Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 81.8%

Time: 20.4s

Alternatives: 7

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% 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: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-278}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -8.5e+54)
     (- t_1)
     (if (<= t -2.8e-188)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt (+ (* 2.0 (* t (+ t (/ t x)))) (* 2.0 (* l (/ l x)))))))
       (if (<= t -5e-217)
         -1.0
         (if (<= t 2.8e-278) (* t (/ (sqrt x) l)) t_1))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -8.5e+54) {
		tmp = -t_1;
	} else if (t <= -2.8e-188) {
		tmp = t * (sqrt(2.0) / sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x))))));
	} else if (t <= -5e-217) {
		tmp = -1.0;
	} else if (t <= 2.8e-278) {
		tmp = t * (sqrt(x) / 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(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-8.5d+54)) then
        tmp = -t_1
    else if (t <= (-2.8d-188)) then
        tmp = t * (sqrt(2.0d0) / sqrt(((2.0d0 * (t * (t + (t / x)))) + (2.0d0 * (l * (l / x))))))
    else if (t <= (-5d-217)) then
        tmp = -1.0d0
    else if (t <= 2.8d-278) then
        tmp = t * (sqrt(x) / 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(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -8.5e+54) {
		tmp = -t_1;
	} else if (t <= -2.8e-188) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x))))));
	} else if (t <= -5e-217) {
		tmp = -1.0;
	} else if (t <= 2.8e-278) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -8.5e+54:
		tmp = -t_1
	elif t <= -2.8e-188:
		tmp = t * (math.sqrt(2.0) / math.sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x))))))
	elif t <= -5e-217:
		tmp = -1.0
	elif t <= 2.8e-278:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -8.5e+54)
		tmp = Float64(-t_1);
	elseif (t <= -2.8e-188)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(2.0 * Float64(l * Float64(l / x)))))));
	elseif (t <= -5e-217)
		tmp = -1.0;
	elseif (t <= 2.8e-278)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -8.5e+54)
		tmp = -t_1;
	elseif (t <= -2.8e-188)
		tmp = t * (sqrt(2.0) / sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x))))));
	elseif (t <= -5e-217)
		tmp = -1.0;
	elseif (t <= 2.8e-278)
		tmp = t * (sqrt(x) / 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[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -8.5e+54], (-t$95$1), If[LessEqual[t, -2.8e-188], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5e-217], -1.0, If[LessEqual[t, 2.8e-278], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -8.4999999999999995e54

   1. Initial program 35.2%

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

      1. associate-*r/ 35.1%

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

      2. fma-neg 35.1%

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

      3. sub-neg 35.1%

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

      4. metadata-eval 35.1%

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

      5. +-commutative 35.1%

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

      6. fma-def 35.1%

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

      7. distribute-rgt-neg-in 35.1%

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

   3. Simplified 35.1%

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

   5. Applied egg-rr 86.3%

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

   6. Taylor expanded in t around -inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. sub-neg 96.8%

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

      3. metadata-eval 96.8%

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

   8. Simplified 96.8%

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

   if -8.4999999999999995e54 < t < -2.8000000000000001e-188

   1. Initial program 55.4%

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

      1. associate-*l/ 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around inf 81.2%

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

      1. associate--l+ 81.2%

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

      2. unpow2 81.2%

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

      3. distribute-lft-out 81.2%

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

      4. unpow2 81.2%

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

      5. unpow2 81.2%

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

      6. associate-*r/ 81.2%

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

      7. mul-1-neg 81.2%

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

      8. +-commutative 81.2%

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

      9. unpow2 81.2%

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

      10. associate-*l* 81.2%

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

      11. unpow2 81.2%

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

      12. fma-udef 81.2%

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

   6. Simplified 81.2%

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

   7. Taylor expanded in t around 0 81.2%

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

      1. associate-*r/ 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unpow2 81.2%

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

   9. Simplified 81.2%

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

       1. *-un-lft-identity 81.2%

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

       2. associate-*l/ 81.2%

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

       3. *-commutative 81.2%

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

       4. +-commutative 81.2%

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

       5. associate-/l* 81.2%

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

       6. distribute-rgt-neg-in 81.2%

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

   11. Applied egg-rr 81.2%

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

       1. *-lft-identity 81.2%

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

       2. +-commutative 81.2%

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

       3. distribute-rgt-neg-out 81.2%

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

       4. unpow2 81.2%

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

       5. distribute-frac-neg 81.2%

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

       6. unpow2 81.2%

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

       7. associate-*r/ 87.9%

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

       8. distribute-lft-neg-in 87.9%

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

       9. cancel-sign-sub 87.9%

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

       10. associate-+l+ 87.9%

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

       11. associate-/r/ 87.9%

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

       12. distribute-rgt-out 87.9%

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

   13. Simplified 87.9%

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

   if -2.8000000000000001e-188 < t < -5.0000000000000002e-217

   1. Initial program 3.1%

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

      1. associate-*l/ 3.1%

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

   3. Simplified 3.1%

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

      1. sqrt-undiv 3.1%

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

      2. fma-neg 3.1%

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

      3. distribute-rgt-neg-out 3.1%

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

      4. fma-udef 3.1%

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

      5. associate-*r* 3.1%

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

      6. fma-def 3.1%

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

      7. add-sqr-sqrt 2.5%

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

      8. sqrt-unprod 3.7%

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

      9. distribute-rgt-neg-out 3.7%

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

      10. distribute-rgt-neg-out 3.7%

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

      11. sqr-neg 3.7%

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

      12. sqrt-unprod 3.7%

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

      13. add-sqr-sqrt 3.7%

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

   5. Applied egg-rr 3.7%

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

   6. Taylor expanded in x around inf 3.7%

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

      1. distribute-lft-out 3.7%

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

      2. unpow2 3.7%

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

      3. unpow2 3.7%

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

   8. Simplified 3.7%

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

   9. Taylor expanded in t around -inf 100.0%

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

   if -5.0000000000000002e-217 < t < 2.80000000000000008e-278

   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 54.1%

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

      1. associate--l+ 54.1%

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

      2. unpow2 54.1%

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

      3. distribute-lft-out 54.1%

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

      4. unpow2 54.1%

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

      5. unpow2 54.1%

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

      6. associate-*r/ 54.1%

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

      7. mul-1-neg 54.1%

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

      8. +-commutative 54.1%

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

      9. unpow2 54.1%

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

      10. associate-*l* 54.1%

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

      11. unpow2 54.1%

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

      12. fma-udef 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in t around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. metadata-eval 54.0%

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

      3. distribute-rgt1-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*l/ 55.8%

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

   9. Simplified 55.8%

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

   10. Taylor expanded in l around 0 63.9%

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

       1. associate-*l/ 64.1%

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

       2. *-lft-identity 64.1%

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

   12. Simplified 64.1%

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

   if 2.80000000000000008e-278 < t

   1. Initial program 41.1%

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

      1. associate-*r/ 41.0%

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

      2. fma-neg 41.0%

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

      3. sub-neg 41.0%

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

      4. metadata-eval 41.0%

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

      5. +-commutative 41.0%

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

      6. fma-def 41.0%

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

      7. distribute-rgt-neg-in 41.0%

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

   3. Simplified 41.0%

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

   5. Applied egg-rr 73.0%

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

   6. Taylor expanded in t around inf 88.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-278}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 78.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-272}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -6.8e-217)
   -1.0
   (if (<= t 1.75e-272) (* t (/ (sqrt x) l)) (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -6.8e-217) {
		tmp = -1.0;
	} else if (t <= 1.75e-272) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = sqrt(((x + -1.0) / (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 <= (-6.8d-217)) then
        tmp = -1.0d0
    else if (t <= 1.75d-272) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = sqrt(((x + (-1.0d0)) / (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 <= -6.8e-217) {
		tmp = -1.0;
	} else if (t <= 1.75e-272) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -6.8e-217:
		tmp = -1.0
	elif t <= 1.75e-272:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -6.8e-217)
		tmp = -1.0;
	elseif (t <= 1.75e-272)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -6.8e-217)
		tmp = -1.0;
	elseif (t <= 1.75e-272)
		tmp = t * (sqrt(x) / l);
	else
		tmp = sqrt(((x + -1.0) / (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, -6.8e-217], -1.0, If[LessEqual[t, 1.75e-272], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.80000000000000032e-217

   1. Initial program 43.3%

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

      1. associate-*l/ 43.3%

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

   3. Simplified 43.3%

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

      1. sqrt-undiv 41.2%

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

      2. fma-neg 41.2%

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

      3. distribute-rgt-neg-out 41.2%

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

      4. fma-udef 41.2%

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

      5. associate-*r* 41.2%

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

      6. fma-def 41.2%

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

      7. add-sqr-sqrt 17.1%

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

      8. sqrt-unprod 40.1%

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

      9. distribute-rgt-neg-out 40.1%

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

      10. distribute-rgt-neg-out 40.1%

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

      11. sqr-neg 40.1%

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

      12. sqrt-unprod 41.2%

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

      13. add-sqr-sqrt 41.2%

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

   5. Applied egg-rr 41.2%

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

   6. Taylor expanded in x around inf 41.2%

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

      1. distribute-lft-out 41.2%

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

      2. unpow2 41.2%

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

      3. unpow2 41.2%

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

   8. Simplified 41.2%

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

   9. Taylor expanded in t around -inf 83.7%

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

   if -6.80000000000000032e-217 < t < 1.7499999999999998e-272

   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 54.1%

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

      1. associate--l+ 54.1%

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

      2. unpow2 54.1%

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

      3. distribute-lft-out 54.1%

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

      4. unpow2 54.1%

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

      5. unpow2 54.1%

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

      6. associate-*r/ 54.1%

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

      7. mul-1-neg 54.1%

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

      8. +-commutative 54.1%

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

      9. unpow2 54.1%

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

      10. associate-*l* 54.1%

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

      11. unpow2 54.1%

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

      12. fma-udef 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in t around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. metadata-eval 54.0%

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

      3. distribute-rgt1-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*l/ 55.8%

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

   9. Simplified 55.8%

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

   10. Taylor expanded in l around 0 63.9%

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

       1. associate-*l/ 64.1%

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

       2. *-lft-identity 64.1%

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

   12. Simplified 64.1%

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

   if 1.7499999999999998e-272 < t

   1. Initial program 41.1%

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

      1. associate-*r/ 41.0%

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

      2. fma-neg 41.0%

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

      3. sub-neg 41.0%

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

      4. metadata-eval 41.0%

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

      5. +-commutative 41.0%

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

      6. fma-def 41.0%

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

      7. distribute-rgt-neg-in 41.0%

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

   3. Simplified 41.0%

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

   5. Applied egg-rr 73.0%

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

   6. Taylor expanded in t around inf 88.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-272}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 78.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{-217}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-279}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -7.6e-217)
     (- t_1)
     (if (<= t 5.5e-279) (* t (/ (sqrt x) l)) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -7.6e-217) {
		tmp = -t_1;
	} else if (t <= 5.5e-279) {
		tmp = t * (sqrt(x) / 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(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-7.6d-217)) then
        tmp = -t_1
    else if (t <= 5.5d-279) then
        tmp = t * (sqrt(x) / 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(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -7.6e-217) {
		tmp = -t_1;
	} else if (t <= 5.5e-279) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -7.6e-217:
		tmp = -t_1
	elif t <= 5.5e-279:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -7.6e-217)
		tmp = Float64(-t_1);
	elseif (t <= 5.5e-279)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -7.6e-217)
		tmp = -t_1;
	elseif (t <= 5.5e-279)
		tmp = t * (sqrt(x) / 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[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7.6e-217], (-t$95$1), If[LessEqual[t, 5.5e-279], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.59999999999999974e-217

   1. Initial program 43.3%

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

      1. associate-*r/ 43.2%

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

      2. fma-neg 43.2%

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

      3. sub-neg 43.2%

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

      4. metadata-eval 43.2%

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

      5. +-commutative 43.2%

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

      6. fma-def 43.2%

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

      7. distribute-rgt-neg-in 43.2%

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

   3. Simplified 43.2%

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

   5. Applied egg-rr 73.4%

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

   6. Taylor expanded in t around -inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. sub-neg 84.6%

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

      3. metadata-eval 84.6%

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

   8. Simplified 84.6%

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

   if -7.59999999999999974e-217 < t < 5.5000000000000002e-279

   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 54.1%

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

      1. associate--l+ 54.1%

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

      2. unpow2 54.1%

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

      3. distribute-lft-out 54.1%

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

      4. unpow2 54.1%

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

      5. unpow2 54.1%

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

      6. associate-*r/ 54.1%

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

      7. mul-1-neg 54.1%

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

      8. +-commutative 54.1%

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

      9. unpow2 54.1%

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

      10. associate-*l* 54.1%

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

      11. unpow2 54.1%

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

      12. fma-udef 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in t around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. metadata-eval 54.0%

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

      3. distribute-rgt1-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*l/ 55.8%

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

   9. Simplified 55.8%

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

   10. Taylor expanded in l around 0 63.9%

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

       1. associate-*l/ 64.1%

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

       2. *-lft-identity 64.1%

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

   12. Simplified 64.1%

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

   if 5.5000000000000002e-279 < t

   1. Initial program 41.1%

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

      1. associate-*r/ 41.0%

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

      2. fma-neg 41.0%

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

      3. sub-neg 41.0%

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

      4. metadata-eval 41.0%

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

      5. +-commutative 41.0%

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

      6. fma-def 41.0%

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

      7. distribute-rgt-neg-in 41.0%

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

   3. Simplified 41.0%

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

   5. Applied egg-rr 73.0%

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

   6. Taylor expanded in t around inf 88.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-217}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-279}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 76.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5.5e-217) -1.0 (if (<= t 1.7e-272) (* (sqrt x) (/ t l)) 1.0)))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.5e-217) {
		tmp = -1.0;
	} else if (t <= 1.7e-272) {
		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 <= (-5.5d-217)) then
        tmp = -1.0d0
    else if (t <= 1.7d-272) 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 <= -5.5e-217) {
		tmp = -1.0;
	} else if (t <= 1.7e-272) {
		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 <= -5.5e-217:
		tmp = -1.0
	elif t <= 1.7e-272:
		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 <= -5.5e-217)
		tmp = -1.0;
	elseif (t <= 1.7e-272)
		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 <= -5.5e-217)
		tmp = -1.0;
	elseif (t <= 1.7e-272)
		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, -5.5e-217], -1.0, If[LessEqual[t, 1.7e-272], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if t < -5.49999999999999975e-217

   1. Initial program 43.3%

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

      1. associate-*l/ 43.3%

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

   3. Simplified 43.3%

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

      1. sqrt-undiv 41.2%

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

      2. fma-neg 41.2%

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

      3. distribute-rgt-neg-out 41.2%

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

      4. fma-udef 41.2%

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

      5. associate-*r* 41.2%

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

      6. fma-def 41.2%

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

      7. add-sqr-sqrt 17.1%

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

      8. sqrt-unprod 40.1%

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

      9. distribute-rgt-neg-out 40.1%

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

      10. distribute-rgt-neg-out 40.1%

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

      11. sqr-neg 40.1%

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

      12. sqrt-unprod 41.2%

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

      13. add-sqr-sqrt 41.2%

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

   5. Applied egg-rr 41.2%

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

   6. Taylor expanded in x around inf 41.2%

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

      1. distribute-lft-out 41.2%

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

      2. unpow2 41.2%

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

      3. unpow2 41.2%

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

   8. Simplified 41.2%

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

   9. Taylor expanded in t around -inf 83.7%

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

   if -5.49999999999999975e-217 < t < 1.7000000000000002e-272

   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 54.1%

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

      1. associate--l+ 54.1%

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

      2. unpow2 54.1%

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

      3. distribute-lft-out 54.1%

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

      4. unpow2 54.1%

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

      5. unpow2 54.1%

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

      6. associate-*r/ 54.1%

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

      7. mul-1-neg 54.1%

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

      8. +-commutative 54.1%

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

      9. unpow2 54.1%

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

      10. associate-*l* 54.1%

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

      11. unpow2 54.1%

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

      12. fma-udef 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in t around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. metadata-eval 54.0%

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

      3. distribute-rgt1-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*l/ 55.8%

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

   9. Simplified 55.8%

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

   10. Taylor expanded in l around 0 60.2%

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

   if 1.7000000000000002e-272 < t

   1. Initial program 41.1%

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

      1. associate-*l/ 41.2%

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

   3. Simplified 41.2%

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

      1. sqrt-undiv 40.6%

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

      2. fma-neg 40.6%

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

      3. distribute-rgt-neg-out 40.6%

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

      4. fma-udef 40.6%

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

      5. associate-*r* 40.6%

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

      6. fma-def 40.6%

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

      7. add-sqr-sqrt 18.7%

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

      8. sqrt-unprod 40.3%

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

      9. distribute-rgt-neg-out 40.3%

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

      10. distribute-rgt-neg-out 40.3%

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

      11. sqr-neg 40.3%

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

      12. sqrt-unprod 40.4%

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

      13. add-sqr-sqrt 40.4%

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

   5. Applied egg-rr 40.4%

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

   6. Taylor expanded in x around inf 39.2%

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

      1. distribute-lft-out 39.2%

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

      2. unpow2 39.2%

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

      3. unpow2 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in l around 0 85.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 77.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-273}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2.9e-217) -1.0 (if (<= t 7.4e-273) (* t (/ (sqrt x) l)) 1.0)))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.9e-217) {
		tmp = -1.0;
	} else if (t <= 7.4e-273) {
		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 <= (-2.9d-217)) then
        tmp = -1.0d0
    else if (t <= 7.4d-273) 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 <= -2.9e-217) {
		tmp = -1.0;
	} else if (t <= 7.4e-273) {
		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 <= -2.9e-217:
		tmp = -1.0
	elif t <= 7.4e-273:
		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 <= -2.9e-217)
		tmp = -1.0;
	elseif (t <= 7.4e-273)
		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 <= -2.9e-217)
		tmp = -1.0;
	elseif (t <= 7.4e-273)
		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, -2.9e-217], -1.0, If[LessEqual[t, 7.4e-273], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if t < -2.89999999999999982e-217

   1. Initial program 43.3%

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

      1. associate-*l/ 43.3%

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

   3. Simplified 43.3%

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

      1. sqrt-undiv 41.2%

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

      2. fma-neg 41.2%

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

      3. distribute-rgt-neg-out 41.2%

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

      4. fma-udef 41.2%

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

      5. associate-*r* 41.2%

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

      6. fma-def 41.2%

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

      7. add-sqr-sqrt 17.1%

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

      8. sqrt-unprod 40.1%

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

      9. distribute-rgt-neg-out 40.1%

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

      10. distribute-rgt-neg-out 40.1%

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

      11. sqr-neg 40.1%

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

      12. sqrt-unprod 41.2%

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

      13. add-sqr-sqrt 41.2%

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

   5. Applied egg-rr 41.2%

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

   6. Taylor expanded in x around inf 41.2%

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

      1. distribute-lft-out 41.2%

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

      2. unpow2 41.2%

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

      3. unpow2 41.2%

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

   8. Simplified 41.2%

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

   9. Taylor expanded in t around -inf 83.7%

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

   if -2.89999999999999982e-217 < t < 7.4000000000000007e-273

   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 54.1%

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

      1. associate--l+ 54.1%

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

      2. unpow2 54.1%

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

      3. distribute-lft-out 54.1%

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

      4. unpow2 54.1%

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

      5. unpow2 54.1%

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

      6. associate-*r/ 54.1%

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

      7. mul-1-neg 54.1%

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

      8. +-commutative 54.1%

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

      9. unpow2 54.1%

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

      10. associate-*l* 54.1%

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

      11. unpow2 54.1%

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

      12. fma-udef 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in t around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. metadata-eval 54.0%

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

      3. distribute-rgt1-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*l/ 55.8%

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

   9. Simplified 55.8%

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

   10. Taylor expanded in l around 0 63.9%

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

       1. associate-*l/ 64.1%

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

       2. *-lft-identity 64.1%

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

   12. Simplified 64.1%

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

   if 7.4000000000000007e-273 < t

   1. Initial program 41.1%

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

      1. associate-*l/ 41.2%

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

   3. Simplified 41.2%

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

      1. sqrt-undiv 40.6%

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

      2. fma-neg 40.6%

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

      3. distribute-rgt-neg-out 40.6%

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

      4. fma-udef 40.6%

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

      5. associate-*r* 40.6%

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

      6. fma-def 40.6%

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

      7. add-sqr-sqrt 18.7%

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

      8. sqrt-unprod 40.3%

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

      9. distribute-rgt-neg-out 40.3%

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

      10. distribute-rgt-neg-out 40.3%

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

      11. sqr-neg 40.3%

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

      12. sqrt-unprod 40.4%

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

      13. add-sqr-sqrt 40.4%

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

   5. Applied egg-rr 40.4%

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

   6. Taylor expanded in x around inf 39.2%

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

      1. distribute-lft-out 39.2%

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

      2. unpow2 39.2%

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

      3. unpow2 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in l around 0 85.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-217}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-273}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 75.0% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.4 \cdot 10^{-307}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -8.4e-307) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -8.4e-307) {
		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 <= (-8.4d-307)) 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 <= -8.4e-307) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -8.4e-307:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -8.4e-307)
		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 <= -8.4e-307)
		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, -8.4e-307], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -8.4000000000000004e-307

   1. Initial program 38.4%

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

      1. associate-*l/ 38.5%

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

   3. Simplified 38.5%

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

      1. sqrt-undiv 36.6%

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

      2. fma-neg 36.6%

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

      3. distribute-rgt-neg-out 36.6%

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

      4. fma-udef 36.6%

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

      5. associate-*r* 36.6%

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

      6. fma-def 36.6%

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

      7. add-sqr-sqrt 15.2%

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

      8. sqrt-unprod 38.5%

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

      9. distribute-rgt-neg-out 38.5%

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

      10. distribute-rgt-neg-out 38.5%

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

      11. sqr-neg 38.5%

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

      12. sqrt-unprod 39.4%

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

      13. add-sqr-sqrt 39.4%

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

   5. Applied egg-rr 39.4%

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

   6. Taylor expanded in x around inf 39.4%

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

      1. distribute-lft-out 39.4%

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

      2. unpow2 39.4%

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

      3. unpow2 39.4%

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

   8. Simplified 39.4%

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

   9. Taylor expanded in t around -inf 77.3%

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

   if -8.4000000000000004e-307 < t

   1. Initial program 39.1%

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

      1. associate-*l/ 39.2%

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

   3. Simplified 39.2%

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

      1. sqrt-undiv 38.6%

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

      2. fma-neg 38.6%

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

      3. distribute-rgt-neg-out 38.6%

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

      4. fma-udef 38.6%

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

      5. associate-*r* 38.6%

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

      6. fma-def 38.6%

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

      7. add-sqr-sqrt 17.8%

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

      8. sqrt-unprod 40.8%

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

      9. distribute-rgt-neg-out 40.8%

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

      10. distribute-rgt-neg-out 40.8%

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

      11. sqr-neg 40.8%

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

      12. sqrt-unprod 41.0%

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

      13. add-sqr-sqrt 41.0%

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

   5. Applied egg-rr 41.0%

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

   6. Taylor expanded in x around inf 39.9%

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

      1. distribute-lft-out 39.9%

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

      2. unpow2 39.9%

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

      3. unpow2 39.9%

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

   8. Simplified 39.9%

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

   9. Taylor expanded in l around 0 82.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.4 \cdot 10^{-307}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 37.9% 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 38.7%

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

   1. associate-*l/ 38.8%

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

3. Simplified 38.8%

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

   1. sqrt-undiv 37.6%

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

   2. fma-neg 37.6%

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

   3. distribute-rgt-neg-out 37.6%

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

   4. fma-udef 37.6%

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

   5. associate-*r* 37.6%

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

   6. fma-def 37.6%

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

   7. add-sqr-sqrt 16.4%

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

   8. sqrt-unprod 39.6%

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

   9. distribute-rgt-neg-out 39.6%

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

   10. distribute-rgt-neg-out 39.6%

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

   11. sqr-neg 39.6%

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

   12. sqrt-unprod 40.2%

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

   13. add-sqr-sqrt 40.2%

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

5. Applied egg-rr 40.2%

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

6. Taylor expanded in x around inf 39.6%

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

   1. distribute-lft-out 39.6%

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

   2. unpow2 39.6%

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

   3. unpow2 39.6%

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

8. Simplified 39.6%

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

9. Taylor expanded in t around -inf 41.9%

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

10. Final simplification 41.9%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023238 
(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)))))