Toniolo and Linder, Equation (7)

Percentage Accurate: 34.0% → 85.3%

Time: 18.9s

Alternatives: 12

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.0% 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: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := \ell \cdot \ell + t\_2\\ t_4 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-156}:\\ \;\;\;\;\frac{t\_4}{t\_4 + \frac{0.5 \cdot \left(2 \cdot t\_3\right)}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{t\_2 + \frac{\left(t\_2 + \left(\ell \cdot \ell + t\_3\right)\right) + \frac{\left(t\_3 + t\_3\right) + \left(\left(\frac{t\_2}{x} + \frac{\ell \cdot \ell}{x}\right) + \frac{t\_3}{x}\right)}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (* t_m t_m)))
        (t_3 (+ (* l l) t_2))
        (t_4 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 7e-156)
      (/ t_4 (+ t_4 (/ (* 0.5 (* 2.0 t_3)) (* t_m (* (sqrt 2.0) x)))))
      (if (<= t_m 3.3e+51)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            t_2
            (/
             (+
              (+ t_2 (+ (* l l) t_3))
              (/ (+ (+ t_3 t_3) (+ (+ (/ t_2 x) (/ (* l l) x)) (/ t_3 x))) x))
             x)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l * l) + t_2;
	double t_4 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 7e-156) {
		tmp = t_4 / (t_4 + ((0.5 * (2.0 * t_3)) / (t_m * (sqrt(2.0) * x))));
	} else if (t_m <= 3.3e+51) {
		tmp = t_m * sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    t_3 = (l * l) + t_2
    t_4 = t_m * sqrt(2.0d0)
    if (t_m <= 7d-156) then
        tmp = t_4 / (t_4 + ((0.5d0 * (2.0d0 * t_3)) / (t_m * (sqrt(2.0d0) * x))))
    else if (t_m <= 3.3d+51) then
        tmp = t_m * sqrt((2.0d0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l * l) + t_2;
	double t_4 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 7e-156) {
		tmp = t_4 / (t_4 + ((0.5 * (2.0 * t_3)) / (t_m * (Math.sqrt(2.0) * x))));
	} else if (t_m <= 3.3e+51) {
		tmp = t_m * Math.sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * (t_m * t_m)
	t_3 = (l * l) + t_2
	t_4 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 7e-156:
		tmp = t_4 / (t_4 + ((0.5 * (2.0 * t_3)) / (t_m * (math.sqrt(2.0) * x))))
	elif t_m <= 3.3e+51:
		tmp = t_m * math.sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * Float64(t_m * t_m))
	t_3 = Float64(Float64(l * l) + t_2)
	t_4 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 7e-156)
		tmp = Float64(t_4 / Float64(t_4 + Float64(Float64(0.5 * Float64(2.0 * t_3)) / Float64(t_m * Float64(sqrt(2.0) * x)))));
	elseif (t_m <= 3.3e+51)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(t_2 + Float64(Float64(Float64(t_2 + Float64(Float64(l * l) + t_3)) + Float64(Float64(Float64(t_3 + t_3) + Float64(Float64(Float64(t_2 / x) + Float64(Float64(l * l) / x)) + Float64(t_3 / x))) / x)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m * t_m);
	t_3 = (l * l) + t_2;
	t_4 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 7e-156)
		tmp = t_4 / (t_4 + ((0.5 * (2.0 * t_3)) / (t_m * (sqrt(2.0) * x))));
	elseif (t_m <= 3.3e+51)
		tmp = t_m * sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * l), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7e-156], N[(t$95$4 / N[(t$95$4 + N[(N[(0.5 * N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+51], N[(t$95$m * N[Sqrt[N[(2.0 / N[(t$95$2 + N[(N[(N[(t$95$2 + N[(N[(l * l), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] + N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 6.9999999999999999e-156

   1. Initial program 28.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 16.4%

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

   if 6.9999999999999999e-156 < t < 3.2999999999999997e51

   1. Initial program 46.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing
   3. 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 46.9%

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

   5. Taylor expanded in x around -inf

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

   7. Simplified 85.9%

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

   if 3.2999999999999997e51 < t

   1. Initial program 30.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 94.5%

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

   5. Simplified 94.5%

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

   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 94.5%

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

   8. Simplified 94.5%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-156}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{0.5 \cdot \left(2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot t\right) + \frac{\left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) + \frac{\left(\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right) + \left(\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := \ell \cdot \ell + t\_2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-171}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x} + \frac{2}{x \cdot x}}}\right)\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{+49}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{t\_2 + \frac{\left(t\_2 + \left(\ell \cdot \ell + t\_3\right)\right) + \frac{\left(t\_3 + t\_3\right) + \left(\left(\frac{t\_2}{x} + \frac{\ell \cdot \ell}{x}\right) + \frac{t\_3}{x}\right)}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (* t_m t_m))) (t_3 (+ (* l l) t_2)))
   (*
    t_s
    (if (<= t_m 9.8e-171)
      (* t_m (* (/ (sqrt 2.0) l) (sqrt (/ 1.0 (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
      (if (<= t_m 8.5e+49)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            t_2
            (/
             (+
              (+ t_2 (+ (* l l) t_3))
              (/ (+ (+ t_3 t_3) (+ (+ (/ t_2 x) (/ (* l l) x)) (/ t_3 x))) x))
             x)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l * l) + t_2;
	double tmp;
	if (t_m <= 9.8e-171) {
		tmp = t_m * ((sqrt(2.0) / l) * sqrt((1.0 / ((2.0 / x) + (2.0 / (x * x))))));
	} else if (t_m <= 8.5e+49) {
		tmp = t_m * sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    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 = (l * l) + t_2
    if (t_m <= 9.8d-171) then
        tmp = t_m * ((sqrt(2.0d0) / l) * sqrt((1.0d0 / ((2.0d0 / x) + (2.0d0 / (x * x))))))
    else if (t_m <= 8.5d+49) then
        tmp = t_m * sqrt((2.0d0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l * l) + t_2;
	double tmp;
	if (t_m <= 9.8e-171) {
		tmp = t_m * ((Math.sqrt(2.0) / l) * Math.sqrt((1.0 / ((2.0 / x) + (2.0 / (x * x))))));
	} else if (t_m <= 8.5e+49) {
		tmp = t_m * Math.sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * (t_m * t_m)
	t_3 = (l * l) + t_2
	tmp = 0
	if t_m <= 9.8e-171:
		tmp = t_m * ((math.sqrt(2.0) / l) * math.sqrt((1.0 / ((2.0 / x) + (2.0 / (x * x))))))
	elif t_m <= 8.5e+49:
		tmp = t_m * math.sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * Float64(t_m * t_m))
	t_3 = Float64(Float64(l * l) + t_2)
	tmp = 0.0
	if (t_m <= 9.8e-171)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l) * sqrt(Float64(1.0 / Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t_m <= 8.5e+49)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(t_2 + Float64(Float64(Float64(t_2 + Float64(Float64(l * l) + t_3)) + Float64(Float64(Float64(t_3 + t_3) + Float64(Float64(Float64(t_2 / x) + Float64(Float64(l * l) / x)) + Float64(t_3 / x))) / x)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m * t_m);
	t_3 = (l * l) + t_2;
	tmp = 0.0;
	if (t_m <= 9.8e-171)
		tmp = t_m * ((sqrt(2.0) / l) * sqrt((1.0 / ((2.0 / x) + (2.0 / (x * x))))));
	elseif (t_m <= 8.5e+49)
		tmp = t_m * sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * l), $MachinePrecision] + t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.8e-171], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+49], N[(t$95$m * N[Sqrt[N[(2.0 / N[(t$95$2 + N[(N[(N[(t$95$2 + N[(N[(l * l), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] + N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 9.79999999999999965e-171

   1. Initial program 29.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 29.1%

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

   5. Taylor expanded in x around inf

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

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

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

   7. Simplified 46.4%

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

   8. Taylor expanded in l around inf

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

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

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

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

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-out N/A

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

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

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

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

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

   10. Simplified 18.8%

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

   if 9.79999999999999965e-171 < t < 8.4999999999999996e49

   1. Initial program 44.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 44.6%

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

   5. Taylor expanded in x around -inf

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

   7. Simplified 81.4%

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

   if 8.4999999999999996e49 < t

   1. Initial program 30.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 94.5%

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

   5. Simplified 94.5%

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

   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 94.5%

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

   8. Simplified 94.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{-171}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\frac{2}{x} + \frac{2}{x \cdot x}}}\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+49}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot t\right) + \frac{\left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) + \frac{\left(\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right) + \left(\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t_3 := \ell \cdot \ell + t\_2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+49}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{t\_2 + \frac{\left(t\_2 + \left(\ell \cdot \ell + t\_3\right)\right) + \frac{\left(t\_3 + t\_3\right) + \left(\left(\frac{t\_2}{x} + \frac{\ell \cdot \ell}{x}\right) + \frac{t\_3}{x}\right)}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (* t_m t_m))) (t_3 (+ (* l l) t_2)))
   (*
    t_s
    (if (<= t_m 1.7e-171)
      (/ (* t_m (sqrt x)) l)
      (if (<= t_m 4e+49)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            t_2
            (/
             (+
              (+ t_2 (+ (* l l) t_3))
              (/ (+ (+ t_3 t_3) (+ (+ (/ t_2 x) (/ (* l l) x)) (/ t_3 x))) x))
             x)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l * l) + t_2;
	double tmp;
	if (t_m <= 1.7e-171) {
		tmp = (t_m * sqrt(x)) / l;
	} else if (t_m <= 4e+49) {
		tmp = t_m * sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    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 = (l * l) + t_2
    if (t_m <= 1.7d-171) then
        tmp = (t_m * sqrt(x)) / l
    else if (t_m <= 4d+49) then
        tmp = t_m * sqrt((2.0d0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double t_3 = (l * l) + t_2;
	double tmp;
	if (t_m <= 1.7e-171) {
		tmp = (t_m * Math.sqrt(x)) / l;
	} else if (t_m <= 4e+49) {
		tmp = t_m * Math.sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * (t_m * t_m)
	t_3 = (l * l) + t_2
	tmp = 0
	if t_m <= 1.7e-171:
		tmp = (t_m * math.sqrt(x)) / l
	elif t_m <= 4e+49:
		tmp = t_m * math.sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * Float64(t_m * t_m))
	t_3 = Float64(Float64(l * l) + t_2)
	tmp = 0.0
	if (t_m <= 1.7e-171)
		tmp = Float64(Float64(t_m * sqrt(x)) / l);
	elseif (t_m <= 4e+49)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(t_2 + Float64(Float64(Float64(t_2 + Float64(Float64(l * l) + t_3)) + Float64(Float64(Float64(t_3 + t_3) + Float64(Float64(Float64(t_2 / x) + Float64(Float64(l * l) / x)) + Float64(t_3 / x))) / x)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m * t_m);
	t_3 = (l * l) + t_2;
	tmp = 0.0;
	if (t_m <= 1.7e-171)
		tmp = (t_m * sqrt(x)) / l;
	elseif (t_m <= 4e+49)
		tmp = t_m * sqrt((2.0 / (t_2 + (((t_2 + ((l * l) + t_3)) + (((t_3 + t_3) + (((t_2 / x) + ((l * l) / x)) + (t_3 / x))) / x)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * l), $MachinePrecision] + t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.7e-171], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$m, 4e+49], N[(t$95$m * N[Sqrt[N[(2.0 / N[(t$95$2 + N[(N[(N[(t$95$2 + N[(N[(l * l), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] + N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 1.69999999999999993e-171

   1. Initial program 29.0%

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

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

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

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

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

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

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

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

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

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

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

   3. Simplified 19.8%

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

   5. Taylor expanded in l around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 17.1%

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

   10. Simplified 17.1%

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

       1. *-lft-identity N/A

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

       2. distribute-rgt-out N/A

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

       3. metadata-eval N/A

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

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

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

       5. *-lowering-*.f64 17.1%

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

   12. Applied egg-rr 17.1%

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

   13. Taylor expanded in t around 0

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   14. 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 18.5%

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

   15. Simplified 18.5%

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

   if 1.69999999999999993e-171 < t < 3.99999999999999979e49

   1. Initial program 44.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 44.6%

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

   5. Taylor expanded in x around -inf

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

   7. Simplified 81.4%

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

   if 3.99999999999999979e49 < t

   1. Initial program 30.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 94.5%

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

   5. Simplified 94.5%

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

   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 94.5%

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

   8. Simplified 94.5%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+49}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot t\right) + \frac{\left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) + \frac{\left(\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right) + \left(\left(\frac{2 \cdot \left(t \cdot t\right)}{x} + \frac{\ell \cdot \ell}{x}\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \left(t\_m \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-171}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+49}:\\ \;\;\;\;t\_m \cdot \frac{1}{\sqrt{\frac{\left(t\_2 + \left(\frac{\ell}{x} \cdot \left(\ell + \frac{\ell}{x}\right) + \frac{t\_m \cdot t\_m}{x} \cdot \left(2 + \frac{2}{x}\right)\right)\right) + \left(\ell \cdot \ell + t\_2\right) \cdot \left(\frac{1}{x} + \frac{1}{x \cdot x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 1.65e-171)
      (/ (* t_m (sqrt x)) l)
      (if (<= t_m 1.05e+49)
        (*
         t_m
         (/
          1.0
          (sqrt
           (/
            (+
             (+
              t_2
              (+
               (* (/ l x) (+ l (/ l x)))
               (* (/ (* t_m t_m) x) (+ 2.0 (/ 2.0 x)))))
             (* (+ (* l l) t_2) (+ (/ 1.0 x) (/ 1.0 (* x x)))))
            2.0))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double tmp;
	if (t_m <= 1.65e-171) {
		tmp = (t_m * sqrt(x)) / l;
	} else if (t_m <= 1.05e+49) {
		tmp = t_m * (1.0 / sqrt((((t_2 + (((l / x) * (l + (l / x))) + (((t_m * t_m) / x) * (2.0 + (2.0 / x))))) + (((l * l) + t_2) * ((1.0 / x) + (1.0 / (x * x))))) / 2.0)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m * t_m)
    if (t_m <= 1.65d-171) then
        tmp = (t_m * sqrt(x)) / l
    else if (t_m <= 1.05d+49) then
        tmp = t_m * (1.0d0 / sqrt((((t_2 + (((l / x) * (l + (l / x))) + (((t_m * t_m) / x) * (2.0d0 + (2.0d0 / x))))) + (((l * l) + t_2) * ((1.0d0 / x) + (1.0d0 / (x * x))))) / 2.0d0)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * (t_m * t_m);
	double tmp;
	if (t_m <= 1.65e-171) {
		tmp = (t_m * Math.sqrt(x)) / l;
	} else if (t_m <= 1.05e+49) {
		tmp = t_m * (1.0 / Math.sqrt((((t_2 + (((l / x) * (l + (l / x))) + (((t_m * t_m) / x) * (2.0 + (2.0 / x))))) + (((l * l) + t_2) * ((1.0 / x) + (1.0 / (x * x))))) / 2.0)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * (t_m * t_m)
	tmp = 0
	if t_m <= 1.65e-171:
		tmp = (t_m * math.sqrt(x)) / l
	elif t_m <= 1.05e+49:
		tmp = t_m * (1.0 / math.sqrt((((t_2 + (((l / x) * (l + (l / x))) + (((t_m * t_m) / x) * (2.0 + (2.0 / x))))) + (((l * l) + t_2) * ((1.0 / x) + (1.0 / (x * x))))) / 2.0)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 1.65e-171)
		tmp = Float64(Float64(t_m * sqrt(x)) / l);
	elseif (t_m <= 1.05e+49)
		tmp = Float64(t_m * Float64(1.0 / sqrt(Float64(Float64(Float64(t_2 + Float64(Float64(Float64(l / x) * Float64(l + Float64(l / x))) + Float64(Float64(Float64(t_m * t_m) / x) * Float64(2.0 + Float64(2.0 / x))))) + Float64(Float64(Float64(l * l) + t_2) * Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x * x))))) / 2.0))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m * t_m);
	tmp = 0.0;
	if (t_m <= 1.65e-171)
		tmp = (t_m * sqrt(x)) / l;
	elseif (t_m <= 1.05e+49)
		tmp = t_m * (1.0 / sqrt((((t_2 + (((l / x) * (l + (l / x))) + (((t_m * t_m) / x) * (2.0 + (2.0 / x))))) + (((l * l) + t_2) * ((1.0 / x) + (1.0 / (x * x))))) / 2.0)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-171], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+49], N[(t$95$m * N[(1.0 / N[Sqrt[N[(N[(N[(t$95$2 + N[(N[(N[(l / x), $MachinePrecision] * N[(l + N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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 < 1.6500000000000001e-171

   1. Initial program 29.0%

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

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

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

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

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

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

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

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

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

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

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

   3. Simplified 19.8%

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

   5. Taylor expanded in l around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 17.1%

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

   10. Simplified 17.1%

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

       1. *-lft-identity N/A

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

       2. distribute-rgt-out N/A

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

       3. metadata-eval N/A

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

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

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

       5. *-lowering-*.f64 17.1%

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

   12. Applied egg-rr 17.1%

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

   13. Taylor expanded in t around 0

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   14. 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 18.5%

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

   15. Simplified 18.5%

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

   if 1.6500000000000001e-171 < t < 1.05000000000000005e49

   1. Initial program 44.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 44.6%

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

   5. Taylor expanded in x around inf

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

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

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

   7. Simplified 81.0%

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

   9. Applied egg-rr 81.3%

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

   if 1.05000000000000005e49 < t

   1. Initial program 30.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 94.5%

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

   5. Simplified 94.5%

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

   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 94.5%

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

   8. Simplified 94.5%

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

Alternative 5: 81.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-170}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+47}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell + 2 \cdot \left(t\_m \cdot t\_m\right)}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t\_m \cdot t\_m + \frac{t\_m \cdot t\_m}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.12e-170)
    (/ (* t_m (sqrt x)) l)
    (if (<= t_m 3.5e+47)
      (*
       t_m
       (sqrt
        (/
         2.0
         (+
          (/ (+ (* l l) (* 2.0 (* t_m t_m))) x)
          (+ (/ (* l l) x) (* 2.0 (+ (* t_m t_m) (/ (* t_m t_m) x))))))))
      (sqrt (/ (+ x -1.0) (+ x 1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.12e-170) {
		tmp = (t_m * sqrt(x)) / l;
	} else if (t_m <= 3.5e+47) {
		tmp = t_m * sqrt((2.0 / ((((l * l) + (2.0 * (t_m * t_m))) / x) + (((l * l) / 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

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.12d-170) then
        tmp = (t_m * sqrt(x)) / l
    else if (t_m <= 3.5d+47) then
        tmp = t_m * sqrt((2.0d0 / ((((l * l) + (2.0d0 * (t_m * t_m))) / x) + (((l * l) / 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

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.12e-170) {
		tmp = (t_m * Math.sqrt(x)) / l;
	} else if (t_m <= 3.5e+47) {
		tmp = t_m * Math.sqrt((2.0 / ((((l * l) + (2.0 * (t_m * t_m))) / x) + (((l * l) / 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

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 1.12e-170:
		tmp = (t_m * math.sqrt(x)) / l
	elif t_m <= 3.5e+47:
		tmp = t_m * math.sqrt((2.0 / ((((l * l) + (2.0 * (t_m * t_m))) / x) + (((l * l) / 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

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.12e-170)
		tmp = Float64(Float64(t_m * sqrt(x)) / l);
	elseif (t_m <= 3.5e+47)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(Float64(Float64(l * l) + Float64(2.0 * Float64(t_m * t_m))) / x) + Float64(Float64(Float64(l * l) / 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

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 1.12e-170)
		tmp = (t_m * sqrt(x)) / l;
	elseif (t_m <= 3.5e+47)
		tmp = t_m * sqrt((2.0 / ((((l * l) + (2.0 * (t_m * t_m))) / x) + (((l * l) / 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

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.12e-170], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$m, 3.5e+47], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(l * l), $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 < 1.12000000000000009e-170

   1. Initial program 29.0%

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

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

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

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

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

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

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

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

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

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

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

   3. Simplified 19.8%

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

   5. Taylor expanded in l around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 17.1%

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

   10. Simplified 17.1%

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

       1. *-lft-identity N/A

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

       2. distribute-rgt-out N/A

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

       3. metadata-eval N/A

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

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

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

       5. *-lowering-*.f64 17.1%

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

   12. Applied egg-rr 17.1%

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

   13. Taylor expanded in t around 0

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   14. 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 18.5%

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

   15. Simplified 18.5%

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

   if 1.12000000000000009e-170 < t < 3.50000000000000015e47

   1. Initial program 44.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 44.6%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

   7. Simplified 80.2%

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

   if 3.50000000000000015e47 < t

   1. Initial program 30.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 94.5%

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

   5. Simplified 94.5%

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

   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 94.5%

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

   8. Simplified 94.5%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-170}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+47}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.32 \cdot 10^{-194}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.32e-194)
    (/ (* t_m (sqrt x)) l)
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.32e-194) {
		tmp = (t_m * sqrt(x)) / l;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.32d-194) then
        tmp = (t_m * sqrt(x)) / l
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.32e-194) {
		tmp = (t_m * Math.sqrt(x)) / l;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 1.32e-194:
		tmp = (t_m * math.sqrt(x)) / l
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.32e-194)
		tmp = Float64(Float64(t_m * sqrt(x)) / l);
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 1.32e-194)
		tmp = (t_m * sqrt(x)) / l;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.32e-194], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.32e-194

   1. Initial program 29.4%

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

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

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

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

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

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

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

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

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

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

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

   3. Simplified 20.0%

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

   5. Taylor expanded in l around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 17.3%

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

   10. Simplified 17.3%

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

       1. *-lft-identity N/A

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

       2. distribute-rgt-out N/A

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

       3. metadata-eval N/A

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

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

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

       5. *-lowering-*.f64 17.3%

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

   12. Applied egg-rr 17.3%

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

   13. Taylor expanded in t around 0

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   14. 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 18.8%

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

   15. Simplified 18.8%

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

   if 1.32e-194 < t

   1. Initial program 36.8%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 78.8%

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

   5. Simplified 78.8%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 78.8%

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

   8. Simplified 78.8%

      \[\leadsto \color{blue}{\sqrt{\frac{x + -1}{x + 1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-191}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x} + 1\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-191)
    (/ (* t_m (sqrt x)) l)
    (+ (/ (+ -1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x) 1.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 8e-191) {
		tmp = (t_m * sqrt(x)) / l;
	} else {
		tmp = ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x) + 1.0;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 8d-191) then
        tmp = (t_m * sqrt(x)) / l
    else
        tmp = (((-1.0d0) + ((0.5d0 + ((-0.5d0) / x)) / x)) / x) + 1.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 8e-191) {
		tmp = (t_m * Math.sqrt(x)) / l;
	} else {
		tmp = ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x) + 1.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 8e-191:
		tmp = (t_m * math.sqrt(x)) / l
	else:
		tmp = ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x) + 1.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 8e-191)
		tmp = Float64(Float64(t_m * sqrt(x)) / l);
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x) + 1.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 8e-191)
		tmp = (t_m * sqrt(x)) / l;
	else
		tmp = ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x) + 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-191], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.0000000000000002e-191

   1. Initial program 29.4%

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

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

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

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

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

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

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

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

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

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

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

   3. Simplified 20.0%

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

   5. Taylor expanded in l around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 17.3%

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

   10. Simplified 17.3%

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

       1. *-lft-identity N/A

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

       2. distribute-rgt-out N/A

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

       3. metadata-eval N/A

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

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

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

       5. *-lowering-*.f64 17.3%

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

   12. Applied egg-rr 17.3%

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

   13. Taylor expanded in t around 0

       \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]
   14. 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 18.8%

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

   15. Simplified 18.8%

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

   if 8.0000000000000002e-191 < t

   1. Initial program 36.8%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 78.8%

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

   5. Simplified 78.8%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-div N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{2}^{0}}{t} \cdot t\right), \left(\sqrt{\frac{1 + x}{x + -1}}\right)\right) \]

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. +-commutative N/A

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

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

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

      18. +-lowering-+.f64 78.6%

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

   7. Applied egg-rr 78.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, t\right), t\right), \color{blue}{\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in x around -inf

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

   13. Simplified 78.6%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-191}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+276}:\\ \;\;\;\;\frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x} + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{\ell}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 1.25e+276)
    (+ (/ (+ -1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x) 1.0)
    (* (sqrt x) (/ t_m l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.25e+276) {
		tmp = ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x) + 1.0;
	} else {
		tmp = sqrt(x) * (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 1.25d+276) then
        tmp = (((-1.0d0) + ((0.5d0 + ((-0.5d0) / x)) / x)) / x) + 1.0d0
    else
        tmp = sqrt(x) * (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.25e+276) {
		tmp = ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x) + 1.0;
	} else {
		tmp = Math.sqrt(x) * (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 1.25e+276:
		tmp = ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x) + 1.0
	else:
		tmp = math.sqrt(x) * (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 1.25e+276)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x) + 1.0);
	else
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 1.25e+276)
		tmp = ((-1.0 + ((0.5 + (-0.5 / x)) / x)) / x) + 1.0;
	else
		tmp = sqrt(x) * (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 1.25e+276], N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.25e276

   1. Initial program 33.4%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. pow1/2 N/A

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

      7. pow1/2 N/A

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

      8. pow-div N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{2}^{0}}{t} \cdot t\right), \left(\sqrt{\frac{1 + x}{x + -1}}\right)\right) \]

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. +-commutative N/A

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

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

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

      18. +-lowering-+.f64 38.9%

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

   7. Applied egg-rr 38.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, t\right), t\right), \color{blue}{\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 38.8%

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

   10. Simplified 38.8%

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

   11. Taylor expanded in x around -inf

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

   13. Simplified 38.9%

       \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]

   if 1.25e276 < 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. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-lowering-+.f64 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 61.9%

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

   10. Simplified 61.9%

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

   11. Taylor expanded in t around 0

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

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

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

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

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

       3. sqrt-lowering-sqrt.f64 80.6%

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

   13. Simplified 80.6%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+276}:\\ \;\;\;\;\frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x} + 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.1% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, t$95$m_] := N[(t$95$s * N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.8%

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

3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-lowering-+.f64 38.6%

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

5. Simplified 38.6%

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

   1. times-frac N/A

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

   2. associate-*r/ N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. pow1/2 N/A

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

   7. pow1/2 N/A

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

   8. pow-div N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{2}^{0}}{t} \cdot t\right), \left(\sqrt{\frac{1 + x}{x + -1}}\right)\right) \]

   10. metadata-eval N/A

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

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

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

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

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

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

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

   14. +-commutative N/A

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

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

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

   16. +-commutative N/A

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

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

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

   18. +-lowering-+.f64 38.5%

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

7. Applied egg-rr 38.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, t\right), t\right), \color{blue}{\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \left(\frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{3}}\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

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

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

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

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

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

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

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

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

   13. cube-mult N/A

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

   14. unpow2 N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 38.5%

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

10. Simplified 38.5%

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

11. Taylor expanded in x around -inf

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

13. Simplified 38.6%

    \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]

14. Final simplification 38.6%

    \[\leadsto \frac{-1 + \frac{0.5 + \frac{-0.5}{x}}{x}}{x} + 1 \]
15. Add Preprocessing

Alternative 10: 76.9% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, t$95$m_] := N[(t$95$s * N[(N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.8%

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

3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-lowering-+.f64 38.6%

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

5. Simplified 38.6%

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

   1. times-frac N/A

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

   2. associate-*r/ N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. pow1/2 N/A

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

   7. pow1/2 N/A

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

   8. pow-div N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{2}^{0}}{t} \cdot t\right), \left(\sqrt{\frac{1 + x}{x + -1}}\right)\right) \]

   10. metadata-eval N/A

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

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

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

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

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

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

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

   14. +-commutative N/A

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

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

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

   16. +-commutative N/A

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

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

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

   18. +-lowering-+.f64 38.5%

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

7. Applied egg-rr 38.5%

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

8. Taylor expanded in x around -inf

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

10. Simplified 38.5%

    \[\leadsto \color{blue}{1 - \frac{\frac{-0.5}{x} + 1}{x}} \]

11. Final simplification 38.5%

    \[\leadsto \frac{-1 - \frac{-0.5}{x}}{x} + 1 \]
12. Add Preprocessing

Alternative 11: 76.7% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, t$95$m_] := N[(t$95$s * N[(N[(-1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.8%

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

3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-lowering-+.f64 38.6%

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

5. Simplified 38.6%

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

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 38.5%

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

8. Simplified 38.5%

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

9. Final simplification 38.5%

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

Alternative 12: 76.0% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 32.8%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \color{blue}{\left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\left(t \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{\frac{1 + x}{x - 1}}}\right)\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{\frac{1 + x}{\color{blue}{x - 1}}}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1 + x}{x - 1}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(x + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 38.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{+.f64}\left(x, -1\right)\right)\right)\right)\right) \]

5. Simplified 38.6%

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 38.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024162 
(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)))))