Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 80.0%

Time: 17.1s

Alternatives: 9

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.3% 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: 80.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;l\_m \leq 1.26 \cdot 10^{+152}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\frac{l\_m \cdot l\_m + 2 \cdot \left(t\_m \cdot t\_m\right)}{x} + \left(\frac{l\_m \cdot l\_m}{x} + 2 \cdot \left(t\_m \cdot t\_m + \frac{t\_m \cdot t\_m}{x}\right)\right)}}\\ \mathbf{elif}\;l\_m \leq 1.15 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{2 \cdot \frac{1}{\frac{1}{x + -1} + \frac{\frac{1}{x} - -1}{x}}} \cdot \frac{1}{l\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 4.3e+68)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (if (<= l_m 1.26e+152)
      (*
       t_m
       (sqrt
        (/
         2.0
         (+
          (/ (+ (* l_m l_m) (* 2.0 (* t_m t_m))) x)
          (+ (/ (* l_m l_m) x) (* 2.0 (+ (* t_m t_m) (/ (* t_m t_m) x))))))))
      (if (<= l_m 1.15e+207)
        (sqrt (+ 1.0 (/ -2.0 x)))
        (*
         t_m
         (*
          (sqrt
           (* 2.0 (/ 1.0 (+ (/ 1.0 (+ x -1.0)) (/ (- (/ 1.0 x) -1.0) x)))))
          (/ 1.0 l_m))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 4.3e+68) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else if (l_m <= 1.26e+152) {
		tmp = t_m * sqrt((2.0 / ((((l_m * l_m) + (2.0 * (t_m * t_m))) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / x)))))));
	} else if (l_m <= 1.15e+207) {
		tmp = sqrt((1.0 + (-2.0 / x)));
	} else {
		tmp = t_m * (sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) * (1.0 / l_m));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 4.3d+68) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (l_m <= 1.26d+152) then
        tmp = t_m * sqrt((2.0d0 / ((((l_m * l_m) + (2.0d0 * (t_m * t_m))) / x) + (((l_m * l_m) / x) + (2.0d0 * ((t_m * t_m) + ((t_m * t_m) / x)))))))
    else if (l_m <= 1.15d+207) then
        tmp = sqrt((1.0d0 + ((-2.0d0) / x)))
    else
        tmp = t_m * (sqrt((2.0d0 * (1.0d0 / ((1.0d0 / (x + (-1.0d0))) + (((1.0d0 / x) - (-1.0d0)) / x))))) * (1.0d0 / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 4.3e+68) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (l_m <= 1.26e+152) {
		tmp = t_m * Math.sqrt((2.0 / ((((l_m * l_m) + (2.0 * (t_m * t_m))) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / x)))))));
	} else if (l_m <= 1.15e+207) {
		tmp = Math.sqrt((1.0 + (-2.0 / x)));
	} else {
		tmp = t_m * (Math.sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) * (1.0 / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 4.3e+68:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	elif l_m <= 1.26e+152:
		tmp = t_m * math.sqrt((2.0 / ((((l_m * l_m) + (2.0 * (t_m * t_m))) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / x)))))))
	elif l_m <= 1.15e+207:
		tmp = math.sqrt((1.0 + (-2.0 / x)))
	else:
		tmp = t_m * (math.sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) * (1.0 / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 4.3e+68)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	elseif (l_m <= 1.26e+152)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(Float64(Float64(l_m * l_m) + Float64(2.0 * Float64(t_m * t_m))) / x) + Float64(Float64(Float64(l_m * l_m) / x) + Float64(2.0 * Float64(Float64(t_m * t_m) + Float64(Float64(t_m * t_m) / x))))))));
	elseif (l_m <= 1.15e+207)
		tmp = sqrt(Float64(1.0 + Float64(-2.0 / x)));
	else
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 * Float64(1.0 / Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(Float64(1.0 / x) - -1.0) / x))))) * Float64(1.0 / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 4.3e+68)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	elseif (l_m <= 1.26e+152)
		tmp = t_m * sqrt((2.0 / ((((l_m * l_m) + (2.0 * (t_m * t_m))) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / x)))))));
	elseif (l_m <= 1.15e+207)
		tmp = sqrt((1.0 + (-2.0 / x)));
	else
		tmp = t_m * (sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) * (1.0 / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 4.3e+68], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.26e+152], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 1.15e+207], N[Sqrt[N[(1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[N[(2.0 * N[(1.0 / N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if l < 4.3000000000000001e68

   1. Initial program 44.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 44.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 44.5%

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

   6. Taylor expanded in t around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

      7. +-lowering-+.f64 44.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

   8. Simplified 44.5%

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

   if 4.3000000000000001e68 < l < 1.2599999999999999e152

   1. Initial program 9.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 9.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{{t}^{2}}{x}\right), \left({t}^{2}\right)\right)\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({t}^{2}\right), x\right), \left({t}^{2}\right)\right)\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(t \cdot t\right), x\right), \left({t}^{2}\right)\right)\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), x\right), \left({t}^{2}\right)\right)\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), x\right), \left(t \cdot t\right)\right)\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), x\right), \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{{\ell}^{2}}{x}\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), x\right), \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\left({\ell}^{2}\right), x\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), x\right), \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), x\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), x\right), \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), x\right)\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)\right)\right)\right)\right) \]

   7. Simplified 78.6%

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

   if 1.2599999999999999e152 < l < 1.14999999999999997e207

   1. Initial program 0.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 30.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 30.1%

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

   6. Taylor expanded in t around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

      7. +-lowering-+.f64 30.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

   8. Simplified 30.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(1 - 2 \cdot \frac{1}{x}\right)}\right) \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{x}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)\right)\right) \]

       5. distribute-neg-frac N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(2\right)}{x}\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-2}{x}\right)\right)\right) \]

       7. /-lowering-/.f64 30.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-2, x\right)\right)\right) \]

   11. Simplified 30.1%

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

   if 1.14999999999999997e207 < l

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in l around inf

      \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}} \cdot \color{blue}{\frac{\sqrt{2}}{\ell}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}\right), \color{blue}{\left(\frac{\sqrt{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 33.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \color{blue}{\left(\frac{1 + \frac{1}{x}}{x}\right)}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{1}{x}\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{x}\right)\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 81.6%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, x\right)\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

   10. Simplified 81.6%

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{\sqrt{\frac{1}{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}} \cdot \sqrt{2}}{\color{blue}{\ell}}\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\left(\sqrt{\frac{1}{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}} \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{1}{\ell}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}} \cdot \sqrt{2}\right), \color{blue}{\left(\frac{1}{\ell}\right)}\right)\right) \]

   12. Applied egg-rr 81.8%

       \[\leadsto t \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{x + -1} - \frac{-1 + \frac{-1}{x}}{x}} \cdot 2} \cdot \frac{1}{\ell}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{+152}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{2 \cdot \frac{1}{\frac{1}{x + -1} + \frac{\frac{1}{x} - -1}{x}}} \cdot \frac{1}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := l\_m \cdot l\_m + t\_2\\ t_4 := 2 \cdot t\_3\\ t_5 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-174}:\\ \;\;\;\;\frac{t\_5}{t\_5 + \frac{0.5 \cdot t\_4}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{t\_5}{\sqrt{t\_2 + \frac{\frac{\left(t\_4 + \left(\frac{t\_2}{x} + \frac{l\_m \cdot l\_m}{x}\right)\right) + \frac{t\_3}{x}}{x} - t\_3 \cdot -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (* t_m t_m)))
        (t_3 (+ (* l_m l_m) t_2))
        (t_4 (* 2.0 t_3))
        (t_5 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.9e-174)
      (/ t_5 (+ t_5 (/ (* 0.5 t_4) (* t_m (* (sqrt 2.0) x)))))
      (if (<= t_m 4e+46)
        (/
         t_5
         (sqrt
          (+
           t_2
           (/
            (-
             (/ (+ (+ t_4 (+ (/ t_2 x) (/ (* l_m l_m) x))) (/ t_3 x)) x)
             (* t_3 -2.0))
            x))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l_m * l_m) + t_2;
	double t_4 = 2.0 * t_3;
	double t_5 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.9e-174) {
		tmp = t_5 / (t_5 + ((0.5 * t_4) / (t_m * (sqrt(2.0) * x))));
	} else if (t_m <= 4e+46) {
		tmp = t_5 / sqrt((t_2 + (((((t_4 + ((t_2 / x) + ((l_m * l_m) / x))) + (t_3 / x)) / x) - (t_3 * -2.0)) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = (l_m * l_m) + t_2
    t_4 = 2.0d0 * t_3
    t_5 = t_m * sqrt(2.0d0)
    if (t_m <= 1.9d-174) then
        tmp = t_5 / (t_5 + ((0.5d0 * t_4) / (t_m * (sqrt(2.0d0) * x))))
    else if (t_m <= 4d+46) then
        tmp = t_5 / sqrt((t_2 + (((((t_4 + ((t_2 / x) + ((l_m * l_m) / x))) + (t_3 / x)) / x) - (t_3 * (-2.0d0))) / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l_m * l_m) + t_2;
	double t_4 = 2.0 * t_3;
	double t_5 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 1.9e-174) {
		tmp = t_5 / (t_5 + ((0.5 * t_4) / (t_m * (Math.sqrt(2.0) * x))));
	} else if (t_m <= 4e+46) {
		tmp = t_5 / Math.sqrt((t_2 + (((((t_4 + ((t_2 / x) + ((l_m * l_m) / x))) + (t_3 / x)) / x) - (t_3 * -2.0)) / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * (t_m * t_m)
	t_3 = (l_m * l_m) + t_2
	t_4 = 2.0 * t_3
	t_5 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 1.9e-174:
		tmp = t_5 / (t_5 + ((0.5 * t_4) / (t_m * (math.sqrt(2.0) * x))))
	elif t_m <= 4e+46:
		tmp = t_5 / math.sqrt((t_2 + (((((t_4 + ((t_2 / x) + ((l_m * l_m) / x))) + (t_3 / x)) / x) - (t_3 * -2.0)) / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * Float64(t_m * t_m))
	t_3 = Float64(Float64(l_m * l_m) + t_2)
	t_4 = Float64(2.0 * t_3)
	t_5 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.9e-174)
		tmp = Float64(t_5 / Float64(t_5 + Float64(Float64(0.5 * t_4) / Float64(t_m * Float64(sqrt(2.0) * x)))));
	elseif (t_m <= 4e+46)
		tmp = Float64(t_5 / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(Float64(t_4 + Float64(Float64(t_2 / x) + Float64(Float64(l_m * l_m) / x))) + Float64(t_3 / x)) / x) - Float64(t_3 * -2.0)) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m * t_m);
	t_3 = (l_m * l_m) + t_2;
	t_4 = 2.0 * t_3;
	t_5 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 1.9e-174)
		tmp = t_5 / (t_5 + ((0.5 * t_4) / (t_m * (sqrt(2.0) * x))));
	elseif (t_m <= 4e+46)
		tmp = t_5 / sqrt((t_2 + (((((t_4 + ((t_2 / x) + ((l_m * l_m) / x))) + (t_3 / x)) / x) - (t_3 * -2.0)) / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-174], N[(t$95$5 / N[(t$95$5 + N[(N[(0.5 * t$95$4), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+46], N[(t$95$5 / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(N[(t$95$4 + N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(t$95$3 * -2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.9000000000000001e-174

   1. Initial program 32.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \left(t \cdot \sqrt{2} + \color{blue}{\frac{1}{2} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{+.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\frac{1}{2} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\frac{\frac{1}{2} \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right)}{\color{blue}{t \cdot \left(x \cdot \sqrt{2}\right)}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right)\right), \color{blue}{\left(t \cdot \left(x \cdot \sqrt{2}\right)\right)}\right)\right)\right) \]

   5. Simplified 15.2%

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

   if 1.9000000000000001e-174 < t < 4e46

   1. Initial program 53.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + -1 \cdot \frac{\left(-1 \cdot \left(-1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right) - \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}{x}}{x} + 2 \cdot {t}^{2}\right)}\right)\right) \]

   5. Simplified 85.3%

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

   if 4e46 < t

   1. Initial program 39.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 92.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 92.8%

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

   6. Taylor expanded in t around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

      7. +-lowering-+.f64 92.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

   8. Simplified 92.8%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-174}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{0.5 \cdot \left(2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) + \left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}}{x} - \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.15 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{2 \cdot \frac{1}{\frac{1}{x + -1} + \frac{\frac{1}{x} - -1}{x}}} \cdot \frac{1}{l\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.15e+207)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (*
     t_m
     (*
      (sqrt (* 2.0 (/ 1.0 (+ (/ 1.0 (+ x -1.0)) (/ (- (/ 1.0 x) -1.0) x)))))
      (/ 1.0 l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.15e+207) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) * (1.0 / l_m));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 1.15d+207) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt((2.0d0 * (1.0d0 / ((1.0d0 / (x + (-1.0d0))) + (((1.0d0 / x) - (-1.0d0)) / x))))) * (1.0d0 / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.15e+207) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) * (1.0 / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 1.15e+207:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) * (1.0 / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 1.15e+207)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 * Float64(1.0 / Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(Float64(1.0 / x) - -1.0) / x))))) * Float64(1.0 / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 1.15e+207)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) * (1.0 / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.15e+207], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[N[(2.0 * N[(1.0 / N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.14999999999999997e207

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 42.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 42.0%

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

   6. Taylor expanded in t around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

      7. +-lowering-+.f64 42.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

   8. Simplified 42.0%

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

   if 1.14999999999999997e207 < l

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in l around inf

      \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}} \cdot \color{blue}{\frac{\sqrt{2}}{\ell}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}\right), \color{blue}{\left(\frac{\sqrt{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 33.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \color{blue}{\left(\frac{1 + \frac{1}{x}}{x}\right)}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{1}{x}\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{x}\right)\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 81.6%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, x\right)\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

   10. Simplified 81.6%

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{\sqrt{\frac{1}{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}} \cdot \sqrt{2}}{\color{blue}{\ell}}\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\left(\sqrt{\frac{1}{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}} \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{1}{\ell}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}} \cdot \sqrt{2}\right), \color{blue}{\left(\frac{1}{\ell}\right)}\right)\right) \]

   12. Applied egg-rr 81.8%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{2 \cdot \frac{1}{\frac{1}{x + -1} + \frac{\frac{1}{x} - -1}{x}}} \cdot \frac{1}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.4 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2 \cdot \frac{1}{\frac{1}{x + -1} + \frac{\frac{1}{x} - -1}{x}}}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.4e+207)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (*
     t_m
     (/
      (sqrt (* 2.0 (/ 1.0 (+ (/ 1.0 (+ x -1.0)) (/ (- (/ 1.0 x) -1.0) x)))))
      l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.4e+207) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 2.4d+207) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt((2.0d0 * (1.0d0 / ((1.0d0 / (x + (-1.0d0))) + (((1.0d0 / x) - (-1.0d0)) / x))))) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.4e+207) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 2.4e+207:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.4e+207)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 * Float64(1.0 / Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(Float64(1.0 / x) - -1.0) / x))))) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 2.4e+207)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt((2.0 * (1.0 / ((1.0 / (x + -1.0)) + (((1.0 / x) - -1.0) / x))))) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.4e+207], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[N[(2.0 * N[(1.0 / N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.4000000000000001e207

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 42.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 42.0%

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

   6. Taylor expanded in t around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

      7. +-lowering-+.f64 42.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

   8. Simplified 42.0%

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

   if 2.4000000000000001e207 < l

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in l around inf

      \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}} \cdot \color{blue}{\frac{\sqrt{2}}{\ell}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}\right), \color{blue}{\left(\frac{\sqrt{2}}{\ell}\right)}\right)\right) \]

   7. Simplified 33.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \color{blue}{\left(\frac{1 + \frac{1}{x}}{x}\right)}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{1}{x}\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{x}\right)\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

      3. /-lowering-/.f64 81.6%

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, x\right)\right), x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \ell\right)\right)\right) \]

   10. Simplified 81.6%

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}} \cdot \frac{\sqrt{2}}{\ell}\right), \color{blue}{t}\right) \]

   12. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\frac{1}{x + -1} - \frac{-1 + \frac{-1}{x}}{x}} \cdot 2}}{\ell} \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \frac{1}{\frac{1}{x + -1} + \frac{\frac{1}{x} - -1}{x}}}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.4 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.4e+207)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.4e+207) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 2.4d+207) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.4e+207) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 2.4e+207:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.4e+207)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 2.4e+207)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.4e+207], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.4000000000000001e207

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 42.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 42.0%

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

   6. Taylor expanded in t around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

      7. +-lowering-+.f64 42.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

   8. Simplified 42.0%

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

   if 2.4000000000000001e207 < l

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{2}{\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1} - {\ell}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1} - {\ell}^{2}\right)\right)\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1}\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2} \cdot \left(1 + x\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + x\right) \cdot {\ell}^{2}\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + x\right), \left({\ell}^{2}\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x + 1\right), \left({\ell}^{2}\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left({\ell}^{2}\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\ell \cdot \ell\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(x + -1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right) \]

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \frac{x}{{\ell}^{2} - -1 \cdot {\ell}^{2}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot x}{{\ell}^{2} - -1 \cdot {\ell}^{2}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot x\right), \left({\ell}^{2} - -1 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 2\right), \left({\ell}^{2} - -1 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left({\ell}^{2} - -1 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left({\ell}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {\ell}^{2}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left({\ell}^{2} + 1 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(1 + 1\right) \cdot {\ell}^{2}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(2 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 20.1%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right) \]

   10. Simplified 20.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{1}{\ell} \cdot \sqrt{x}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{1 \cdot \sqrt{x}}{\color{blue}{\ell}}\right)\right) \]

       2. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{\sqrt{x}}{\ell}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(\sqrt{x}\right), \color{blue}{\ell}\right)\right) \]

       4. sqrt-lowering-sqrt.f64 78.4%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \ell\right)\right) \]

   13. Simplified 78.4%

       \[\leadsto t \cdot \color{blue}{\frac{\sqrt{x}}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.1 \cdot 10^{+207}:\\ \;\;\;\;\frac{0.5}{x \cdot x} + \left(1 + \frac{-1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.1e+207)
    (+ (/ 0.5 (* x x)) (+ 1.0 (/ -1.0 x)))
    (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.1e+207) {
		tmp = (0.5 / (x * x)) + (1.0 + (-1.0 / x));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 2.1d+207) then
        tmp = (0.5d0 / (x * x)) + (1.0d0 + ((-1.0d0) / x))
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.1e+207) {
		tmp = (0.5 / (x * x)) + (1.0 + (-1.0 / x));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 2.1e+207:
		tmp = (0.5 / (x * x)) + (1.0 + (-1.0 / x))
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.1e+207)
		tmp = Float64(Float64(0.5 / Float64(x * x)) + Float64(1.0 + Float64(-1.0 / x)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 2.1e+207)
		tmp = (0.5 / (x * x)) + (1.0 + (-1.0 / x));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.1e+207], N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.0999999999999999e207

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 42.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

   5. Simplified 42.0%

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

   6. Taylor expanded in t around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

      7. +-lowering-+.f64 42.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

   8. Simplified 42.0%

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

   9. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(1 - \frac{1}{x}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right), \left(\color{blue}{1} - \frac{1}{x}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right), \left(1 - \frac{1}{x}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(1 - \frac{1}{x}\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{x}\right)\right)\right) \]

       11. /-lowering-/.f64 42.0%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right)\right) \]

   11. Simplified 42.0%

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

   if 2.0999999999999999e207 < l

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{2}{\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1} - {\ell}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1} - {\ell}^{2}\right)\right)\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\frac{{\ell}^{2} \cdot \left(1 + x\right)}{x - 1}\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2} \cdot \left(1 + x\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + x\right) \cdot {\ell}^{2}\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + x\right), \left({\ell}^{2}\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x + 1\right), \left({\ell}^{2}\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left({\ell}^{2}\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\ell \cdot \ell\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(x - 1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(x + -1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right) \]

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \frac{x}{{\ell}^{2} - -1 \cdot {\ell}^{2}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot x}{{\ell}^{2} - -1 \cdot {\ell}^{2}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot x\right), \left({\ell}^{2} - -1 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 2\right), \left({\ell}^{2} - -1 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left({\ell}^{2} - -1 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left({\ell}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {\ell}^{2}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left({\ell}^{2} + 1 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(1 + 1\right) \cdot {\ell}^{2}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(2 \cdot {\ell}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 20.1%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right) \]

   10. Simplified 20.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{1}{\ell} \cdot \sqrt{x}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{1 \cdot \sqrt{x}}{\color{blue}{\ell}}\right)\right) \]

       2. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(t, \left(\frac{\sqrt{x}}{\ell}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(\sqrt{x}\right), \color{blue}{\ell}\right)\right) \]

       4. sqrt-lowering-sqrt.f64 78.4%

          \[\leadsto \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \ell\right)\right) \]

   13. Simplified 78.4%

       \[\leadsto t \cdot \color{blue}{\frac{\sqrt{x}}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{-1}{x}\right)\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ (/ 0.5 (* x x)) (+ 1.0 (/ -1.0 x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * ((0.5 / (x * x)) + (1.0 + (-1.0 / x)));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * ((0.5d0 / (x * x)) + (1.0d0 + ((-1.0d0) / x)))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * ((0.5 / (x * x)) + (1.0 + (-1.0 / x)));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * ((0.5 / (x * x)) + (1.0 + (-1.0 / x)))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(Float64(0.5 / Float64(x * x)) + Float64(1.0 + Float64(-1.0 / x))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * ((0.5 / (x * x)) + (1.0 + (-1.0 / x)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 39.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 39.5%

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

6. Taylor expanded in t around 0

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{x - 1}{1 + x}\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), \left(1 + x\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 + x\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), \left(1 + x\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(1 + x\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \left(x + 1\right)\right)\right) \]

   7. +-lowering-+.f64 39.5%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), \mathsf{+.f64}\left(x, 1\right)\right)\right) \]

8. Simplified 39.5%

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

9. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{{x}^{2}}\right), \color{blue}{\left(1 - \frac{1}{x}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right), \left(\color{blue}{1} - \frac{1}{x}\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right), \left(1 - \frac{1}{x}\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(1 - \frac{1}{x}\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right) \]

    9. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right)\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{x}\right)\right)\right) \]

    11. /-lowering-/.f64 39.5%

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right)\right) \]

11. Simplified 39.5%

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

Alternative 8: 76.7% accurate, 45.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((-1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + (-1.0 / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + (-1.0 / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 39.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 39.5%

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

6. Taylor expanded in x around inf

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{x}\right)}\right) \]

   2. /-lowering-/.f64 39.5%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right) \]

8. Simplified 39.5%

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

9. Final simplification 39.5%

   \[\leadsto 1 + \frac{-1}{x} \]
10. Add Preprocessing

Alternative 9: 75.9% accurate, 225.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 37.8%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 39.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 39.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 39.3%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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)))))