tmp = 1.0;
	} else if (t <= 2.45e-163) {
		tmp = t_1;
	} else if (t <= 5.9e+59) {
		tmp = t_3;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(2.0) * (t / (l * math.sqrt((2.0 / x))))
	t_2 = (l * l) / x
	t_3 = t * (math.sqrt(2.0) / math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))))
	tmp = 0
	if t <= -5.6e+136:
		tmp = -math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif t <= -4e-155:
		tmp = t_3
	elif t <= 1.25e-216:
		tmp = t_1
	elif t <= 2.3e-190:
		tmp = 1.0
	elif t <= 2.45e-163:
		tmp = t_1
	elif t <= 5.9e+59:
		tmp = t_3
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) * Float64(t / Float64(l * sqrt(Float64(2.0 / x)))))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))))
	tmp = 0.0
	if (t <= -5.6e+136)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= -4e-155)
		tmp = t_3;
	elseif (t <= 1.25e-216)
		tmp = t_1;
	elseif (t <= 2.3e-190)
		tmp = 1.0;
	elseif (t <= 2.45e-163)
		tmp = t_1;
	elseif (t <= 5.9e+59)
		tmp = t_3;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(2.0) * (t / (l * sqrt((2.0 / x))));
	t_2 = (l * l) / x;
	t_3 = t * (sqrt(2.0) / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	tmp = 0.0;
	if (t <= -5.6e+136)
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	elseif (t <= -4e-155)
		tmp = t_3;
	elseif (t <= 1.25e-216)
		tmp = t_1;
	elseif (t <= 2.3e-190)
		tmp = 1.0;
	elseif (t <= 2.45e-163)
		tmp = t_1;
	elseif (t <= 5.9e+59)
		tmp = t_3;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(t$95$2 + 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, -5.6e+136], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -4e-155], t$95$3, If[LessEqual[t, 1.25e-216], t$95$1, If[LessEqual[t, 2.3e-190], 1.0, If[LessEqual[t, 2.45e-163], t$95$1, If[LessEqual[t, 5.9e+59], t$95$3, 1.0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.6000000000000004e136

   1. Initial program 3.9%

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

      1. associate-*r/ 3.9%

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

      2. fma-neg 3.9%

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

      3. sub-neg 3.9%

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

      4. metadata-eval 3.9%

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

      5. +-commutative 3.9%

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

      6. fma-def 3.9%

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

      7. distribute-rgt-neg-in 3.9%

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

   3. Simplified 3.9%

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

   5. Applied egg-rr 83.5%

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

   6. Taylor expanded in t around -inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

      4. +-commutative 94.5%

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

   8. Simplified 94.5%

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

   if -5.6000000000000004e136 < t < -4.00000000000000006e-155 or 2.4500000000000001e-163 < t < 5.90000000000000038e59

   1. Initial program 62.2%

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

      1. associate-*l/ 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. associate--l+ 88.2%

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

      2. unpow2 88.2%

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

      3. distribute-lft-out 88.2%

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

      4. unpow2 88.2%

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

      5. unpow2 88.2%

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

      6. associate-*r/ 88.2%

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

      7. mul-1-neg 88.2%

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

      8. unpow2 88.2%

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

      9. fma-udef 88.2%

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

      10. unpow2 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in l around inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unpow2 87.7%

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

      4. distribute-rgt-neg-in 87.7%

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

   9. Simplified 87.7%

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

   if -4.00000000000000006e-155 < t < 1.25000000000000005e-216 or 2.29999999999999992e-190 < t < 2.4500000000000001e-163

   1. Initial program 1.7%

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

      1. associate-*r/ 1.7%

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

      2. fma-neg 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. fma-def 1.7%

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

      7. distribute-rgt-neg-in 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in l around inf 8.8%

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

   5. Taylor expanded in x around inf 52.4%

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

   if 1.25000000000000005e-216 < t < 2.29999999999999992e-190 or 5.90000000000000038e59 < t

   1. Initial program 34.5%

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

      1. associate-*l/ 34.5%

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

   3. Simplified 34.5%

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

   4. Taylor expanded in x around inf 94.6%

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

      1. sqrt-unprod 96.1%

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

      2. metadata-eval 96.1%

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

      3. metadata-eval 96.1%

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

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+136}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+59}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -1.04 \cdot 10^{-151}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-216}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -1.04e-151)
     (- t_1)
     (if (<= t 2.55e-216)
       (/ (* t (sqrt 2.0)) (* l (sqrt (+ (/ 2.0 (* x x)) (/ 2.0 x)))))
       (if (<= t 1.75e-189)
         1.0
         (if (<= t 1.02e-164)
           (* (sqrt 2.0) (/ t (* l (sqrt (/ 2.0 x)))))
           t_1))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -1.04e-151) {
		tmp = -t_1;
	} else if (t <= 2.55e-216) {
		tmp = (t * sqrt(2.0)) / (l * sqrt(((2.0 / (x * x)) + (2.0 / x))));
	} else if (t <= 1.75e-189) {
		tmp = 1.0;
	} else if (t <= 1.02e-164) {
		tmp = sqrt(2.0) * (t / (l * sqrt((2.0 / x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    if (t <= (-1.04d-151)) then
        tmp = -t_1
    else if (t <= 2.55d-216) then
        tmp = (t * sqrt(2.0d0)) / (l * sqrt(((2.0d0 / (x * x)) + (2.0d0 / x))))
    else if (t <= 1.75d-189) then
        tmp = 1.0d0
    else if (t <= 1.02d-164) then
        tmp = sqrt(2.0d0) * (t / (l * sqrt((2.0d0 / x))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -1.04e-151) {
		tmp = -t_1;
	} else if (t <= 2.55e-216) {
		tmp = (t * Math.sqrt(2.0)) / (l * Math.sqrt(((2.0 / (x * x)) + (2.0 / x))));
	} else if (t <= 1.75e-189) {
		tmp = 1.0;
	} else if (t <= 1.02e-164) {
		tmp = Math.sqrt(2.0) * (t / (l * Math.sqrt((2.0 / x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((-1.0 + x) / (x + 1.0)))
	tmp = 0
	if t <= -1.04e-151:
		tmp = -t_1
	elif t <= 2.55e-216:
		tmp = (t * math.sqrt(2.0)) / (l * math.sqrt(((2.0 / (x * x)) + (2.0 / x))))
	elif t <= 1.75e-189:
		tmp = 1.0
	elif t <= 1.02e-164:
		tmp = math.sqrt(2.0) * (t / (l * math.sqrt((2.0 / x))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.04e-151)
		tmp = Float64(-t_1);
	elseif (t <= 2.55e-216)
		tmp = Float64(Float64(t * sqrt(2.0)) / Float64(l * sqrt(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x)))));
	elseif (t <= 1.75e-189)
		tmp = 1.0;
	elseif (t <= 1.02e-164)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(l * sqrt(Float64(2.0 / x)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1.04e-151)
		tmp = -t_1;
	elseif (t <= 2.55e-216)
		tmp = (t * sqrt(2.0)) / (l * sqrt(((2.0 / (x * x)) + (2.0 / x))));
	elseif (t <= 1.75e-189)
		tmp = 1.0;
	elseif (t <= 1.02e-164)
		tmp = sqrt(2.0) * (t / (l * sqrt((2.0 / x))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.04e-151], (-t$95$1), If[LessEqual[t, 2.55e-216], N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.75e-189], 1.0, If[LessEqual[t, 1.02e-164], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.04000000000000005e-151

   1. Initial program 34.1%

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

      1. associate-*r/ 34.1%

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

      2. fma-neg 34.1%

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

      3. sub-neg 34.1%

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

      4. metadata-eval 34.1%

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

      5. +-commutative 34.1%

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

      6. fma-def 34.1%

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

      7. distribute-rgt-neg-in 34.1%

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

   3. Simplified 34.1%

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

   5. Applied egg-rr 74.2%

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

   6. Taylor expanded in t around -inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. sub-neg 84.3%

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

      3. metadata-eval 84.3%

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

      4. +-commutative 84.3%

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

   8. Simplified 84.3%

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

   if -1.04000000000000005e-151 < t < 2.5500000000000001e-216

   1. Initial program 1.8%

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

      1. associate-*r/ 1.8%

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

      2. fma-neg 1.8%

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

      3. sub-neg 1.8%

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

      4. metadata-eval 1.8%

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

      5. +-commutative 1.8%

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

      6. fma-def 1.8%

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

      7. distribute-rgt-neg-in 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in l around inf 7.0%

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

   5. Taylor expanded in x around inf 52.4%

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

      1. associate-*r/ 52.4%

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

      2. metadata-eval 52.4%

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

      3. unpow2 52.4%

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

      4. associate-*r/ 52.4%

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

      5. metadata-eval 52.4%

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

   7. Simplified 52.4%

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

      1. associate-*r/ 52.5%

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

      2. *-commutative 52.5%

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

   9. Applied egg-rr 52.5%

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

   if 2.5500000000000001e-216 < t < 1.7500000000000001e-189

   1. Initial program 2.7%

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

      1. associate-*l/ 2.7%

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

   3. Simplified 2.7%

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

   4. Taylor expanded in x around inf 71.4%

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

      1. sqrt-unprod 72.5%

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

      2. metadata-eval 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 72.5%

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

   if 1.7500000000000001e-189 < t < 1.02e-164

   1. Initial program 1.5%

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

      1. associate-*r/ 1.5%

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

      2. fma-neg 1.5%

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

      3. sub-neg 1.5%

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

      4. metadata-eval 1.5%

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

      5. +-commutative 1.5%

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

      6. fma-def 1.5%

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

      7. distribute-rgt-neg-in 1.5%

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

   3. Simplified 1.5%

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

   4. Taylor expanded in l around inf 21.1%

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

   5. Taylor expanded in x around inf 60.9%

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

   if 1.02e-164 < t

   1. Initial program 49.0%

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

      1. associate-*r/ 48.9%

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

      2. fma-neg 48.9%

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

      3. sub-neg 48.9%

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

      4. metadata-eval 48.9%

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

      5. +-commutative 48.9%

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

      6. fma-def 48.9%

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

      7. distribute-rgt-neg-in 48.9%

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

   3. Simplified 48.9%

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

   5. Applied egg-rr 79.6%

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

   6. Taylor expanded in l around 0 90.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.04 \cdot 10^{-151}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-216}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 7: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ t_2 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-151}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) (/ t (* l (sqrt (/ 2.0 x))))))
        (t_2 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -1.2e-151)
     (- t_2)
     (if (<= t 2.4e-216)
       t_1
       (if (<= t 1.16e-189) 1.0 (if (<= t 7e-165) t_1 t_2))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) * (t / (l * sqrt((2.0 / x))));
	double t_2 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -1.2e-151) {
		tmp = -t_2;
	} else if (t <= 2.4e-216) {
		tmp = t_1;
	} else if (t <= 1.16e-189) {
		tmp = 1.0;
	} else if (t <= 7e-165) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(2.0d0) * (t / (l * sqrt((2.0d0 / x))))
    t_2 = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    if (t <= (-1.2d-151)) then
        tmp = -t_2
    else if (t <= 2.4d-216) then
        tmp = t_1
    else if (t <= 1.16d-189) then
        tmp = 1.0d0
    else if (t <= 7d-165) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(2.0) * (t / (l * Math.sqrt((2.0 / x))));
	double t_2 = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -1.2e-151) {
		tmp = -t_2;
	} else if (t <= 2.4e-216) {
		tmp = t_1;
	} else if (t <= 1.16e-189) {
		tmp = 1.0;
	} else if (t <= 7e-165) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(2.0) * (t / (l * math.sqrt((2.0 / x))))
	t_2 = math.sqrt(((-1.0 + x) / (x + 1.0)))
	tmp = 0
	if t <= -1.2e-151:
		tmp = -t_2
	elif t <= 2.4e-216:
		tmp = t_1
	elif t <= 1.16e-189:
		tmp = 1.0
	elif t <= 7e-165:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) * Float64(t / Float64(l * sqrt(Float64(2.0 / x)))))
	t_2 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.2e-151)
		tmp = Float64(-t_2);
	elseif (t <= 2.4e-216)
		tmp = t_1;
	elseif (t <= 1.16e-189)
		tmp = 1.0;
	elseif (t <= 7e-165)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(2.0) * (t / (l * sqrt((2.0 / x))));
	t_2 = sqrt(((-1.0 + x) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1.2e-151)
		tmp = -t_2;
	elseif (t <= 2.4e-216)
		tmp = t_1;
	elseif (t <= 1.16e-189)
		tmp = 1.0;
	elseif (t <= 7e-165)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.2e-151], (-t$95$2), If[LessEqual[t, 2.4e-216], t$95$1, If[LessEqual[t, 1.16e-189], 1.0, If[LessEqual[t, 7e-165], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.2e-151

   1. Initial program 34.1%

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

      1. associate-*r/ 34.1%

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

      2. fma-neg 34.1%

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

      3. sub-neg 34.1%

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

      4. metadata-eval 34.1%

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

      5. +-commutative 34.1%

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

      6. fma-def 34.1%

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

      7. distribute-rgt-neg-in 34.1%

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

   3. Simplified 34.1%

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

   5. Applied egg-rr 74.2%

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

   6. Taylor expanded in t around -inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. sub-neg 84.3%

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

      3. metadata-eval 84.3%

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

      4. +-commutative 84.3%

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

   8. Simplified 84.3%

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

   if -1.2e-151 < t < 2.40000000000000004e-216 or 1.1600000000000001e-189 < t < 7.0000000000000003e-165

   1. Initial program 1.7%

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

      1. associate-*r/ 1.7%

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

      2. fma-neg 1.7%

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

      3. sub-neg 1.7%

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

      4. metadata-eval 1.7%

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

      5. +-commutative 1.7%

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

      6. fma-def 1.7%

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

      7. distribute-rgt-neg-in 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in l around inf 8.7%

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

   5. Taylor expanded in x around inf 53.4%

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

   if 2.40000000000000004e-216 < t < 1.1600000000000001e-189

   1. Initial program 2.7%

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

      1. associate-*l/ 2.7%

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

   3. Simplified 2.7%

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

   4. Taylor expanded in x around inf 71.4%

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

      1. sqrt-unprod 72.5%

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

      2. metadata-eval 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 72.5%

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

   if 7.0000000000000003e-165 < t

   1. Initial program 49.0%

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

      1. associate-*r/ 48.9%

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

      2. fma-neg 48.9%

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

      3. sub-neg 48.9%

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

      4. metadata-eval 48.9%

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

      5. +-commutative 48.9%

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

      6. fma-def 48.9%

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

      7. distribute-rgt-neg-in 48.9%

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

   3. Simplified 48.9%

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

   5. Applied egg-rr 79.6%

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

   6. Taylor expanded in l around 0 90.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-151}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 8: 79.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{if}\;t \leq -2.65 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (/ (sqrt x) l))))
   (if (<= t -2.65e-151)
     (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
     (if (<= t 4e-217)
       t_1
       (if (<= t 4e-189)
         1.0
         (if (<= t 3.8e-164) t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(x) / l);
	double tmp;
	if (t <= -2.65e-151) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 4e-217) {
		tmp = t_1;
	} else if (t <= 4e-189) {
		tmp = 1.0;
	} else if (t <= 3.8e-164) {
		tmp = t_1;
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (sqrt(x) / l)
    if (t <= (-2.65d-151)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if (t <= 4d-217) then
        tmp = t_1
    else if (t <= 4d-189) then
        tmp = 1.0d0
    else if (t <= 3.8d-164) then
        tmp = t_1
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(x) / l);
	double tmp;
	if (t <= -2.65e-151) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 4e-217) {
		tmp = t_1;
	} else if (t <= 4e-189) {
		tmp = 1.0;
	} else if (t <= 3.8e-164) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * (math.sqrt(x) / l)
	tmp = 0
	if t <= -2.65e-151:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif t <= 4e-217:
		tmp = t_1
	elif t <= 4e-189:
		tmp = 1.0
	elif t <= 3.8e-164:
		tmp = t_1
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(x) / l))
	tmp = 0.0
	if (t <= -2.65e-151)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif (t <= 4e-217)
		tmp = t_1;
	elseif (t <= 4e-189)
		tmp = 1.0;
	elseif (t <= 3.8e-164)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(x) / l);
	tmp = 0.0;
	if (t <= -2.65e-151)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif (t <= 4e-217)
		tmp = t_1;
	elseif (t <= 4e-189)
		tmp = 1.0;
	elseif (t <= 3.8e-164)
		tmp = t_1;
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.65e-151], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-217], t$95$1, If[LessEqual[t, 4e-189], 1.0, If[LessEqual[t, 3.8e-164], t$95$1, N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.64999999999999989e-151

   1. Initial program 34.1%

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

      1. associate-*r/ 34.1%

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

      2. fma-neg 34.1%

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

      3. sub-neg 34.1%

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

      4. metadata-eval 34.1%

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

      5. +-commutative 34.1%

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

      6. fma-def 34.1%

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

      7. distribute-rgt-neg-in 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in t around -inf 84.2%

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

      1. associate-*r* 84.2%

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

      2. neg-mul-1 84.2%

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

      3. distribute-rgt-neg-out 84.2%

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

      4. +-commutative 84.2%

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

      5. sub-neg 84.2%

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

      6. metadata-eval 84.2%

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

      7. +-commutative 84.2%

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

   6. Simplified 84.2%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-*r/ 84.0%

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

      2. metadata-eval 84.0%

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

      3. unpow2 84.0%

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

   9. Simplified 84.0%

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

   if -2.64999999999999989e-151 < t < 4.00000000000000033e-217 or 4.00000000000000027e-189 < t < 3.79999999999999989e-164

   1. Initial program 1.7%

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

      1. associate-*l/ 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in x around inf 65.0%

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

      1. associate--l+ 65.0%

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

      2. unpow2 65.0%

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

      3. distribute-lft-out 65.0%

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

      4. unpow2 65.0%

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

      5. unpow2 65.0%

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

      6. associate-*r/ 65.0%

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

      7. mul-1-neg 65.0%

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

      8. unpow2 65.0%

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

      9. fma-udef 65.0%

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

      10. unpow2 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in t around 0 64.7%

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

      1. cancel-sign-sub-inv 64.7%

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

      2. metadata-eval 64.7%

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

      3. distribute-rgt1-in 64.7%

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

      4. metadata-eval 64.7%

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

      5. unpow2 64.7%

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

      6. associate-*l/ 69.1%

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

      7. *-commutative 69.1%

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

   9. Simplified 69.1%

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

   10. Taylor expanded in l around 0 53.3%

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

       1. associate-*l/ 53.4%

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

       2. *-lft-identity 53.4%

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

   12. Simplified 53.4%

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

   if 4.00000000000000033e-217 < t < 4.00000000000000027e-189

   1. Initial program 2.7%

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

      1. associate-*l/ 2.7%

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

   3. Simplified 2.7%

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

   4. Taylor expanded in x around inf 71.4%

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

      1. sqrt-unprod 72.5%

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

      2. metadata-eval 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 72.5%

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

   if 3.79999999999999989e-164 < t

   1. Initial program 49.0%

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

      1. associate-*r/ 48.9%

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

      2. fma-neg 48.9%

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

      3. sub-neg 48.9%

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

      4. metadata-eval 48.9%

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

      5. +-commutative 48.9%

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

      6. fma-def 48.9%

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

      7. distribute-rgt-neg-in 48.9%

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

   3. Simplified 48.9%

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

   5. Applied egg-rr 79.6%

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

   6. Taylor expanded in l around 0 90.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 9: 79.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.38 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2.3e-151)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (<= t 2.4e-216)
     (* t (/ (sqrt x) l))
     (if (<= t 1.02e-189)
       1.0
       (if (<= t 1.38e-163)
         (* t (* (sqrt x) (/ 1.0 l)))
         (sqrt (/ (+ -1.0 x) (+ x 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.3e-151) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 2.4e-216) {
		tmp = t * (sqrt(x) / l);
	} else if (t <= 1.02e-189) {
		tmp = 1.0;
	} else if (t <= 1.38e-163) {
		tmp = t * (sqrt(x) * (1.0 / l));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.3d-151)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if (t <= 2.4d-216) then
        tmp = t * (sqrt(x) / l)
    else if (t <= 1.02d-189) then
        tmp = 1.0d0
    else if (t <= 1.38d-163) then
        tmp = t * (sqrt(x) * (1.0d0 / l))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.3e-151) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 2.4e-216) {
		tmp = t * (Math.sqrt(x) / l);
	} else if (t <= 1.02e-189) {
		tmp = 1.0;
	} else if (t <= 1.38e-163) {
		tmp = t * (Math.sqrt(x) * (1.0 / l));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2.3e-151:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif t <= 2.4e-216:
		tmp = t * (math.sqrt(x) / l)
	elif t <= 1.02e-189:
		tmp = 1.0
	elif t <= 1.38e-163:
		tmp = t * (math.sqrt(x) * (1.0 / l))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2.3e-151)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif (t <= 2.4e-216)
		tmp = Float64(t * Float64(sqrt(x) / l));
	elseif (t <= 1.02e-189)
		tmp = 1.0;
	elseif (t <= 1.38e-163)
		tmp = Float64(t * Float64(sqrt(x) * Float64(1.0 / l)));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2.3e-151)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif (t <= 2.4e-216)
		tmp = t * (sqrt(x) / l);
	elseif (t <= 1.02e-189)
		tmp = 1.0;
	elseif (t <= 1.38e-163)
		tmp = t * (sqrt(x) * (1.0 / l));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2.3e-151], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-216], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.02e-189], 1.0, If[LessEqual[t, 1.38e-163], N[(t * N[(N[Sqrt[x], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.29999999999999996e-151

   1. Initial program 34.1%

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

      1. associate-*r/ 34.1%

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

      2. fma-neg 34.1%

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

      3. sub-neg 34.1%

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

      4. metadata-eval 34.1%

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

      5. +-commutative 34.1%

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

      6. fma-def 34.1%

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

      7. distribute-rgt-neg-in 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in t around -inf 84.2%

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

      1. associate-*r* 84.2%

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

      2. neg-mul-1 84.2%

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

      3. distribute-rgt-neg-out 84.2%

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

      4. +-commutative 84.2%

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

      5. sub-neg 84.2%

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

      6. metadata-eval 84.2%

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

      7. +-commutative 84.2%

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

   6. Simplified 84.2%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-*r/ 84.0%

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

      2. metadata-eval 84.0%

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

      3. unpow2 84.0%

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

   9. Simplified 84.0%

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

   if -2.29999999999999996e-151 < t < 2.40000000000000004e-216

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in x around inf 63.0%

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

      1. associate--l+ 63.0%

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

      2. unpow2 63.0%

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

      3. distribute-lft-out 63.0%

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

      4. unpow2 63.0%

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

      5. unpow2 63.0%

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

      6. associate-*r/ 63.0%

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

      7. mul-1-neg 63.0%

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

      8. unpow2 63.0%

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

      9. fma-udef 63.0%

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

      10. unpow2 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in t around 0 62.7%

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

      1. cancel-sign-sub-inv 62.7%

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

      2. metadata-eval 62.7%

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

      3. distribute-rgt1-in 62.7%

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

      4. metadata-eval 62.7%

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

      5. unpow2 62.7%

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

      6. associate-*l/ 67.7%

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

      7. *-commutative 67.7%

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

   9. Simplified 67.7%

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

   10. Taylor expanded in l around 0 52.3%

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

       1. associate-*l/ 52.4%

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

       2. *-lft-identity 52.4%

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

   12. Simplified 52.4%

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

   if 2.40000000000000004e-216 < t < 1.01999999999999999e-189

   1. Initial program 2.7%

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

      1. associate-*l/ 2.7%

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

   3. Simplified 2.7%

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

   4. Taylor expanded in x around inf 71.4%

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

      1. sqrt-unprod 72.5%

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

      2. metadata-eval 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 72.5%

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

   if 1.01999999999999999e-189 < t < 1.37999999999999999e-163

   1. Initial program 1.5%

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

      1. associate-*l/ 1.5%

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

   3. Simplified 1.5%

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

   4. Taylor expanded in x around inf 80.0%

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

      1. associate--l+ 80.0%

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

      2. unpow2 80.0%

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

      3. distribute-lft-out 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

      6. associate-*r/ 80.0%

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

      7. mul-1-neg 80.0%

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

      8. unpow2 80.0%

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

      9. fma-udef 80.0%

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

      10. unpow2 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in t around 0 80.0%

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

      1. cancel-sign-sub-inv 80.0%

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

      2. metadata-eval 80.0%

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

      3. distribute-rgt1-in 80.0%

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

      4. metadata-eval 80.0%

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

      5. unpow2 80.0%

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

      6. associate-*l/ 80.0%

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

      7. *-commutative 80.0%

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

   9. Simplified 80.0%

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

   10. Taylor expanded in l around 0 60.9%

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

   if 1.37999999999999999e-163 < t

   1. Initial program 49.0%

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

      1. associate-*r/ 48.9%

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

      2. fma-neg 48.9%

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

      3. sub-neg 48.9%

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

      4. metadata-eval 48.9%

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

      5. +-commutative 48.9%

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

      6. fma-def 48.9%

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

      7. distribute-rgt-neg-in 48.9%

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

   3. Simplified 48.9%

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

   5. Applied egg-rr 79.6%

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

   6. Taylor expanded in l around 0 90.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.38 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 10: 79.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-151}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -1e-151)
     (- t_1)
     (if (<= t 1.3e-216)
       (* t (/ (sqrt x) l))
       (if (<= t 3.6e-190)
         1.0
         (if (<= t 1.02e-164) (* t (* (sqrt x) (/ 1.0 l))) t_1))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -1e-151) {
		tmp = -t_1;
	} else if (t <= 1.3e-216) {
		tmp = t * (sqrt(x) / l);
	} else if (t <= 3.6e-190) {
		tmp = 1.0;
	} else if (t <= 1.02e-164) {
		tmp = t * (sqrt(x) * (1.0 / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    if (t <= (-1d-151)) then
        tmp = -t_1
    else if (t <= 1.3d-216) then
        tmp = t * (sqrt(x) / l)
    else if (t <= 3.6d-190) then
        tmp = 1.0d0
    else if (t <= 1.02d-164) then
        tmp = t * (sqrt(x) * (1.0d0 / l))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -1e-151) {
		tmp = -t_1;
	} else if (t <= 1.3e-216) {
		tmp = t * (Math.sqrt(x) / l);
	} else if (t <= 3.6e-190) {
		tmp = 1.0;
	} else if (t <= 1.02e-164) {
		tmp = t * (Math.sqrt(x) * (1.0 / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((-1.0 + x) / (x + 1.0)))
	tmp = 0
	if t <= -1e-151:
		tmp = -t_1
	elif t <= 1.3e-216:
		tmp = t * (math.sqrt(x) / l)
	elif t <= 3.6e-190:
		tmp = 1.0
	elif t <= 1.02e-164:
		tmp = t * (math.sqrt(x) * (1.0 / l))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1e-151)
		tmp = Float64(-t_1);
	elseif (t <= 1.3e-216)
		tmp = Float64(t * Float64(sqrt(x) / l));
	elseif (t <= 3.6e-190)
		tmp = 1.0;
	elseif (t <= 1.02e-164)
		tmp = Float64(t * Float64(sqrt(x) * Float64(1.0 / l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1e-151)
		tmp = -t_1;
	elseif (t <= 1.3e-216)
		tmp = t * (sqrt(x) / l);
	elseif (t <= 3.6e-190)
		tmp = 1.0;
	elseif (t <= 1.02e-164)
		tmp = t * (sqrt(x) * (1.0 / l));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1e-151], (-t$95$1), If[LessEqual[t, 1.3e-216], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e-190], 1.0, If[LessEqual[t, 1.02e-164], N[(t * N[(N[Sqrt[x], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -9.9999999999999994e-152

   1. Initial program 34.1%

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

      1. associate-*r/ 34.1%

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

      2. fma-neg 34.1%

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

      3. sub-neg 34.1%

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

      4. metadata-eval 34.1%

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

      5. +-commutative 34.1%

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

      6. fma-def 34.1%

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

      7. distribute-rgt-neg-in 34.1%

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

   3. Simplified 34.1%

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

   5. Applied egg-rr 74.2%

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

   6. Taylor expanded in t around -inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. sub-neg 84.3%

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

      3. metadata-eval 84.3%

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

      4. +-commutative 84.3%

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

   8. Simplified 84.3%

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

   if -9.9999999999999994e-152 < t < 1.2999999999999999e-216

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in x around inf 63.0%

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

      1. associate--l+ 63.0%

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

      2. unpow2 63.0%

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

      3. distribute-lft-out 63.0%

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

      4. unpow2 63.0%

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

      5. unpow2 63.0%

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

      6. associate-*r/ 63.0%

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

      7. mul-1-neg 63.0%

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

      8. unpow2 63.0%

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

      9. fma-udef 63.0%

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

      10. unpow2 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in t around 0 62.7%

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

      1. cancel-sign-sub-inv 62.7%

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

      2. metadata-eval 62.7%

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

      3. distribute-rgt1-in 62.7%

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

      4. metadata-eval 62.7%

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

      5. unpow2 62.7%

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

      6. associate-*l/ 67.7%

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

      7. *-commutative 67.7%

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

   9. Simplified 67.7%

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

   10. Taylor expanded in l around 0 52.3%

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

       1. associate-*l/ 52.4%

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

       2. *-lft-identity 52.4%

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

   12. Simplified 52.4%

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

   if 1.2999999999999999e-216 < t < 3.60000000000000007e-190

   1. Initial program 2.7%

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

      1. associate-*l/ 2.7%

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

   3. Simplified 2.7%

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

   4. Taylor expanded in x around inf 71.4%

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

      1. sqrt-unprod 72.5%

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

      2. metadata-eval 72.5%

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

      3. metadata-eval 72.5%

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

   6. Applied egg-rr 72.5%

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

   if 3.60000000000000007e-190 < t < 1.02e-164

   1. Initial program 1.5%

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

      1. associate-*l/ 1.5%

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

   3. Simplified 1.5%

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

   4. Taylor expanded in x around inf 80.0%

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

      1. associate--l+ 80.0%

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

      2. unpow2 80.0%

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

      3. distribute-lft-out 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

      6. associate-*r/ 80.0%

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

      7. mul-1-neg 80.0%

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

      8. unpow2 80.0%

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

      9. fma-udef 80.0%

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

      10. unpow2 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in t around 0 80.0%

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

      1. cancel-sign-sub-inv 80.0%

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

      2. metadata-eval 80.0%

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

      3. distribute-rgt1-in 80.0%

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

      4. metadata-eval 80.0%

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

      5. unpow2 80.0%

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

      6. associate-*l/ 80.0%

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

      7. *-commutative 80.0%

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

   9. Simplified 80.0%

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

   10. Taylor expanded in l around 0 60.9%

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

   if 1.02e-164 < t

   1. Initial program 49.0%

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

      1. associate-*r/ 48.9%

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

      2. fma-neg 48.9%

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

      3. sub-neg 48.9%

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

      4. metadata-eval 48.9%

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

      5. +-commutative 48.9%

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

      6. fma-def 48.9%

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

      7. distribute-rgt-neg-in 48.9%

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

   3. Simplified 48.9%

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

   5. Applied egg-rr 79.6%

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

   6. Taylor expanded in l around 0 90.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-151}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 11: 77.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-234} \lor \neg \left(t \leq 8.5 \cdot 10^{-189}\right) \land t \leq 1.12 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-151)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (or (<= t 2.1e-234) (and (not (<= t 8.5e-189)) (<= t 1.12e-164)))
     (* (sqrt x) (/ t l))
     1.0)))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-151) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if ((t <= 2.1e-234) || (!(t <= 8.5e-189) && (t <= 1.12e-164))) {
		tmp = sqrt(x) * (t / l);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1d-151)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if ((t <= 2.1d-234) .or. (.not. (t <= 8.5d-189)) .and. (t <= 1.12d-164)) then
        tmp = sqrt(x) * (t / l)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-151) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if ((t <= 2.1e-234) || (!(t <= 8.5e-189) && (t <= 1.12e-164))) {
		tmp = Math.sqrt(x) * (t / l);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-151:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif (t <= 2.1e-234) or (not (t <= 8.5e-189) and (t <= 1.12e-164)):
		tmp = math.sqrt(x) * (t / l)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-151)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif ((t <= 2.1e-234) || (!(t <= 8.5e-189) && (t <= 1.12e-164)))
		tmp = Float64(sqrt(x) * Float64(t / l));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-151)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif ((t <= 2.1e-234) || (~((t <= 8.5e-189)) && (t <= 1.12e-164)))
		tmp = sqrt(x) * (t / l);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-151], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 2.1e-234], And[N[Not[LessEqual[t, 8.5e-189]], $MachinePrecision], LessEqual[t, 1.12e-164]]], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if t < -9.9999999999999994e-152

   1. Initial program 34.1%

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

      1. associate-*r/ 34.1%

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

      2. fma-neg 34.1%

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

      3. sub-neg 34.1%

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

      4. metadata-eval 34.1%

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

      5. +-commutative 34.1%

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

      6. fma-def 34.1%

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

      7. distribute-rgt-neg-in 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in t around -inf 84.2%

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

      1. associate-*r* 84.2%

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

      2. neg-mul-1 84.2%

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

      3. distribute-rgt-neg-out 84.2%

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

      4. +-commutative 84.2%

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

      5. sub-neg 84.2%

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

      6. metadata-eval 84.2%

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

      7. +-commutative 84.2%

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

   6. Simplified 84.2%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-*r/ 84.0%

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

      2. metadata-eval 84.0%

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

      3. unpow2 84.0%

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

   9. Simplified 84.0%

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

   if -9.9999999999999994e-152 < t < 2.09999999999999991e-234 or 8.50000000000000068e-189 < t < 1.12e-164

   1. Initial program 1.7%

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

      1. associate-*l/ 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in x around inf 66.7%

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

      1. associate--l+ 66.7%

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

      2. unpow2 66.7%

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

      3. distribute-lft-out 66.7%

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

      4. unpow2 66.7%

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

      5. unpow2 66.7%

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

      6. associate-*r/ 66.7%

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

      7. mul-1-neg 66.7%

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

      8. unpow2 66.7%

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

      9. fma-udef 66.7%

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

      10. unpow2 66.7%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in t around 0 66.5%

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

      1. cancel-sign-sub-inv 66.5%

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

      2. metadata-eval 66.5%

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

      3. distribute-rgt1-in 66.5%

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

      4. metadata-eval 66.5%

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

      5. unpow2 66.5%

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

      6. associate-*l/ 71.3%

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

      7. *-commutative 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in l around 0 50.3%

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

   if 2.09999999999999991e-234 < t < 8.50000000000000068e-189 or 1.12e-164 < t

   1. Initial program 43.5%

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

      1. associate-*l/ 43.5%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in x around inf 85.6%

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

      1. sqrt-unprod 86.9%

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

      2. metadata-eval 86.9%

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

      3. metadata-eval 86.9%

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

   6. Applied egg-rr 86.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-234} \lor \neg \left(t \leq 8.5 \cdot 10^{-189}\right) \land t \leq 1.12 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-216} \lor \neg \left(t \leq 1.52 \cdot 10^{-189}\right) \land t \leq 5.7 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.1e-151)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (or (<= t 2.55e-216) (and (not (<= t 1.52e-189)) (<= t 5.7e-164)))
     (* t (/ (sqrt x) l))
     1.0)))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.1e-151) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if ((t <= 2.55e-216) || (!(t <= 1.52e-189) && (t <= 5.7e-164))) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.1d-151)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if ((t <= 2.55d-216) .or. (.not. (t <= 1.52d-189)) .and. (t <= 5.7d-164)) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.1e-151) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if ((t <= 2.55e-216) || (!(t <= 1.52e-189) && (t <= 5.7e-164))) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.1e-151:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif (t <= 2.55e-216) or (not (t <= 1.52e-189) and (t <= 5.7e-164)):
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.1e-151)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif ((t <= 2.55e-216) || (!(t <= 1.52e-189) && (t <= 5.7e-164)))
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.1e-151)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif ((t <= 2.55e-216) || (~((t <= 1.52e-189)) && (t <= 5.7e-164)))
		tmp = t * (sqrt(x) / l);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.1e-151], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 2.55e-216], And[N[Not[LessEqual[t, 1.52e-189]], $MachinePrecision], LessEqual[t, 5.7e-164]]], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1e-151

   1. Initial program 34.1%

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

      1. associate-*r/ 34.1%

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

      2. fma-neg 34.1%

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

      3. sub-neg 34.1%

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

      4. metadata-eval 34.1%

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

      5. +-commutative 34.1%

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

      6. fma-def 34.1%

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

      7. distribute-rgt-neg-in 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in t around -inf 84.2%

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

      1. associate-*r* 84.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 84.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-\sqrt{2} \cdot t\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. distribute-rgt-neg-out 84.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      4. +-commutative 84.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      5. sub-neg 84.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      6. metadata-eval 84.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      7. +-commutative 84.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 84.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 84.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 84.0%

         \[\leadsto \frac{1}{x} - \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) \]

      2. metadata-eval 84.0%

         \[\leadsto \frac{1}{x} - \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) \]

      3. unpow2 84.0%

         \[\leadsto \frac{1}{x} - \left(1 + \frac{0.5}{\color{blue}{x \cdot x}}\right) \]

   9. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)} \]

   if -1.1e-151 < t < 2.5500000000000001e-216 or 1.5199999999999999e-189 < t < 5.70000000000000023e-164

   1. Initial program 1.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 1.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 1.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 65.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
   5. Step-by-step derivation

      1. associate--l+ 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow2 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow2 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow2 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/ 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow2 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. fma-udef 65.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      10. unpow2 65.0%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(\ell, \ell, 2 \cdot \color{blue}{\left(t \cdot t\right)}\right)}{x}\right)}} \cdot t \]

   6. Simplified 65.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}} \cdot t \]

   7. Taylor expanded in t around 0 64.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 64.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(--1\right) \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]

      2. metadata-eval 64.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \color{blue}{1} \cdot \frac{{\ell}^{2}}{x}}} \cdot t \]

      3. distribute-rgt1-in 64.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(1 + 1\right) \cdot \frac{{\ell}^{2}}{x}}}} \cdot t \]

      4. metadata-eval 64.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2} \cdot \frac{{\ell}^{2}}{x}}} \cdot t \]

      5. unpow2 64.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{x}}} \cdot t \]

      6. associate-*l/ 69.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]

      7. *-commutative 69.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]

   9. Simplified 69.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]

   10. Taylor expanded in l around 0 53.3%

       \[\leadsto \color{blue}{\left(\frac{1}{\ell} \cdot \sqrt{x}\right)} \cdot t \]
   11. Step-by-step derivation

       1. associate-*l/ 53.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x}}{\ell}} \cdot t \]

       2. *-lft-identity 53.4%

          \[\leadsto \frac{\color{blue}{\sqrt{x}}}{\ell} \cdot t \]

   12. Simplified 53.4%

       \[\leadsto \color{blue}{\frac{\sqrt{x}}{\ell}} \cdot t \]

   if 2.5500000000000001e-216 < t < 1.5199999999999999e-189 or 5.70000000000000023e-164 < t

   1. Initial program 45.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 45.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 87.2%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 88.5%

         \[\leadsto \color{blue}{\sqrt{2 \cdot 0.5}} \]

      2. metadata-eval 88.5%

         \[\leadsto \sqrt{\color{blue}{1}} \]

      3. metadata-eval 88.5%

         \[\leadsto \color{blue}{1} \]

   6. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-216} \lor \neg \left(t \leq 1.52 \cdot 10^{-189}\right) \land t \leq 5.7 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 75.8% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x)))) 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 26.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 26.7%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. sub-neg 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

      4. metadata-eval 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

      5. +-commutative 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]

      6. fma-def 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]

      7. distribute-rgt-neg-in 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Simplified 26.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in t around -inf 71.3%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-\sqrt{2} \cdot t\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. distribute-rgt-neg-out 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      4. +-commutative 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      5. sub-neg 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      6. metadata-eval 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      7. +-commutative 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 71.3%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 71.2%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 71.2%

         \[\leadsto \frac{1}{x} - \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) \]

      2. metadata-eval 71.2%

         \[\leadsto \frac{1}{x} - \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) \]

      3. unpow2 71.2%

         \[\leadsto \frac{1}{x} - \left(1 + \frac{0.5}{\color{blue}{x \cdot x}}\right) \]

   9. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)} \]

   if -4.999999999999985e-310 < t

   1. Initial program 40.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 40.6%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 81.9%

         \[\leadsto \color{blue}{\sqrt{2 \cdot 0.5}} \]

      2. metadata-eval 81.9%

         \[\leadsto \sqrt{\color{blue}{1}} \]

      3. metadata-eval 81.9%

         \[\leadsto \color{blue}{1} \]

   6. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 75.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-310) (+ -1.0 (/ 1.0 x)) 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 26.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 26.7%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. sub-neg 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

      4. metadata-eval 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

      5. +-commutative 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]

      6. fma-def 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]

      7. distribute-rgt-neg-in 26.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Simplified 26.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in t around -inf 71.3%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-\sqrt{2} \cdot t\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. distribute-rgt-neg-out 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      4. +-commutative 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      5. sub-neg 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      6. metadata-eval 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      7. +-commutative 71.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 71.3%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 70.8%

      \[\leadsto \color{blue}{\frac{1}{x} - 1} \]

   if -4.999999999999985e-310 < t

   1. Initial program 40.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 40.6%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 81.9%

         \[\leadsto \color{blue}{\sqrt{2 \cdot 0.5}} \]

      2. metadata-eval 81.9%

         \[\leadsto \sqrt{\color{blue}{1}} \]

      3. metadata-eval 81.9%

         \[\leadsto \color{blue}{1} \]

   6. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15: 75.4% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.75 \cdot 10^{-304}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -2.75e-304) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.75e-304) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.75d-304)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.75e-304) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2.75e-304:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2.75e-304)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2.75e-304)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2.75e-304], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -2.75000000000000017e-304

   1. Initial program 27.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 27.0%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 27.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. sub-neg 27.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

      4. metadata-eval 27.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

      5. +-commutative 27.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]

      6. fma-def 27.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]

      7. distribute-rgt-neg-in 27.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Simplified 27.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in t around -inf 71.8%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 71.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 71.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-\sqrt{2} \cdot t\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. distribute-rgt-neg-out 71.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      4. +-commutative 71.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      5. sub-neg 71.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      6. metadata-eval 71.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      7. +-commutative 71.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 71.8%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 70.7%

      \[\leadsto \color{blue}{-1} \]

   if -2.75000000000000017e-304 < t

   1. Initial program 40.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 40.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 40.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 80.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 81.2%

         \[\leadsto \color{blue}{\sqrt{2 \cdot 0.5}} \]

      2. metadata-eval 81.2%

         \[\leadsto \sqrt{\color{blue}{1}} \]

      3. metadata-eval 81.2%

         \[\leadsto \color{blue}{1} \]

   6. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.75 \cdot 10^{-304}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16: 38.3% accurate, 225.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ -1 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 -1.0)

C

l = abs(l);
double code(double x, double l, double t) {
	return -1.0;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return -1.0;
}

Python

l = abs(l)
def code(x, l, t):
	return -1.0

Julia

l = abs(l)
function code(x, l, t)
	return -1.0
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := -1.0

Derivation

1. Initial program 33.7%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*r/ 33.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

   2. fma-neg 33.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

   3. sub-neg 33.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

   4. metadata-eval 33.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]

   5. +-commutative 33.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]

   6. fma-def 33.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]

   7. distribute-rgt-neg-in 33.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

3. Simplified 33.6%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

4. Taylor expanded in t around -inf 36.5%

   \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 36.5%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

   2. neg-mul-1 36.5%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-\sqrt{2} \cdot t\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

   3. distribute-rgt-neg-out 36.5%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

   4. +-commutative 36.5%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

   5. sub-neg 36.5%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

   6. metadata-eval 36.5%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

   7. +-commutative 36.5%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

6. Simplified 36.5%

   \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

7. Taylor expanded in x around inf 36.0%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 36.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023240 
(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)))))