Toniolo and Linder, Equation (7)

Percentage Accurate: 34.3% → 84.4%

Time: 22.5s

Alternatives: 11

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: 34.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \frac{t}{-t}\\ t_3 := \frac{t}{\sqrt{\frac{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{if}\;t \leq -1.26 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.16 \cdot 10^{-170}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.66 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x))
        (t_2 (/ t (- t)))
        (t_3
         (/
          t
          (sqrt (/ (+ t_1 (+ t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))) 2.0)))))
   (if (<= t -1.26e+61)
     t_2
     (if (<= t -1.16e-170)
       t_3
       (if (<= t -1.66e-206)
         t_2
         (if (<= t 3.2e-217)
           (* t (* (sqrt x) (/ 1.0 l)))
           (if (<= t 2.55e-147)
             1.0
             (if (<= t 6.2e-12) t_3 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double t_2 = t / -t;
	double t_3 = t / sqrt(((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x))))) / 2.0));
	double tmp;
	if (t <= -1.26e+61) {
		tmp = t_2;
	} else if (t <= -1.16e-170) {
		tmp = t_3;
	} else if (t <= -1.66e-206) {
		tmp = t_2;
	} else if (t <= 3.2e-217) {
		tmp = t * (sqrt(x) * (1.0 / l));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 6.2e-12) {
		tmp = t_3;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (l * l) / x
    t_2 = t / -t
    t_3 = t / sqrt(((t_1 + (t_1 + (2.0d0 * ((t * t) + ((t * t) / x))))) / 2.0d0))
    if (t <= (-1.26d+61)) then
        tmp = t_2
    else if (t <= (-1.16d-170)) then
        tmp = t_3
    else if (t <= (-1.66d-206)) then
        tmp = t_2
    else if (t <= 3.2d-217) then
        tmp = t * (sqrt(x) * (1.0d0 / l))
    else if (t <= 2.55d-147) then
        tmp = 1.0d0
    else if (t <= 6.2d-12) then
        tmp = t_3
    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 t_1 = (l * l) / x;
	double t_2 = t / -t;
	double t_3 = t / Math.sqrt(((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x))))) / 2.0));
	double tmp;
	if (t <= -1.26e+61) {
		tmp = t_2;
	} else if (t <= -1.16e-170) {
		tmp = t_3;
	} else if (t <= -1.66e-206) {
		tmp = t_2;
	} else if (t <= 3.2e-217) {
		tmp = t * (Math.sqrt(x) * (1.0 / l));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 6.2e-12) {
		tmp = t_3;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (l * l) / x
	t_2 = t / -t
	t_3 = t / math.sqrt(((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x))))) / 2.0))
	tmp = 0
	if t <= -1.26e+61:
		tmp = t_2
	elif t <= -1.16e-170:
		tmp = t_3
	elif t <= -1.66e-206:
		tmp = t_2
	elif t <= 3.2e-217:
		tmp = t * (math.sqrt(x) * (1.0 / l))
	elif t <= 2.55e-147:
		tmp = 1.0
	elif t <= 6.2e-12:
		tmp = t_3
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	t_2 = Float64(t / Float64(-t))
	t_3 = Float64(t / sqrt(Float64(Float64(t_1 + Float64(t_1 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))) / 2.0)))
	tmp = 0.0
	if (t <= -1.26e+61)
		tmp = t_2;
	elseif (t <= -1.16e-170)
		tmp = t_3;
	elseif (t <= -1.66e-206)
		tmp = t_2;
	elseif (t <= 3.2e-217)
		tmp = Float64(t * Float64(sqrt(x) * Float64(1.0 / l)));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 6.2e-12)
		tmp = t_3;
	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)
	t_1 = (l * l) / x;
	t_2 = t / -t;
	t_3 = t / sqrt(((t_1 + (t_1 + (2.0 * ((t * t) + ((t * t) / x))))) / 2.0));
	tmp = 0.0;
	if (t <= -1.26e+61)
		tmp = t_2;
	elseif (t <= -1.16e-170)
		tmp = t_3;
	elseif (t <= -1.66e-206)
		tmp = t_2;
	elseif (t <= 3.2e-217)
		tmp = t * (sqrt(x) * (1.0 / l));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 6.2e-12)
		tmp = t_3;
	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_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(t / (-t)), $MachinePrecision]}, Block[{t$95$3 = N[(t / N[Sqrt[N[(N[(t$95$1 + N[(t$95$1 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.26e+61], t$95$2, If[LessEqual[t, -1.16e-170], t$95$3, If[LessEqual[t, -1.66e-206], t$95$2, If[LessEqual[t, 3.2e-217], N[(t * N[(N[Sqrt[x], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 6.2e-12], t$95$3, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.2600000000000001e61 or -1.1599999999999999e-170 < t < -1.66000000000000013e-206

   1. Initial program 27.3%

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

      1. associate-*l/ 27.3%

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

   3. Simplified 27.3%

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

      1. *-commutative 27.3%

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

      2. clear-num 27.3%

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

      3. un-div-inv 27.3%

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

      4. sqrt-undiv 27.4%

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

      5. metadata-eval 27.4%

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

      6. sub-neg 27.4%

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

      7. associate-*l/ 5.5%

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

      8. sub-neg 5.5%

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

      9. metadata-eval 5.5%

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

   5. Applied egg-rr 5.5%

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

      1. +-commutative 5.5%

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

      2. *-commutative 5.5%

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

      3. +-commutative 5.5%

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

      4. +-commutative 5.5%

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

   7. Simplified 5.5%

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

   8. Taylor expanded in t around inf 7.9%

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

      1. associate-*r/ 7.9%

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

      2. +-commutative 7.9%

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

      3. *-commutative 7.9%

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

      4. unpow2 7.9%

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

      5. sub-neg 7.9%

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

      6. metadata-eval 7.9%

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

   10. Simplified 7.9%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 95.4%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t} \]

       3. mul-1-neg 95.4%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 95.4%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -1.2600000000000001e61 < t < -1.1599999999999999e-170 or 2.5499999999999998e-147 < t < 6.2000000000000002e-12

   1. Initial program 45.3%

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

      1. associate-*l/ 45.3%

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

   3. Simplified 45.3%

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

      1. *-commutative 45.3%

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

      2. clear-num 45.3%

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

      3. un-div-inv 45.3%

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

      4. sqrt-undiv 45.6%

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

      5. metadata-eval 45.6%

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

      6. sub-neg 45.6%

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

      7. associate-*l/ 43.0%

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

      8. sub-neg 43.0%

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

      9. metadata-eval 43.0%

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

   5. Applied egg-rr 43.0%

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

      1. +-commutative 43.0%

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

      2. *-commutative 43.0%

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

      3. +-commutative 43.0%

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

      4. +-commutative 43.0%

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

   7. Simplified 43.0%

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

   8. Taylor expanded in x around inf 85.2%

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

      1. distribute-lft-in 85.2%

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

      2. associate--l+ 85.2%

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

      3. unpow2 85.2%

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

      4. +-commutative 85.2%

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

      5. unpow2 85.2%

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

      6. unpow2 85.2%

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

      7. mul-1-neg 85.2%

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

      8. +-commutative 85.2%

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

      9. unpow2 85.2%

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

      10. unpow2 85.2%

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

      11. fma-udef 85.2%

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

   10. Simplified 85.2%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. unpow2 84.4%

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

   13. Simplified 84.4%

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

   if -1.66000000000000013e-206 < t < 3.2000000000000001e-217

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in x around inf 55.4%

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

      1. associate--l+ 55.4%

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

      2. unpow2 55.4%

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

      3. distribute-lft-out 55.4%

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

      4. unpow2 55.4%

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

      5. unpow2 55.4%

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

      6. associate-*r/ 55.4%

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

      7. mul-1-neg 55.4%

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

      8. +-commutative 55.4%

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

      9. unpow2 55.4%

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

      10. unpow2 55.4%

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

      11. fma-udef 55.4%

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

   6. Simplified 55.4%

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

   7. Taylor expanded in t around 0 55.0%

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

      1. sub-neg 55.0%

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

      2. unpow2 55.0%

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

      3. mul-1-neg 55.0%

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

      4. remove-double-neg 55.0%

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

      5. unpow2 55.0%

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

   9. Simplified 55.0%

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

   10. Taylor expanded in l around 0 47.6%

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

   if 3.2000000000000001e-217 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

   4. Taylor expanded in t around inf 25.2%

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

      1. *-commutative 25.2%

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

      2. +-commutative 25.2%

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

      3. associate-*r/ 25.2%

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

      4. sub-neg 25.2%

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

      5. metadata-eval 25.2%

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

      6. associate-*r* 25.2%

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

      7. unpow2 25.2%

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

      8. +-commutative 25.2%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in x around inf 70.9%

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

   if 6.2000000000000002e-12 < t

   1. Initial program 43.1%

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

      1. associate-*l/ 43.1%

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

   3. Simplified 43.1%

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

   4. Taylor expanded in t around inf 21.3%

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

      1. *-commutative 21.3%

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

      2. +-commutative 21.3%

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

      3. associate-*r/ 50.6%

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

      4. sub-neg 50.6%

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

      5. metadata-eval 50.6%

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

      6. associate-*r* 50.6%

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

      7. unpow2 50.6%

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

      8. +-commutative 50.6%

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

   6. Simplified 50.6%

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

   7. Taylor expanded in t around 0 92.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{+61}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq -1.16 \cdot 10^{-170}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{elif}\;t \leq -1.66 \cdot 10^{-206}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_3 := \frac{t}{-t}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+60}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-179}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(t_2 + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-209}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(t_1 + t_2\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x))
        (t_2 (* 2.0 (+ (* t t) (/ (* t t) x))))
        (t_3 (/ t (- t))))
   (if (<= t -4.8e+60)
     t_3
     (if (<= t -1e-179)
       (/ t (sqrt (/ (+ t_1 (+ t_2 (/ (fma 2.0 (* t t) (* l l)) x))) 2.0)))
       (if (<= t -1.55e-209)
         t_3
         (if (<= t 6.2e-218)
           (* t (/ (sqrt x) l))
           (if (<= t 2.55e-147)
             1.0
             (if (<= t 3.2e-12)
               (/ t (sqrt (/ (+ t_1 (+ t_1 t_2)) 2.0)))
               (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double t_2 = 2.0 * ((t * t) + ((t * t) / x));
	double t_3 = t / -t;
	double tmp;
	if (t <= -4.8e+60) {
		tmp = t_3;
	} else if (t <= -1e-179) {
		tmp = t / sqrt(((t_1 + (t_2 + (fma(2.0, (t * t), (l * l)) / x))) / 2.0));
	} else if (t <= -1.55e-209) {
		tmp = t_3;
	} else if (t <= 6.2e-218) {
		tmp = t * (sqrt(x) / l);
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 3.2e-12) {
		tmp = t / sqrt(((t_1 + (t_1 + t_2)) / 2.0));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	t_2 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	t_3 = Float64(t / Float64(-t))
	tmp = 0.0
	if (t <= -4.8e+60)
		tmp = t_3;
	elseif (t <= -1e-179)
		tmp = Float64(t / sqrt(Float64(Float64(t_1 + Float64(t_2 + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))) / 2.0)));
	elseif (t <= -1.55e-209)
		tmp = t_3;
	elseif (t <= 6.2e-218)
		tmp = Float64(t * Float64(sqrt(x) / l));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 3.2e-12)
		tmp = Float64(t / sqrt(Float64(Float64(t_1 + Float64(t_1 + t_2)) / 2.0)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t / (-t)), $MachinePrecision]}, If[LessEqual[t, -4.8e+60], t$95$3, If[LessEqual[t, -1e-179], N[(t / N[Sqrt[N[(N[(t$95$1 + N[(t$95$2 + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.55e-209], t$95$3, If[LessEqual[t, 6.2e-218], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 3.2e-12], N[(t / N[Sqrt[N[(N[(t$95$1 + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -4.8e60 or -1e-179 < t < -1.55e-209

   1. Initial program 26.9%

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

      1. associate-*l/ 26.9%

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

   3. Simplified 26.9%

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

      1. *-commutative 26.9%

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

      2. clear-num 26.9%

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

      3. un-div-inv 26.9%

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

      4. sqrt-undiv 27.0%

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

      5. metadata-eval 27.0%

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

      6. sub-neg 27.0%

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

      7. associate-*l/ 5.5%

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

      8. sub-neg 5.5%

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

      9. metadata-eval 5.5%

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

   5. Applied egg-rr 5.5%

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

      1. +-commutative 5.5%

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

      2. *-commutative 5.5%

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

      3. +-commutative 5.5%

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

      4. +-commutative 5.5%

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

   7. Simplified 5.5%

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

   8. Taylor expanded in t around inf 7.8%

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

      1. associate-*r/ 7.8%

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

      2. +-commutative 7.8%

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

      3. *-commutative 7.8%

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

      4. unpow2 7.8%

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

      5. sub-neg 7.8%

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

      6. metadata-eval 7.8%

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

   10. Simplified 7.8%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 94.1%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t} \]

       3. mul-1-neg 94.1%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 94.1%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -4.8e60 < t < -1e-179

   1. Initial program 48.8%

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

      1. associate-*l/ 48.8%

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

   3. Simplified 48.8%

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

      1. *-commutative 48.8%

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

      2. clear-num 48.8%

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

      3. un-div-inv 48.8%

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

      4. sqrt-undiv 49.1%

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

      5. metadata-eval 49.1%

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

      6. sub-neg 49.1%

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

      7. associate-*l/ 45.1%

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

      8. sub-neg 45.1%

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

      9. metadata-eval 45.1%

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

   5. Applied egg-rr 45.1%

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

      1. +-commutative 45.1%

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

      2. *-commutative 45.1%

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

      3. +-commutative 45.1%

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

      4. +-commutative 45.1%

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

   7. Simplified 45.1%

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

   8. Taylor expanded in x around inf 87.5%

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

      1. distribute-lft-in 87.5%

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

      2. associate--l+ 87.5%

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

      3. unpow2 87.5%

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

      4. +-commutative 87.5%

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

      5. unpow2 87.5%

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

      6. unpow2 87.5%

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

      7. mul-1-neg 87.5%

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

      8. +-commutative 87.5%

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

      9. unpow2 87.5%

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

      10. unpow2 87.5%

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

      11. fma-udef 87.5%

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

   10. Simplified 87.5%

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

   if -1.55e-209 < t < 6.19999999999999994e-218

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in x around inf 57.5%

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

      1. associate--l+ 57.5%

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

      2. unpow2 57.5%

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

      3. distribute-lft-out 57.5%

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

      4. unpow2 57.5%

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

      5. unpow2 57.5%

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

      6. associate-*r/ 57.5%

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

      7. mul-1-neg 57.5%

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

      8. +-commutative 57.5%

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

      9. unpow2 57.5%

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

      10. unpow2 57.5%

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

      11. fma-udef 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in t around 0 57.1%

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

      1. sub-neg 57.1%

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

      2. unpow2 57.1%

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

      3. mul-1-neg 57.1%

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

      4. remove-double-neg 57.1%

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

      5. unpow2 57.1%

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

   9. Simplified 57.1%

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

   10. Taylor expanded in l around 0 49.4%

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

       1. associate-*l/ 49.4%

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

       2. *-lft-identity 49.4%

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

   12. Simplified 49.4%

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

   if 6.19999999999999994e-218 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

   4. Taylor expanded in t around inf 25.2%

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

      1. *-commutative 25.2%

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

      2. +-commutative 25.2%

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

      3. associate-*r/ 25.2%

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

      4. sub-neg 25.2%

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

      5. metadata-eval 25.2%

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

      6. associate-*r* 25.2%

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

      7. unpow2 25.2%

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

      8. +-commutative 25.2%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in x around inf 70.9%

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

   if 2.5499999999999998e-147 < t < 3.2000000000000001e-12

   1. Initial program 38.3%

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

      1. associate-*l/ 38.3%

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

   3. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. clear-num 38.3%

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

      3. un-div-inv 38.2%

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

      4. sqrt-undiv 38.5%

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

      5. metadata-eval 38.5%

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

      6. sub-neg 38.5%

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

      7. associate-*l/ 38.7%

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

      8. sub-neg 38.7%

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

      9. metadata-eval 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. +-commutative 38.7%

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

      2. *-commutative 38.7%

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

      3. +-commutative 38.7%

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

      4. +-commutative 38.7%

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

   7. Simplified 38.7%

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

   8. Taylor expanded in x around inf 80.6%

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

      1. distribute-lft-in 80.6%

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

      2. associate--l+ 80.6%

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

      3. unpow2 80.6%

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

      4. +-commutative 80.6%

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

      5. unpow2 80.6%

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

      6. unpow2 80.6%

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

      7. mul-1-neg 80.6%

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

      8. +-commutative 80.6%

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

      9. unpow2 80.6%

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

      10. unpow2 80.6%

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

      11. fma-udef 80.6%

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

   10. Simplified 80.6%

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

   11. Taylor expanded in t around 0 80.6%

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

       1. unpow2 80.6%

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

   13. Simplified 80.6%

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

   if 3.2000000000000001e-12 < t

   1. Initial program 43.1%

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

      1. associate-*l/ 43.1%

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

   3. Simplified 43.1%

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

   4. Taylor expanded in t around inf 21.3%

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

      1. *-commutative 21.3%

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

      2. +-commutative 21.3%

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

      3. associate-*r/ 50.6%

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

      4. sub-neg 50.6%

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

      5. metadata-eval 50.6%

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

      6. associate-*r* 50.6%

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

      7. unpow2 50.6%

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

      8. +-commutative 50.6%

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

   6. Simplified 50.6%

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

   7. Taylor expanded in t around 0 92.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-179}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-209}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 79.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (/ (sqrt x) l))))
   (if (<= t -3.6e-151)
     (/ t (- (/ (- t) x) t))
     (if (<= t 4.2e-217)
       t_1
       (if (<= t 8.6e-147)
         1.0
         (if (<= t 1.32e-92) t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(x) / l);
	double tmp;
	if (t <= -3.6e-151) {
		tmp = t / ((-t / x) - t);
	} else if (t <= 4.2e-217) {
		tmp = t_1;
	} else if (t <= 8.6e-147) {
		tmp = 1.0;
	} else if (t <= 1.32e-92) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = t * (sqrt(x) / l)
    if (t <= (-3.6d-151)) then
        tmp = t / ((-t / x) - t)
    else if (t <= 4.2d-217) then
        tmp = t_1
    else if (t <= 8.6d-147) then
        tmp = 1.0d0
    else if (t <= 1.32d-92) then
        tmp = t_1
    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 t_1 = t * (Math.sqrt(x) / l);
	double tmp;
	if (t <= -3.6e-151) {
		tmp = t / ((-t / x) - t);
	} else if (t <= 4.2e-217) {
		tmp = t_1;
	} else if (t <= 8.6e-147) {
		tmp = 1.0;
	} else if (t <= 1.32e-92) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * (math.sqrt(x) / l)
	tmp = 0
	if t <= -3.6e-151:
		tmp = t / ((-t / x) - t)
	elif t <= 4.2e-217:
		tmp = t_1
	elif t <= 8.6e-147:
		tmp = 1.0
	elif t <= 1.32e-92:
		tmp = t_1
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(x) / l))
	tmp = 0.0
	if (t <= -3.6e-151)
		tmp = Float64(t / Float64(Float64(Float64(-t) / x) - t));
	elseif (t <= 4.2e-217)
		tmp = t_1;
	elseif (t <= 8.6e-147)
		tmp = 1.0;
	elseif (t <= 1.32e-92)
		tmp = t_1;
	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)
	t_1 = t * (sqrt(x) / l);
	tmp = 0.0;
	if (t <= -3.6e-151)
		tmp = t / ((-t / x) - t);
	elseif (t <= 4.2e-217)
		tmp = t_1;
	elseif (t <= 8.6e-147)
		tmp = 1.0;
	elseif (t <= 1.32e-92)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.6e-151], N[(t / N[(N[((-t) / x), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-217], t$95$1, If[LessEqual[t, 8.6e-147], 1.0, If[LessEqual[t, 1.32e-92], t$95$1, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.60000000000000032e-151

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

      1. *-commutative 39.0%

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

      2. clear-num 39.0%

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

      3. un-div-inv 39.0%

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

      4. sqrt-undiv 39.2%

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

      5. metadata-eval 39.2%

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

      6. sub-neg 39.2%

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

      7. associate-*l/ 24.1%

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

      8. sub-neg 24.1%

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

      9. metadata-eval 24.1%

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

   5. Applied egg-rr 24.1%

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

      1. +-commutative 24.1%

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

      2. *-commutative 24.1%

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

      3. +-commutative 24.1%

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

      4. +-commutative 24.1%

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

   7. Simplified 24.1%

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

   8. Taylor expanded in t around inf 37.5%

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

      1. associate-*r/ 37.5%

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

      2. +-commutative 37.5%

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

      3. *-commutative 37.5%

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

      4. unpow2 37.5%

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

      5. sub-neg 37.5%

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

      6. metadata-eval 37.5%

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

   10. Simplified 37.5%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. fma-def 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 89.0%

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

       5. mul-1-neg 89.0%

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

       6. fma-neg 89.0%

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

       7. mul-1-neg 89.0%

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

   13. Simplified 89.0%

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

   if -3.60000000000000032e-151 < t < 4.2e-217 or 8.6000000000000002e-147 < t < 1.3200000000000001e-92

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in x around inf 57.8%

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

      1. associate--l+ 57.8%

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

      2. unpow2 57.8%

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

      3. distribute-lft-out 57.8%

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

      4. unpow2 57.8%

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

      5. unpow2 57.8%

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

      6. associate-*r/ 57.8%

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

      7. mul-1-neg 57.8%

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

      8. +-commutative 57.8%

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

      9. unpow2 57.8%

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

      10. unpow2 57.8%

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

      11. fma-udef 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in t around 0 57.5%

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

      1. sub-neg 57.5%

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

      2. unpow2 57.5%

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

      3. mul-1-neg 57.5%

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

      4. remove-double-neg 57.5%

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

      5. unpow2 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in l around 0 47.7%

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

       1. associate-*l/ 47.7%

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

       2. *-lft-identity 47.7%

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

   12. Simplified 47.7%

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

   if 4.2e-217 < t < 8.6000000000000002e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

   4. Taylor expanded in t around inf 25.2%

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

      1. *-commutative 25.2%

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

      2. +-commutative 25.2%

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

      3. associate-*r/ 25.2%

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

      4. sub-neg 25.2%

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

      5. metadata-eval 25.2%

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

      6. associate-*r* 25.2%

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

      7. unpow2 25.2%

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

      8. +-commutative 25.2%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in x around inf 70.9%

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

   if 1.3200000000000001e-92 < t

   1. Initial program 45.3%

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

      1. associate-*l/ 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around inf 29.0%

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

      1. *-commutative 29.0%

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

      2. +-commutative 29.0%

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

      3. associate-*r/ 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

      6. associate-*r* 53.5%

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

      7. unpow2 53.5%

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

      8. +-commutative 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in t around 0 88.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-92}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 79.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{x}}{\ell}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{-150}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (/ (sqrt x) l))) (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.75e-150)
     (- t_2)
     (if (<= t 2.2e-218)
       t_1
       (if (<= t 1.05e-144) 1.0 (if (<= t 1.15e-92) t_1 t_2))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(x) / l);
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.75e-150) {
		tmp = -t_2;
	} else if (t <= 2.2e-218) {
		tmp = t_1;
	} else if (t <= 1.05e-144) {
		tmp = 1.0;
	} else if (t <= 1.15e-92) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (sqrt(x) / l)
    t_2 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-1.75d-150)) then
        tmp = -t_2
    else if (t <= 2.2d-218) then
        tmp = t_1
    else if (t <= 1.05d-144) then
        tmp = 1.0d0
    else if (t <= 1.15d-92) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(x) / l);
	double t_2 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.75e-150) {
		tmp = -t_2;
	} else if (t <= 2.2e-218) {
		tmp = t_1;
	} else if (t <= 1.05e-144) {
		tmp = 1.0;
	} else if (t <= 1.15e-92) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * (math.sqrt(x) / l)
	t_2 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -1.75e-150:
		tmp = -t_2
	elif t <= 2.2e-218:
		tmp = t_1
	elif t <= 1.05e-144:
		tmp = 1.0
	elif t <= 1.15e-92:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(x) / l))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.75e-150)
		tmp = Float64(-t_2);
	elseif (t <= 2.2e-218)
		tmp = t_1;
	elseif (t <= 1.05e-144)
		tmp = 1.0;
	elseif (t <= 1.15e-92)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(x) / l);
	t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1.75e-150)
		tmp = -t_2;
	elseif (t <= 2.2e-218)
		tmp = t_1;
	elseif (t <= 1.05e-144)
		tmp = 1.0;
	elseif (t <= 1.15e-92)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.75e-150], (-t$95$2), If[LessEqual[t, 2.2e-218], t$95$1, If[LessEqual[t, 1.05e-144], 1.0, If[LessEqual[t, 1.15e-92], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.7499999999999999e-150

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around inf 37.4%

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

      1. *-commutative 37.4%

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

      2. +-commutative 37.4%

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

      3. associate-*r/ 56.1%

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

      4. sub-neg 56.1%

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

      5. metadata-eval 56.1%

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

      6. associate-*r* 56.1%

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

      7. unpow2 56.1%

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

      8. +-commutative 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in t around -inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. +-commutative 89.3%

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

      3. sub-neg 89.3%

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

      4. metadata-eval 89.3%

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

   9. Simplified 89.3%

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

   if -1.7499999999999999e-150 < t < 2.20000000000000007e-218 or 1.0500000000000001e-144 < t < 1.15000000000000008e-92

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in x around inf 57.8%

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

      1. associate--l+ 57.8%

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

      2. unpow2 57.8%

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

      3. distribute-lft-out 57.8%

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

      4. unpow2 57.8%

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

      5. unpow2 57.8%

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

      6. associate-*r/ 57.8%

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

      7. mul-1-neg 57.8%

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

      8. +-commutative 57.8%

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

      9. unpow2 57.8%

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

      10. unpow2 57.8%

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

      11. fma-udef 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in t around 0 57.5%

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

      1. sub-neg 57.5%

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

      2. unpow2 57.5%

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

      3. mul-1-neg 57.5%

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

      4. remove-double-neg 57.5%

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

      5. unpow2 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in l around 0 47.7%

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

       1. associate-*l/ 47.7%

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

       2. *-lft-identity 47.7%

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

   12. Simplified 47.7%

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

   if 2.20000000000000007e-218 < t < 1.0500000000000001e-144

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

   4. Taylor expanded in t around inf 25.2%

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

      1. *-commutative 25.2%

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

      2. +-commutative 25.2%

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

      3. associate-*r/ 25.2%

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

      4. sub-neg 25.2%

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

      5. metadata-eval 25.2%

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

      6. associate-*r* 25.2%

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

      7. unpow2 25.2%

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

      8. +-commutative 25.2%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in x around inf 70.9%

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

   if 1.15000000000000008e-92 < t

   1. Initial program 45.3%

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

      1. associate-*l/ 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around inf 29.0%

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

      1. *-commutative 29.0%

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

      2. +-commutative 29.0%

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

      3. associate-*r/ 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

      6. associate-*r* 53.5%

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

      7. unpow2 53.5%

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

      8. +-commutative 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in t around 0 88.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-150}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 5: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (sqrt x) (/ t l))))
   (if (<= t -2.2e-151)
     (/ t (- (/ (- t) x) t))
     (if (<= t 3.8e-217)
       t_1
       (if (<= t 1.95e-144)
         1.0
         (if (<= t 8.5e-94) t_1 (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(x) * (t / l);
	double tmp;
	if (t <= -2.2e-151) {
		tmp = t / ((-t / x) - t);
	} else if (t <= 3.8e-217) {
		tmp = t_1;
	} else if (t <= 1.95e-144) {
		tmp = 1.0;
	} else if (t <= 8.5e-94) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	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) * (t / l)
    if (t <= (-2.2d-151)) then
        tmp = t / ((-t / x) - t)
    else if (t <= 3.8d-217) then
        tmp = t_1
    else if (t <= 1.95d-144) then
        tmp = 1.0d0
    else if (t <= 8.5d-94) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    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) * (t / l);
	double tmp;
	if (t <= -2.2e-151) {
		tmp = t / ((-t / x) - t);
	} else if (t <= 3.8e-217) {
		tmp = t_1;
	} else if (t <= 1.95e-144) {
		tmp = 1.0;
	} else if (t <= 8.5e-94) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(x) * (t / l)
	tmp = 0
	if t <= -2.2e-151:
		tmp = t / ((-t / x) - t)
	elif t <= 3.8e-217:
		tmp = t_1
	elif t <= 1.95e-144:
		tmp = 1.0
	elif t <= 8.5e-94:
		tmp = t_1
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(x) * Float64(t / l))
	tmp = 0.0
	if (t <= -2.2e-151)
		tmp = Float64(t / Float64(Float64(Float64(-t) / x) - t));
	elseif (t <= 3.8e-217)
		tmp = t_1;
	elseif (t <= 1.95e-144)
		tmp = 1.0;
	elseif (t <= 8.5e-94)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(x) * (t / l);
	tmp = 0.0;
	if (t <= -2.2e-151)
		tmp = t / ((-t / x) - t);
	elseif (t <= 3.8e-217)
		tmp = t_1;
	elseif (t <= 1.95e-144)
		tmp = 1.0;
	elseif (t <= 8.5e-94)
		tmp = t_1;
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.2e-151], N[(t / N[(N[((-t) / x), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-217], t$95$1, If[LessEqual[t, 1.95e-144], 1.0, If[LessEqual[t, 8.5e-94], t$95$1, N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.1999999999999999e-151

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

      1. *-commutative 39.0%

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

      2. clear-num 39.0%

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

      3. un-div-inv 39.0%

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

      4. sqrt-undiv 39.2%

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

      5. metadata-eval 39.2%

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

      6. sub-neg 39.2%

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

      7. associate-*l/ 24.1%

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

      8. sub-neg 24.1%

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

      9. metadata-eval 24.1%

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

   5. Applied egg-rr 24.1%

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

      1. +-commutative 24.1%

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

      2. *-commutative 24.1%

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

      3. +-commutative 24.1%

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

      4. +-commutative 24.1%

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

   7. Simplified 24.1%

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

   8. Taylor expanded in t around inf 37.5%

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

      1. associate-*r/ 37.5%

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

      2. +-commutative 37.5%

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

      3. *-commutative 37.5%

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

      4. unpow2 37.5%

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

      5. sub-neg 37.5%

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

      6. metadata-eval 37.5%

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

   10. Simplified 37.5%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. fma-def 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 89.0%

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

       5. mul-1-neg 89.0%

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

       6. fma-neg 89.0%

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

       7. mul-1-neg 89.0%

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

   13. Simplified 89.0%

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

   if -2.1999999999999999e-151 < t < 3.79999999999999987e-217 or 1.95000000000000007e-144 < t < 8.50000000000000003e-94

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in x around inf 57.8%

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

      1. associate--l+ 57.8%

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

      2. unpow2 57.8%

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

      3. distribute-lft-out 57.8%

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

      4. unpow2 57.8%

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

      5. unpow2 57.8%

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

      6. associate-*r/ 57.8%

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

      7. mul-1-neg 57.8%

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

      8. +-commutative 57.8%

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

      9. unpow2 57.8%

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

      10. unpow2 57.8%

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

      11. fma-udef 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in t around 0 57.5%

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

      1. sub-neg 57.5%

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

      2. unpow2 57.5%

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

      3. mul-1-neg 57.5%

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

      4. remove-double-neg 57.5%

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

      5. unpow2 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in l around 0 36.6%

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

   if 3.79999999999999987e-217 < t < 1.95000000000000007e-144

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

   4. Taylor expanded in t around inf 25.2%

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

      1. *-commutative 25.2%

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

      2. +-commutative 25.2%

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

      3. associate-*r/ 25.2%

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

      4. sub-neg 25.2%

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

      5. metadata-eval 25.2%

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

      6. associate-*r* 25.2%

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

      7. unpow2 25.2%

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

      8. +-commutative 25.2%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in x around inf 70.9%

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

   if 8.50000000000000003e-94 < t

   1. Initial program 45.3%

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

      1. associate-*l/ 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around inf 29.0%

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

      1. *-commutative 29.0%

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

      2. +-commutative 29.0%

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

      3. associate-*r/ 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

      6. associate-*r* 53.5%

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

      7. unpow2 53.5%

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

      8. +-commutative 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in x around inf 88.2%

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

      1. associate--l+ 88.2%

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

      2. associate-*r/ 88.2%

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

      3. metadata-eval 88.2%

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

      4. unpow2 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 6: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (/ (sqrt x) l))))
   (if (<= t -3.4e-150)
     (/ t (- (/ (- t) x) t))
     (if (<= t 6e-217)
       t_1
       (if (<= t 3.3e-147)
         1.0
         (if (<= t 4.2e-92) t_1 (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(x) / l);
	double tmp;
	if (t <= -3.4e-150) {
		tmp = t / ((-t / x) - t);
	} else if (t <= 6e-217) {
		tmp = t_1;
	} else if (t <= 3.3e-147) {
		tmp = 1.0;
	} else if (t <= 4.2e-92) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (sqrt(x) / l)
    if (t <= (-3.4d-150)) then
        tmp = t / ((-t / x) - t)
    else if (t <= 6d-217) then
        tmp = t_1
    else if (t <= 3.3d-147) then
        tmp = 1.0d0
    else if (t <= 4.2d-92) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(x) / l);
	double tmp;
	if (t <= -3.4e-150) {
		tmp = t / ((-t / x) - t);
	} else if (t <= 6e-217) {
		tmp = t_1;
	} else if (t <= 3.3e-147) {
		tmp = 1.0;
	} else if (t <= 4.2e-92) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * (math.sqrt(x) / l)
	tmp = 0
	if t <= -3.4e-150:
		tmp = t / ((-t / x) - t)
	elif t <= 6e-217:
		tmp = t_1
	elif t <= 3.3e-147:
		tmp = 1.0
	elif t <= 4.2e-92:
		tmp = t_1
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(x) / l))
	tmp = 0.0
	if (t <= -3.4e-150)
		tmp = Float64(t / Float64(Float64(Float64(-t) / x) - t));
	elseif (t <= 6e-217)
		tmp = t_1;
	elseif (t <= 3.3e-147)
		tmp = 1.0;
	elseif (t <= 4.2e-92)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(x) / l);
	tmp = 0.0;
	if (t <= -3.4e-150)
		tmp = t / ((-t / x) - t);
	elseif (t <= 6e-217)
		tmp = t_1;
	elseif (t <= 3.3e-147)
		tmp = 1.0;
	elseif (t <= 4.2e-92)
		tmp = t_1;
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.4e-150], N[(t / N[(N[((-t) / x), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-217], t$95$1, If[LessEqual[t, 3.3e-147], 1.0, If[LessEqual[t, 4.2e-92], t$95$1, N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.39999999999999999e-150

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

      1. *-commutative 39.0%

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

      2. clear-num 39.0%

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

      3. un-div-inv 39.0%

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

      4. sqrt-undiv 39.2%

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

      5. metadata-eval 39.2%

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

      6. sub-neg 39.2%

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

      7. associate-*l/ 24.1%

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

      8. sub-neg 24.1%

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

      9. metadata-eval 24.1%

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

   5. Applied egg-rr 24.1%

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

      1. +-commutative 24.1%

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

      2. *-commutative 24.1%

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

      3. +-commutative 24.1%

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

      4. +-commutative 24.1%

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

   7. Simplified 24.1%

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

   8. Taylor expanded in t around inf 37.5%

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

      1. associate-*r/ 37.5%

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

      2. +-commutative 37.5%

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

      3. *-commutative 37.5%

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

      4. unpow2 37.5%

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

      5. sub-neg 37.5%

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

      6. metadata-eval 37.5%

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

   10. Simplified 37.5%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. fma-def 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 89.0%

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

       5. mul-1-neg 89.0%

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

       6. fma-neg 89.0%

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

       7. mul-1-neg 89.0%

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

   13. Simplified 89.0%

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

   if -3.39999999999999999e-150 < t < 6.00000000000000009e-217 or 3.29999999999999987e-147 < t < 4.2e-92

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in x around inf 57.8%

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

      1. associate--l+ 57.8%

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

      2. unpow2 57.8%

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

      3. distribute-lft-out 57.8%

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

      4. unpow2 57.8%

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

      5. unpow2 57.8%

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

      6. associate-*r/ 57.8%

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

      7. mul-1-neg 57.8%

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

      8. +-commutative 57.8%

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

      9. unpow2 57.8%

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

      10. unpow2 57.8%

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

      11. fma-udef 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in t around 0 57.5%

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

      1. sub-neg 57.5%

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

      2. unpow2 57.5%

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

      3. mul-1-neg 57.5%

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

      4. remove-double-neg 57.5%

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

      5. unpow2 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in l around 0 47.7%

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

       1. associate-*l/ 47.7%

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

       2. *-lft-identity 47.7%

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

   12. Simplified 47.7%

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

   if 6.00000000000000009e-217 < t < 3.29999999999999987e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

   4. Taylor expanded in t around inf 25.2%

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

      1. *-commutative 25.2%

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

      2. +-commutative 25.2%

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

      3. associate-*r/ 25.2%

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

      4. sub-neg 25.2%

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

      5. metadata-eval 25.2%

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

      6. associate-*r* 25.2%

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

      7. unpow2 25.2%

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

      8. +-commutative 25.2%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in x around inf 70.9%

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

   if 4.2e-92 < t

   1. Initial program 45.3%

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

      1. associate-*l/ 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around inf 29.0%

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

      1. *-commutative 29.0%

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

      2. +-commutative 29.0%

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

      3. associate-*r/ 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

      6. associate-*r* 53.5%

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

      7. unpow2 53.5%

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

      8. +-commutative 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in x around inf 88.2%

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

      1. associate--l+ 88.2%

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

      2. associate-*r/ 88.2%

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

      3. metadata-eval 88.2%

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

      4. unpow2 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-92}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 7: 77.1% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2e-310)
   (/ t (- (/ (- t) x) t))
   (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = t / ((-t / x) - t);
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	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 <= (-2d-310)) then
        tmp = t / ((-t / x) - t)
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    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 <= -2e-310) {
		tmp = t / ((-t / x) - t);
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = t / ((-t / x) - t)
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(t / Float64(Float64(Float64(-t) / x) - t));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = t / ((-t / x) - t);
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2e-310], N[(t / N[(N[((-t) / x), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 33.0%

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

      1. associate-*l/ 33.0%

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

   3. Simplified 33.0%

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

      1. *-commutative 33.0%

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

      2. clear-num 33.0%

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

      3. un-div-inv 33.0%

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

      4. sqrt-undiv 33.2%

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

      5. metadata-eval 33.2%

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

      6. sub-neg 33.2%

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

      7. associate-*l/ 22.3%

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

      8. sub-neg 22.3%

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

      9. metadata-eval 22.3%

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

   5. Applied egg-rr 22.3%

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

      1. +-commutative 22.3%

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

      2. *-commutative 22.3%

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

      3. +-commutative 22.3%

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

      4. +-commutative 22.3%

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

   7. Simplified 22.3%

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

   8. Taylor expanded in t around inf 31.8%

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

      1. associate-*r/ 31.8%

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

      2. +-commutative 31.8%

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

      3. *-commutative 31.8%

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

      4. unpow2 31.8%

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

      5. sub-neg 31.8%

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

      6. metadata-eval 31.8%

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

   10. Simplified 31.8%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. fma-def 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 80.4%

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

       5. mul-1-neg 80.4%

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

       6. fma-neg 80.4%

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

       7. mul-1-neg 80.4%

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

   13. Simplified 80.4%

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

   if -1.999999999999994e-310 < t

   1. Initial program 36.4%

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

      1. associate-*l/ 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around inf 24.6%

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

      1. *-commutative 24.6%

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

      2. +-commutative 24.6%

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

      3. associate-*r/ 42.8%

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

      4. sub-neg 42.8%

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

      5. metadata-eval 42.8%

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

      6. associate-*r* 42.8%

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

      7. unpow2 42.8%

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

      8. +-commutative 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in x around inf 75.5%

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

      1. associate--l+ 75.5%

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

      2. associate-*r/ 75.5%

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

      3. metadata-eval 75.5%

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

      4. unpow2 75.5%

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

   9. Simplified 75.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 8: 77.0% accurate, 22.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2e-310) (/ t (- (/ (- t) x) t)) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = t / ((-t / x) - t);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	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 <= (-2d-310)) then
        tmp = t / ((-t / x) - t)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    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 <= -2e-310) {
		tmp = t / ((-t / x) - t);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = t / ((-t / x) - t)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(t / Float64(Float64(Float64(-t) / x) - t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = t / ((-t / x) - t);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2e-310], N[(t / N[(N[((-t) / x), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 33.0%

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

      1. associate-*l/ 33.0%

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

   3. Simplified 33.0%

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

      1. *-commutative 33.0%

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

      2. clear-num 33.0%

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

      3. un-div-inv 33.0%

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

      4. sqrt-undiv 33.2%

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

      5. metadata-eval 33.2%

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

      6. sub-neg 33.2%

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

      7. associate-*l/ 22.3%

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

      8. sub-neg 22.3%

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

      9. metadata-eval 22.3%

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

   5. Applied egg-rr 22.3%

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

      1. +-commutative 22.3%

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

      2. *-commutative 22.3%

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

      3. +-commutative 22.3%

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

      4. +-commutative 22.3%

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

   7. Simplified 22.3%

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

   8. Taylor expanded in t around inf 31.8%

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

      1. associate-*r/ 31.8%

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

      2. +-commutative 31.8%

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

      3. *-commutative 31.8%

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

      4. unpow2 31.8%

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

      5. sub-neg 31.8%

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

      6. metadata-eval 31.8%

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

   10. Simplified 31.8%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. fma-def 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 80.4%

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

       5. mul-1-neg 80.4%

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

       6. fma-neg 80.4%

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

       7. mul-1-neg 80.4%

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

   13. Simplified 80.4%

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

   if -1.999999999999994e-310 < t

   1. Initial program 36.4%

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

      1. associate-*l/ 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around inf 24.6%

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

      1. *-commutative 24.6%

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

      2. +-commutative 24.6%

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

      3. associate-*r/ 42.8%

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

      4. sub-neg 42.8%

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

      5. metadata-eval 42.8%

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

      6. associate-*r* 42.8%

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

      7. unpow2 42.8%

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

      8. +-commutative 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in x around inf 75.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{\frac{-t}{x} - t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 9: 76.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2e-310) (/ t (- t)) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = t / -t;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	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 <= (-2d-310)) then
        tmp = t / -t
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    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 <= -2e-310) {
		tmp = t / -t;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = t / -t
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(t / Float64(-t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = t / -t;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2e-310], N[(t / (-t)), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 33.0%

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

      1. associate-*l/ 33.0%

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

   3. Simplified 33.0%

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

      1. *-commutative 33.0%

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

      2. clear-num 33.0%

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

      3. un-div-inv 33.0%

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

      4. sqrt-undiv 33.2%

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

      5. metadata-eval 33.2%

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

      6. sub-neg 33.2%

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

      7. associate-*l/ 22.3%

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

      8. sub-neg 22.3%

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

      9. metadata-eval 22.3%

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

   5. Applied egg-rr 22.3%

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

      1. +-commutative 22.3%

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

      2. *-commutative 22.3%

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

      3. +-commutative 22.3%

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

      4. +-commutative 22.3%

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

   7. Simplified 22.3%

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

   8. Taylor expanded in t around inf 31.8%

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

      1. associate-*r/ 31.8%

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

      2. +-commutative 31.8%

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

      3. *-commutative 31.8%

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

      4. unpow2 31.8%

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

      5. sub-neg 31.8%

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

      6. metadata-eval 31.8%

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

   10. Simplified 31.8%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 79.8%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t} \]

       3. mul-1-neg 79.8%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 79.8%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -1.999999999999994e-310 < t

   1. Initial program 36.4%

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

      1. associate-*l/ 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around inf 24.6%

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

      1. *-commutative 24.6%

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

      2. +-commutative 24.6%

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

      3. associate-*r/ 42.8%

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

      4. sub-neg 42.8%

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

      5. metadata-eval 42.8%

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

      6. associate-*r* 42.8%

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

      7. unpow2 42.8%

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

      8. +-commutative 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in x around inf 75.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 10: 76.3% accurate, 37.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -2e-310) (/ t (- t)) 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = t / -t;
	} 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 <= (-2d-310)) then
        tmp = t / -t
    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 <= -2e-310) {
		tmp = t / -t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = t / -t
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(t / Float64(-t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = t / -t;
	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, -2e-310], N[(t / (-t)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 33.0%

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

      1. associate-*l/ 33.0%

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

   3. Simplified 33.0%

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

      1. *-commutative 33.0%

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

      2. clear-num 33.0%

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

      3. un-div-inv 33.0%

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

      4. sqrt-undiv 33.2%

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

      5. metadata-eval 33.2%

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

      6. sub-neg 33.2%

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

      7. associate-*l/ 22.3%

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

      8. sub-neg 22.3%

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

      9. metadata-eval 22.3%

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

   5. Applied egg-rr 22.3%

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

      1. +-commutative 22.3%

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

      2. *-commutative 22.3%

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

      3. +-commutative 22.3%

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

      4. +-commutative 22.3%

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

   7. Simplified 22.3%

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

   8. Taylor expanded in t around inf 31.8%

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

      1. associate-*r/ 31.8%

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

      2. +-commutative 31.8%

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

      3. *-commutative 31.8%

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

      4. unpow2 31.8%

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

      5. sub-neg 31.8%

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

      6. metadata-eval 31.8%

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

   10. Simplified 31.8%

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

   11. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 79.8%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t} \]

       3. mul-1-neg 79.8%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 79.8%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -1.999999999999994e-310 < t

   1. Initial program 36.4%

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

      1. associate-*l/ 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around inf 24.6%

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

      1. *-commutative 24.6%

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

      2. +-commutative 24.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{{t}^{2} \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \cdot t \]

      3. associate-*r/ 42.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{x + 1}{x - 1}\right)}}} \cdot t \]

      4. sub-neg 42.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{\color{blue}{x + \left(-1\right)}}\right)}} \cdot t \]

      5. metadata-eval 42.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{x + \color{blue}{-1}}\right)}} \cdot t \]

      6. associate-*r* 42.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{x + 1}{x + -1}}}} \cdot t \]

      7. unpow2 42.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \frac{x + 1}{x + -1}}} \cdot t \]

      8. +-commutative 42.8%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   6. Simplified 42.8%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{-1 + x}}}} \cdot t \]

   7. Taylor expanded in x around inf 74.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 39.0% 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 34.8%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 34.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 34.8%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

4. Taylor expanded in t around inf 28.0%

   \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}} \cdot t \]
5. Step-by-step derivation

   1. *-commutative 28.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{{t}^{2} \cdot \left(1 + x\right)}}{x - 1}}} \cdot t \]

   2. +-commutative 28.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{{t}^{2} \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \cdot t \]

   3. associate-*r/ 45.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{x + 1}{x - 1}\right)}}} \cdot t \]

   4. sub-neg 45.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{\color{blue}{x + \left(-1\right)}}\right)}} \cdot t \]

   5. metadata-eval 45.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{x + \color{blue}{-1}}\right)}} \cdot t \]

   6. associate-*r* 45.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{x + 1}{x + -1}}}} \cdot t \]

   7. unpow2 45.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \frac{x + 1}{x + -1}}} \cdot t \]

   8. +-commutative 45.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

6. Simplified 45.0%

   \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{-1 + x}}}} \cdot t \]

7. Taylor expanded in x around inf 39.3%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 39.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023279 
(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)))))