Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 84.9%

Time: 18.2s

Alternatives: 13

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: 33.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := t\_2 + l\_m \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-157}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 2.95 \cdot 10^{+69}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{t\_2 + \frac{\left(\frac{t\_2}{x} + \frac{l\_m \cdot l\_m}{x}\right) + \left(\frac{t\_3}{x} - t\_3 \cdot -2\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (+ t_2 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 1.15e-157)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 2.95e+69)
        (/
         (* t_m (sqrt 2.0))
         (sqrt
          (+
           t_2
           (/
            (+ (+ (/ t_2 x) (/ (* l_m l_m) x)) (- (/ t_3 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 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 1.15e-157) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 2.95e+69) {
		tmp = (t_m * sqrt(2.0)) / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / 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) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = t_2 + (l_m * l_m)
    if (t_m <= 1.15d-157) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 2.95d+69) then
        tmp = (t_m * sqrt(2.0d0)) / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / 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 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 1.15e-157) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 2.95e+69) {
		tmp = (t_m * Math.sqrt(2.0)) / Math.sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / 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 = t_2 + (l_m * l_m)
	tmp = 0
	if t_m <= 1.15e-157:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 2.95e+69:
		tmp = (t_m * math.sqrt(2.0)) / math.sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / 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(t_2 + Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 1.15e-157)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 2.95e+69)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(t_2 / x) + Float64(Float64(l_m * l_m) / x)) + Float64(Float64(t_3 / 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 = t_2 + (l_m * l_m);
	tmp = 0.0;
	if (t_m <= 1.15e-157)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 2.95e+69)
		tmp = (t_m * sqrt(2.0)) / sqrt((t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / 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[(t$95$2 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.15e-157], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 2.95e+69], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / x), $MachinePrecision] - N[(t$95$3 * -2.0), $MachinePrecision]), $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 < 1.14999999999999994e-157

   1. Initial program 31.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 32.0%

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

   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 3.8%

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

   7. Simplified 3.8%

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

   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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 18.6%

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

   10. Simplified 18.6%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 21.0%

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

   13. Simplified 21.0%

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

   if 1.14999999999999994e-157 < t < 2.95000000000000002e69

   1. Initial program 63.2%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   5. Simplified 80.9%

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

   if 2.95000000000000002e69 < t

   1. Initial program 37.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 96.5%

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

   5. Simplified 96.5%

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

   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 96.5%

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

   8. Simplified 96.5%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-157}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \left(\frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x} - \left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right) \cdot -2\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.5% accurate, 0.3× speedup

Math

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

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)) (* l_m l_m)))
        (t_3
         (/
          (* t_m (sqrt 2.0))
          (sqrt (- (* t_2 (/ (+ x 1.0) (+ x -1.0))) (* l_m l_m))))))
   (*
    t_s
    (if (<= t_3 2.0)
      (sqrt (/ (+ x -1.0) (+ x 1.0)))
      (if (<= t_3 INFINITY)
        (*
         (sqrt 2.0)
         (*
          t_m
          (pow
           (+
            (* 2.0 (* t_m (+ t_m (/ t_m x))))
            (* (/ 1.0 x) (+ (* l_m l_m) t_2)))
           -0.5)))
        (* t_m (* (sqrt (/ 2.0 l_m)) (sqrt (/ x (* l_m 2.0))))))))))

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)) + (l_m * l_m);
	double t_3 = (t_m * sqrt(2.0)) / sqrt(((t_2 * ((x + 1.0) / (x + -1.0))) - (l_m * l_m)));
	double tmp;
	if (t_3 <= 2.0) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(2.0) * (t_m * pow(((2.0 * (t_m * (t_m + (t_m / x)))) + ((1.0 / x) * ((l_m * l_m) + t_2))), -0.5));
	} else {
		tmp = t_m * (sqrt((2.0 / l_m)) * sqrt((x / (l_m * 2.0))));
	}
	return t_s * tmp;
}

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)) + (l_m * l_m);
	double t_3 = (t_m * Math.sqrt(2.0)) / Math.sqrt(((t_2 * ((x + 1.0) / (x + -1.0))) - (l_m * l_m)));
	double tmp;
	if (t_3 <= 2.0) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(2.0) * (t_m * Math.pow(((2.0 * (t_m * (t_m + (t_m / x)))) + ((1.0 / x) * ((l_m * l_m) + t_2))), -0.5));
	} else {
		tmp = t_m * (Math.sqrt((2.0 / l_m)) * Math.sqrt((x / (l_m * 2.0))));
	}
	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)) + (l_m * l_m)
	t_3 = (t_m * math.sqrt(2.0)) / math.sqrt(((t_2 * ((x + 1.0) / (x + -1.0))) - (l_m * l_m)))
	tmp = 0
	if t_3 <= 2.0:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t_3 <= math.inf:
		tmp = math.sqrt(2.0) * (t_m * math.pow(((2.0 * (t_m * (t_m + (t_m / x)))) + ((1.0 / x) * ((l_m * l_m) + t_2))), -0.5))
	else:
		tmp = t_m * (math.sqrt((2.0 / l_m)) * math.sqrt((x / (l_m * 2.0))))
	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(Float64(2.0 * Float64(t_m * t_m)) + Float64(l_m * l_m))
	t_3 = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(Float64(t_2 * Float64(Float64(x + 1.0) / Float64(x + -1.0))) - Float64(l_m * l_m))))
	tmp = 0.0
	if (t_3 <= 2.0)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(2.0) * Float64(t_m * (Float64(Float64(2.0 * Float64(t_m * Float64(t_m + Float64(t_m / x)))) + Float64(Float64(1.0 / x) * Float64(Float64(l_m * l_m) + t_2))) ^ -0.5)));
	else
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 / l_m)) * sqrt(Float64(x / Float64(l_m * 2.0)))));
	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)) + (l_m * l_m);
	t_3 = (t_m * sqrt(2.0)) / sqrt(((t_2 * ((x + 1.0) / (x + -1.0))) - (l_m * l_m)));
	tmp = 0.0;
	if (t_3 <= 2.0)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t_3 <= Inf)
		tmp = sqrt(2.0) * (t_m * (((2.0 * (t_m * (t_m + (t_m / x)))) + ((1.0 / x) * ((l_m * l_m) + t_2))) ^ -0.5));
	else
		tmp = t_m * (sqrt((2.0 / l_m)) * sqrt((x / (l_m * 2.0))));
	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[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(t$95$2 * N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 2.0], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[(N[(2.0 * N[(t$95$m * N[(t$95$m + N[(t$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[N[(2.0 / l$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(x / N[(l$95$m * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < 2

   1. Initial program 53.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 37.9%

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

   5. Simplified 37.9%

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

   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 37.9%

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

   8. Simplified 37.9%

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

   if 2 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 2.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

   5. Simplified 55.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

   7. Applied egg-rr 55.1%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l 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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 23.7%

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

   10. Simplified 23.7%

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

       1. *-commutative N/A

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

       2. distribute-lft-out N/A

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

       3. times-frac N/A

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

       4. sqrt-prod N/A

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

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

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

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

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

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

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

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

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

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

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

       10. count-2 N/A

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

       11. *-lowering-*.f64 35.6%

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

   12. Applied egg-rr 35.6%

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

4. Final simplification 39.3%

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

Alternative 3: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-157}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 2.85 \cdot 10^{+69}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{\frac{2 \cdot \left(t\_m \cdot t\_m\right) + l\_m \cdot l\_m}{x} + \left(\frac{l\_m \cdot l\_m}{x} + 2 \cdot \left(t\_m \cdot t\_m + \frac{t\_m \cdot t\_m}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 (<= t_m 1.15e-157)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 2.85e+69)
      (/
       (* t_m (sqrt 2.0))
       (sqrt
        (+
         (/ (+ (* 2.0 (* t_m t_m)) (* l_m l_m)) x)
         (+ (/ (* l_m l_m) x) (* 2.0 (+ (* t_m t_m) (/ (* t_m t_m) 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 tmp;
	if (t_m <= 1.15e-157) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 2.85e+69) {
		tmp = (t_m * sqrt(2.0)) / sqrt(((((2.0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / 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) :: tmp
    if (t_m <= 1.15d-157) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 2.85d+69) then
        tmp = (t_m * sqrt(2.0d0)) / sqrt(((((2.0d0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0d0 * ((t_m * t_m) + ((t_m * t_m) / 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 tmp;
	if (t_m <= 1.15e-157) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 2.85e+69) {
		tmp = (t_m * Math.sqrt(2.0)) / Math.sqrt(((((2.0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / 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):
	tmp = 0
	if t_m <= 1.15e-157:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 2.85e+69:
		tmp = (t_m * math.sqrt(2.0)) / math.sqrt(((((2.0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / 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)
	tmp = 0.0
	if (t_m <= 1.15e-157)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 2.85e+69)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(l_m * l_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)))))));
	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)
	tmp = 0.0;
	if (t_m <= 1.15e-157)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 2.85e+69)
		tmp = (t_m * sqrt(2.0)) / sqrt(((((2.0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.15e-157], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 2.85e+69], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * l$95$m), $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], 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 < 1.14999999999999994e-157

   1. Initial program 31.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 32.0%

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

   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 3.8%

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

   7. Simplified 3.8%

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

   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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 18.6%

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

   10. Simplified 18.6%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 21.0%

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

   13. Simplified 21.0%

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

   if 1.14999999999999994e-157 < t < 2.85e69

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

   5. Simplified 80.9%

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

   if 2.85e69 < t

   1. Initial program 37.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 96.5%

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

   5. Simplified 96.5%

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

   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 96.5%

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

   8. Simplified 96.5%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-157}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := t\_2 + l\_m \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{t\_2 + \frac{\left(\frac{t\_2}{x} + \frac{l\_m \cdot l\_m}{x}\right) + \left(\frac{t\_3}{x} + \left(t\_2 + \left(l\_m \cdot l\_m + t\_3\right)\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (+ t_2 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 7.4e-155)
      (/ (* t_m (sqrt x)) l_m)
      (if (<= t_m 3.3e+69)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            t_2
            (/
             (+
              (+ (/ t_2 x) (/ (* l_m l_m) x))
              (+ (/ t_3 x) (+ t_2 (+ (* l_m l_m) t_3))))
             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 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 7.4e-155) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 3.3e+69) {
		tmp = t_m * sqrt((2.0 / (t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / x) + (t_2 + ((l_m * l_m) + t_3)))) / 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) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = t_2 + (l_m * l_m)
    if (t_m <= 7.4d-155) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 3.3d+69) then
        tmp = t_m * sqrt((2.0d0 / (t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / x) + (t_2 + ((l_m * l_m) + t_3)))) / 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 = t_2 + (l_m * l_m);
	double tmp;
	if (t_m <= 7.4e-155) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 3.3e+69) {
		tmp = t_m * Math.sqrt((2.0 / (t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / x) + (t_2 + ((l_m * l_m) + t_3)))) / 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 = t_2 + (l_m * l_m)
	tmp = 0
	if t_m <= 7.4e-155:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 3.3e+69:
		tmp = t_m * math.sqrt((2.0 / (t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / x) + (t_2 + ((l_m * l_m) + t_3)))) / 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(t_2 + Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 7.4e-155)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 3.3e+69)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(t_2 + Float64(Float64(Float64(Float64(t_2 / x) + Float64(Float64(l_m * l_m) / x)) + Float64(Float64(t_3 / x) + Float64(t_2 + Float64(Float64(l_m * l_m) + t_3)))) / 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 = t_2 + (l_m * l_m);
	tmp = 0.0;
	if (t_m <= 7.4e-155)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 3.3e+69)
		tmp = t_m * sqrt((2.0 / (t_2 + ((((t_2 / x) + ((l_m * l_m) / x)) + ((t_3 / x) + (t_2 + ((l_m * l_m) + t_3)))) / 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[(t$95$2 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.4e-155], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+69], N[(t$95$m * N[Sqrt[N[(2.0 / N[(t$95$2 + N[(N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / x), $MachinePrecision] + N[(t$95$2 + N[(N[(l$95$m * l$95$m), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $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 < 7.4000000000000001e-155

   1. Initial program 31.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 32.0%

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

   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 3.8%

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

   7. Simplified 3.8%

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

   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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 18.6%

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

   10. Simplified 18.6%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 21.0%

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

   13. Simplified 21.0%

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

   if 7.4000000000000001e-155 < t < 3.2999999999999999e69

   1. Initial program 63.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 63.4%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

   7. Simplified 81.1%

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

   if 3.2999999999999999e69 < t

   1. Initial program 37.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 96.5%

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

   5. Simplified 96.5%

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

   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 96.5%

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

   8. Simplified 96.5%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+69}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot t\right) + \frac{\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \left(\frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \ell + \left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)\right)\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 2.85 \cdot 10^{+69}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\frac{2 \cdot \left(t\_m \cdot t\_m\right) + l\_m \cdot l\_m}{x} + \left(\frac{l\_m \cdot l\_m}{x} + 2 \cdot \left(t\_m \cdot t\_m + \frac{t\_m \cdot t\_m}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 (<= t_m 7.4e-155)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 2.85e+69)
      (*
       t_m
       (sqrt
        (/
         2.0
         (+
          (/ (+ (* 2.0 (* t_m t_m)) (* l_m l_m)) x)
          (+ (/ (* l_m l_m) x) (* 2.0 (+ (* t_m t_m) (/ (* t_m t_m) 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 tmp;
	if (t_m <= 7.4e-155) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 2.85e+69) {
		tmp = t_m * sqrt((2.0 / ((((2.0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / 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) :: tmp
    if (t_m <= 7.4d-155) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 2.85d+69) then
        tmp = t_m * sqrt((2.0d0 / ((((2.0d0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0d0 * ((t_m * t_m) + ((t_m * t_m) / 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 tmp;
	if (t_m <= 7.4e-155) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 2.85e+69) {
		tmp = t_m * Math.sqrt((2.0 / ((((2.0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / 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):
	tmp = 0
	if t_m <= 7.4e-155:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 2.85e+69:
		tmp = t_m * math.sqrt((2.0 / ((((2.0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / 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)
	tmp = 0.0
	if (t_m <= 7.4e-155)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 2.85e+69)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(l_m * l_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))))))));
	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)
	tmp = 0.0;
	if (t_m <= 7.4e-155)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 2.85e+69)
		tmp = t_m * sqrt((2.0 / ((((2.0 * (t_m * t_m)) + (l_m * l_m)) / x) + (((l_m * l_m) / x) + (2.0 * ((t_m * t_m) + ((t_m * t_m) / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 7.4e-155], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 2.85e+69], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * l$95$m), $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], 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 < 7.4000000000000001e-155

   1. Initial program 31.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 32.0%

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

   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 3.8%

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

   7. Simplified 3.8%

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

   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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 18.6%

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

   10. Simplified 18.6%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 21.0%

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

   13. Simplified 21.0%

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

   if 7.4000000000000001e-155 < t < 2.85e69

   1. Initial program 63.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 63.4%

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

   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

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

   7. Simplified 81.1%

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

   if 2.85e69 < t

   1. Initial program 37.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 96.5%

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

   5. Simplified 96.5%

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

   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 96.5%

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

   8. Simplified 96.5%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+69}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-157}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 130000000000:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\frac{2 \cdot \left(l\_m \cdot l\_m\right) + \left(t\_m \cdot t\_m\right) \cdot \left(4 + x \cdot 2\right)}{x}}{2}}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 (<= t_m 1.25e-157)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 130000000000.0)
      (/
       1.0
       (/
        (sqrt
         (/
          (/ (+ (* 2.0 (* l_m l_m)) (* (* t_m t_m) (+ 4.0 (* x 2.0)))) x)
          2.0))
        t_m))
      (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 tmp;
	if (t_m <= 1.25e-157) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 130000000000.0) {
		tmp = 1.0 / (sqrt(((((2.0 * (l_m * l_m)) + ((t_m * t_m) * (4.0 + (x * 2.0)))) / x) / 2.0)) / t_m);
	} 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) :: tmp
    if (t_m <= 1.25d-157) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 130000000000.0d0) then
        tmp = 1.0d0 / (sqrt(((((2.0d0 * (l_m * l_m)) + ((t_m * t_m) * (4.0d0 + (x * 2.0d0)))) / x) / 2.0d0)) / t_m)
    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 tmp;
	if (t_m <= 1.25e-157) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 130000000000.0) {
		tmp = 1.0 / (Math.sqrt(((((2.0 * (l_m * l_m)) + ((t_m * t_m) * (4.0 + (x * 2.0)))) / x) / 2.0)) / t_m);
	} 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):
	tmp = 0
	if t_m <= 1.25e-157:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 130000000000.0:
		tmp = 1.0 / (math.sqrt(((((2.0 * (l_m * l_m)) + ((t_m * t_m) * (4.0 + (x * 2.0)))) / x) / 2.0)) / t_m)
	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)
	tmp = 0.0
	if (t_m <= 1.25e-157)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 130000000000.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) + Float64(Float64(t_m * t_m) * Float64(4.0 + Float64(x * 2.0)))) / x) / 2.0)) / t_m));
	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)
	tmp = 0.0;
	if (t_m <= 1.25e-157)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 130000000000.0)
		tmp = 1.0 / (sqrt(((((2.0 * (l_m * l_m)) + ((t_m * t_m) * (4.0 + (x * 2.0)))) / x) / 2.0)) / t_m);
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.25e-157], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 130000000000.0], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(4.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] / t$95$m), $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 < 1.25000000000000005e-157

   1. Initial program 31.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 32.0%

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

   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 3.8%

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

   7. Simplified 3.8%

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

   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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 18.6%

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

   10. Simplified 18.6%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 21.0%

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

   13. Simplified 21.0%

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

   if 1.25000000000000005e-157 < t < 1.3e11

   1. Initial program 59.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

   5. Simplified 79.1%

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

   6. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left({t}^{2} \cdot x\right) + \left(2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}\right)\right), x\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}\right) + 2 \cdot \left({t}^{2} \cdot x\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(4 \cdot {t}^{2}\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(4 \cdot {t}^{2}\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(4 \cdot {t}^{2}\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(4 \cdot {t}^{2}\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \left({t}^{2}\right)\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \left(t \cdot t\right)\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \left(x \cdot {t}^{2}\right)\right)\right), x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \left({t}^{2}\right)\right)\right)\right), x\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \left(t \cdot t\right)\right)\right)\right), x\right)\right)\right) \]

      15. *-lowering-*.f64 79.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), x\right)\right)\right) \]

   8. Simplified 79.2%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}}{\sqrt{2} \cdot t}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}}{\sqrt{2} \cdot t}\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\sqrt{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}}{\sqrt{2}}}{\color{blue}{t}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\sqrt{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}}{\sqrt{2}}\right), \color{blue}{t}\right)\right) \]

   10. Applied egg-rr 79.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + \left(t \cdot t\right) \cdot \left(4 + x \cdot 2\right)}{x}}{2}}}{t}}} \]

   if 1.3e11 < t

   1. Initial program 42.0%

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

   3. 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 94.0%

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

   5. Simplified 94.0%

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

   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 94.0%

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

   8. Simplified 94.0%

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

Alternative 7: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 90000000000:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\frac{2 \cdot \left(l\_m \cdot l\_m\right) + \left(t\_m \cdot t\_m\right) \cdot \left(4 + x \cdot 2\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 (<= t_m 7.4e-155)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 90000000000.0)
      (*
       t_m
       (sqrt
        (/
         2.0
         (/ (+ (* 2.0 (* l_m l_m)) (* (* t_m t_m) (+ 4.0 (* x 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 tmp;
	if (t_m <= 7.4e-155) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 90000000000.0) {
		tmp = t_m * sqrt((2.0 / (((2.0 * (l_m * l_m)) + ((t_m * t_m) * (4.0 + (x * 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) :: tmp
    if (t_m <= 7.4d-155) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 90000000000.0d0) then
        tmp = t_m * sqrt((2.0d0 / (((2.0d0 * (l_m * l_m)) + ((t_m * t_m) * (4.0d0 + (x * 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 tmp;
	if (t_m <= 7.4e-155) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 90000000000.0) {
		tmp = t_m * Math.sqrt((2.0 / (((2.0 * (l_m * l_m)) + ((t_m * t_m) * (4.0 + (x * 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):
	tmp = 0
	if t_m <= 7.4e-155:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 90000000000.0:
		tmp = t_m * math.sqrt((2.0 / (((2.0 * (l_m * l_m)) + ((t_m * t_m) * (4.0 + (x * 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)
	tmp = 0.0
	if (t_m <= 7.4e-155)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 90000000000.0)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(Float64(2.0 * Float64(l_m * l_m)) + Float64(Float64(t_m * t_m) * Float64(4.0 + Float64(x * 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)
	tmp = 0.0;
	if (t_m <= 7.4e-155)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 90000000000.0)
		tmp = t_m * sqrt((2.0 / (((2.0 * (l_m * l_m)) + ((t_m * t_m) * (4.0 + (x * 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_] := N[(t$95$s * If[LessEqual[t$95$m, 7.4e-155], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 90000000000.0], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(4.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $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 < 7.4000000000000001e-155

   1. Initial program 31.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 32.0%

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

   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 3.8%

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

   7. Simplified 3.8%

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

   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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 18.6%

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

   10. Simplified 18.6%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 21.0%

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

   13. Simplified 21.0%

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

   if 7.4000000000000001e-155 < t < 9e10

   1. Initial program 59.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

   5. Simplified 79.1%

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

   6. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left({t}^{2} \cdot x\right) + \left(2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}\right)\right), x\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}\right) + 2 \cdot \left({t}^{2} \cdot x\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(4 \cdot {t}^{2}\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(4 \cdot {t}^{2}\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(4 \cdot {t}^{2}\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(4 \cdot {t}^{2}\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \left({t}^{2}\right)\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \left(t \cdot t\right)\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(2 \cdot \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2} \cdot x\right)\right)\right), x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \left(x \cdot {t}^{2}\right)\right)\right), x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \left({t}^{2}\right)\right)\right)\right), x\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \left(t \cdot t\right)\right)\right)\right), x\right)\right)\right) \]

      15. *-lowering-*.f64 79.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), x\right)\right)\right) \]

   8. Simplified 79.2%

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

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}}} \]

      2. associate-/l* N/A

         \[\leadsto t \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}}\right)}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{*.f64}\left(t, \left(\sqrt{\frac{2}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)}{x}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(2 \cdot \left(\ell \cdot \ell\right) + 4 \cdot \left(t \cdot t\right)\right) + 2 \cdot \left(x \cdot \left(t \cdot t\right)\right)\right), x\right)\right)\right)\right) \]

   10. Applied egg-rr 79.2%

       \[\leadsto \color{blue}{t \cdot \sqrt{\frac{2}{\frac{2 \cdot \left(\ell \cdot \ell\right) + \left(t \cdot t\right) \cdot \left(4 + x \cdot 2\right)}{x}}}} \]

   if 9e10 < t

   1. Initial program 42.0%

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

   3. 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 94.0%

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

   5. Simplified 94.0%

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

   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 94.0%

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

   8. Simplified 94.0%

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

Alternative 8: 79.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

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 l_m) 4e+291)
    (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 * l_m) <= 4e+291) {
		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 * l_m) <= 4d+291) 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 * l_m) <= 4e+291) {
		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 * l_m) <= 4e+291:
		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 (Float64(l_m * l_m) <= 4e+291)
		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 * l_m) <= 4e+291)
		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[N[(l$95$m * l$95$m), $MachinePrecision], 4e+291], 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 (*.f64 l l) < 3.9999999999999998e291

   1. Initial program 46.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 39.0%

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

   5. Simplified 39.0%

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

   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 39.0%

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

   8. Simplified 39.0%

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

   if 3.9999999999999998e291 < (*.f64 l 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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 23.7%

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

   10. Simplified 23.7%

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

   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 48.7%

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

   13. Simplified 48.7%

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

Alternative 9: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 7.1 \cdot 10^{+155}:\\ \;\;\;\;1 + \left(\frac{-1 + \frac{0.5}{x}}{x} - \frac{\frac{0.5}{x \cdot x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

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 7.1e+155)
    (+ 1.0 (- (/ (+ -1.0 (/ 0.5 x)) x) (/ (/ 0.5 (* x x)) 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 <= 7.1e+155) {
		tmp = 1.0 + (((-1.0 + (0.5 / x)) / x) - ((0.5 / (x * x)) / 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 <= 7.1d+155) then
        tmp = 1.0d0 + ((((-1.0d0) + (0.5d0 / x)) / x) - ((0.5d0 / (x * x)) / 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 <= 7.1e+155) {
		tmp = 1.0 + (((-1.0 + (0.5 / x)) / x) - ((0.5 / (x * x)) / 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 <= 7.1e+155:
		tmp = 1.0 + (((-1.0 + (0.5 / x)) / x) - ((0.5 / (x * x)) / 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 <= 7.1e+155)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 + Float64(0.5 / x)) / x) - Float64(Float64(0.5 / Float64(x * x)) / 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 <= 7.1e+155)
		tmp = 1.0 + (((-1.0 + (0.5 / x)) / x) - ((0.5 / (x * x)) / 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, 7.1e+155], N[(1.0 + N[(N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / 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 < 7.09999999999999992e155

   1. Initial program 42.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 37.1%

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

   5. Simplified 37.1%

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

   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 37.1%

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

   8. Simplified 37.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

      6. +-lowering-+.f64 37.0%

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

   10. Applied egg-rr 37.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ N/A

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

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

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{2}} - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right)}\right) \]

       3. associate--r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2}}{{x}^{2}} - \frac{1}{x}\right) - \color{blue}{\frac{1}{2} \cdot \frac{1}{{x}^{3}}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2}}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot 1}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \left(2 + -1\right)}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \left(2 + \frac{1}{-1}\right)}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

       8. rem-square-sqrt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \left(2 + \frac{1}{\sqrt{-1} \cdot \sqrt{-1}}\right)}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}\right)}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

       10. associate-/l/ N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{\frac{1}{2} \cdot \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}\right)}{x}}{x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}}{x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

       12. div-sub N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x} - 1}{x} - \color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\left(\frac{\frac{1}{2} \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x} - 1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

   13. Simplified 37.1%

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

   if 7.09999999999999992e155 < 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. *-lft-identity N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 17.1%

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

   10. Simplified 17.1%

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

   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 60.2%

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

   13. Simplified 60.2%

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

Alternative 10: 76.4% accurate, 13.2× speedup

Math

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

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 (+ 1.0 (- (/ (+ -1.0 (/ 0.5 x)) x) (/ (/ 0.5 (* x x)) 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 + (0.5 / x)) / x) - ((0.5 / (x * x)) / 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) + (0.5d0 / x)) / x) - ((0.5d0 / (x * x)) / 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 + (0.5 / x)) / x) - ((0.5 / (x * x)) / 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 + (0.5 / x)) / x) - ((0.5 / (x * x)) / 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(Float64(Float64(-1.0 + Float64(0.5 / x)) / x) - Float64(Float64(0.5 / Float64(x * x)) / 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 + (0.5 / x)) / x) - ((0.5 / (x * x)) / 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[(N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 35.5%

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

5. Simplified 35.5%

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

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 35.5%

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

8. Simplified 35.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

   6. +-lowering-+.f64 35.4%

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

10. Applied egg-rr 35.4%

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

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)} \]
12. Step-by-step derivation

    1. associate--l+ N/A

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

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

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{2}} - \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right)}\right) \]

    3. associate--r+ N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2}}{{x}^{2}} - \frac{1}{x}\right) - \color{blue}{\frac{1}{2} \cdot \frac{1}{{x}^{3}}}\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2}}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot 1}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \left(2 + -1\right)}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \left(2 + \frac{1}{-1}\right)}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

    8. rem-square-sqrt N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \left(2 + \frac{1}{\sqrt{-1} \cdot \sqrt{-1}}\right)}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}\right)}{x \cdot x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

    10. associate-/l/ N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{\frac{1}{2} \cdot \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}\right)}{x}}{x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

    11. associate-*r/ N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{2} \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}}{x} - \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right) \]

    12. div-sub N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x} - 1}{x} - \color{blue}{\frac{1}{2}} \cdot \frac{1}{{x}^{3}}\right)\right) \]

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

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{\_.f64}\left(\left(\frac{\frac{1}{2} \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x} - 1}{x}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

13. Simplified 35.5%

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

Alternative 11: 76.3% accurate, 25.0× speedup

Math

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

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 (/ 0.5 x)) 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 + (0.5 / x)) / 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) + (0.5d0 / x)) / 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 + (0.5 / x)) / 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 + (0.5 / x)) / 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(Float64(-1.0 + Float64(0.5 / x)) / 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 + (0.5 / x)) / 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[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 35.5%

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

5. Simplified 35.5%

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

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 35.5%

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

8. Simplified 35.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

   6. +-lowering-+.f64 35.4%

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

10. Applied egg-rr 35.4%

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

11. Taylor expanded in x around inf

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

    1. associate--l+ N/A

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

    2. unpow2 N/A

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

    3. metadata-eval N/A

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. rem-square-sqrt N/A

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

    7. unpow2 N/A

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

    8. associate-/l/ N/A

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

    9. associate-*r/ N/A

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

    10. div-sub N/A

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

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

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

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

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

13. Simplified 35.5%

    \[\leadsto \color{blue}{1 + \frac{-1 + \frac{0.5}{x}}{x}} \]
14. Add Preprocessing

Alternative 12: 76.1% 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 37.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 35.5%

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

5. Simplified 35.5%

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

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 35.5%

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

8. Simplified 35.5%

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

9. Final simplification 35.5%

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

Alternative 13: 75.4% 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 37.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 35.5%

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

5. Simplified 35.5%

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

6. Taylor expanded in x around inf

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

8. Simplified 35.4%

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

Reproduce

herbie shell --seed 2024154 
(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)))))