Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 85.6%

Time: 16.3s

Alternatives: 10

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-226}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot l\_m\right)}\\ \mathbf{elif}\;t\_m \leq 3.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{2 \cdot t\_3}{t\_2 \cdot x}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 0.0038:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \frac{l\_m \cdot l\_m}{x}\right) - \mathsf{fma}\left(t\_3, -2, \frac{t\_3}{0 - x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))) (t_3 (fma 2.0 (* t_m t_m) (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 1.2e-226)
      (/ t_2 (* (sqrt (/ 1.0 x)) (* (sqrt 2.0) l_m)))
      (if (<= t_m 3.8e-158)
        (/ t_2 (fma 0.5 (/ (* 2.0 t_3) (* t_2 x)) t_2))
        (if (<= t_m 0.0038)
          (/
           t_2
           (sqrt
            (fma
             2.0
             (* t_m t_m)
             (/
              (-
               (fma 2.0 (/ (* t_m t_m) x) (/ (* l_m l_m) x))
               (fma t_3 -2.0 (/ t_3 (- 0.0 x))))
              x))))
          (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = fma(2.0, (t_m * t_m), (l_m * l_m));
	double tmp;
	if (t_m <= 1.2e-226) {
		tmp = t_2 / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	} else if (t_m <= 3.8e-158) {
		tmp = t_2 / fma(0.5, ((2.0 * t_3) / (t_2 * x)), t_2);
	} else if (t_m <= 0.0038) {
		tmp = t_2 / sqrt(fma(2.0, (t_m * t_m), ((fma(2.0, ((t_m * t_m) / x), ((l_m * l_m) / x)) - fma(t_3, -2.0, (t_3 / (0.0 - x)))) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 1.2e-226)
		tmp = Float64(t_2 / Float64(sqrt(Float64(1.0 / x)) * Float64(sqrt(2.0) * l_m)));
	elseif (t_m <= 3.8e-158)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(2.0 * t_3) / Float64(t_2 * x)), t_2));
	elseif (t_m <= 0.0038)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(t_m * t_m), Float64(Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), Float64(Float64(l_m * l_m) / x)) - fma(t_3, -2.0, Float64(t_3 / Float64(0.0 - x)))) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.2e-226], N[(t$95$2 / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.8e-158], N[(t$95$2 / N[(0.5 * N[(N[(2.0 * t$95$3), $MachinePrecision] / N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.0038], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * -2.0 + N[(t$95$3 / N[(0.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.2e-226

   1. Initial program 31.1%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 2.2

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

   5. Simplified 2.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 24.7

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

   8. Simplified 24.7%

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

   if 1.2e-226 < t < 3.7999999999999999e-158

   1. Initial program 8.1%

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

   3. Taylor expanded in x around inf

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

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

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

   5. Simplified 74.8%

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

   if 3.7999999999999999e-158 < t < 0.00379999999999999999

   1. Initial program 45.5%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

   5. Simplified 86.2%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), -2, 0 - \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right) - \mathsf{fma}\left(2, \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right)}{-x}\right)}}} \]

   if 0.00379999999999999999 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 90.5

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

   5. Simplified 90.5%

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

   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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 90.5

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

   8. Simplified 90.5%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-226}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 0.0038:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right) - \mathsf{fma}\left(\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), -2, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{0 - x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-227}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot l\_m\right)}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{2 \cdot t\_3}{t\_2 \cdot x}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 0.0023:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m, t\_m, \frac{t\_3}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))) (t_3 (fma 2.0 (* t_m t_m) (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 4.4e-227)
      (/ t_2 (* (sqrt (/ 1.0 x)) (* (sqrt 2.0) l_m)))
      (if (<= t_m 8.6e-159)
        (/ t_2 (fma 0.5 (/ (* 2.0 t_3) (* t_2 x)) t_2))
        (if (<= t_m 0.0023)
          (/ t_m (sqrt (fma t_m t_m (/ t_3 x))))
          (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = fma(2.0, (t_m * t_m), (l_m * l_m));
	double tmp;
	if (t_m <= 4.4e-227) {
		tmp = t_2 / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	} else if (t_m <= 8.6e-159) {
		tmp = t_2 / fma(0.5, ((2.0 * t_3) / (t_2 * x)), t_2);
	} else if (t_m <= 0.0023) {
		tmp = t_m / sqrt(fma(t_m, t_m, (t_3 / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 4.4e-227)
		tmp = Float64(t_2 / Float64(sqrt(Float64(1.0 / x)) * Float64(sqrt(2.0) * l_m)));
	elseif (t_m <= 8.6e-159)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(2.0 * t_3) / Float64(t_2 * x)), t_2));
	elseif (t_m <= 0.0023)
		tmp = Float64(t_m / sqrt(fma(t_m, t_m, Float64(t_3 / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.4e-227], N[(t$95$2 / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e-159], N[(t$95$2 / N[(0.5 * N[(N[(2.0 * t$95$3), $MachinePrecision] / N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.0023], N[(t$95$m / N[Sqrt[N[(t$95$m * t$95$m + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.39999999999999962e-227

   1. Initial program 31.1%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 2.2

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

   5. Simplified 2.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 24.7

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

   8. Simplified 24.7%

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

   if 4.39999999999999962e-227 < t < 8.6e-159

   1. Initial program 8.1%

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

   3. Taylor expanded in x around inf

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

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

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

   5. Simplified 74.8%

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

   if 8.6e-159 < t < 0.0023

   1. Initial program 45.5%

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

      1. sub-neg N/A

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

   4. Applied egg-rr 7.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}, \left(x + 1\right) \cdot \left(x + 1\right), 0 - \ell \cdot \ell\right)}}} \]

   5. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 86.2

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

   7. Simplified 86.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. times-frac N/A

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

      6. *-inverses N/A

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

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

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

   9. Applied egg-rr 87.0%

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

   if 0.0023 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 90.5

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

   5. Simplified 90.5%

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

   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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 90.5

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

   8. Simplified 90.5%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-227}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 0.0023:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot l\_m\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-225}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{1}{x}} \cdot t\_2}\\ \mathbf{elif}\;t\_m \leq 5.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{l\_m \cdot t\_2}{t\_m \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 0.0195:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) l_m)) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 3.9e-225)
      (/ t_3 (* (sqrt (/ 1.0 x)) t_2))
      (if (<= t_m 5.4e-158)
        (/ t_3 (fma 0.5 (/ (* l_m t_2) (* t_m x)) t_3))
        (if (<= t_m 0.0195)
          (/ t_m (sqrt (fma t_m t_m (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x))))
          (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * l_m;
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 3.9e-225) {
		tmp = t_3 / (sqrt((1.0 / x)) * t_2);
	} else if (t_m <= 5.4e-158) {
		tmp = t_3 / fma(0.5, ((l_m * t_2) / (t_m * x)), t_3);
	} else if (t_m <= 0.0195) {
		tmp = t_m / sqrt(fma(t_m, t_m, (fma(2.0, (t_m * t_m), (l_m * l_m)) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * l_m)
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 3.9e-225)
		tmp = Float64(t_3 / Float64(sqrt(Float64(1.0 / x)) * t_2));
	elseif (t_m <= 5.4e-158)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(l_m * t_2) / Float64(t_m * x)), t_3));
	elseif (t_m <= 0.0195)
		tmp = Float64(t_m / sqrt(fma(t_m, t_m, Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.9e-225], N[(t$95$3 / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.4e-158], N[(t$95$3 / N[(0.5 * N[(N[(l$95$m * t$95$2), $MachinePrecision] / N[(t$95$m * x), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.0195], N[(t$95$m / N[Sqrt[N[(t$95$m * t$95$m + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.9e-225

   1. Initial program 31.1%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 2.2

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

   5. Simplified 2.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 24.7

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

   8. Simplified 24.7%

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

   if 3.9e-225 < t < 5.3999999999999997e-158

   1. Initial program 8.1%

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

      1. sub-neg N/A

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

   4. Applied egg-rr 0.6%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}, \left(x + 1\right) \cdot \left(x + 1\right), 0 - \ell \cdot \ell\right)}}} \]

   5. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 25.6

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

   7. Simplified 25.6%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 74.8

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

   10. Simplified 74.8%

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

   11. Taylor expanded in t around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 74.8

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

   13. Simplified 74.8%

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

   if 5.3999999999999997e-158 < t < 0.0195

   1. Initial program 45.5%

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

      1. sub-neg N/A

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

   4. Applied egg-rr 7.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}, \left(x + 1\right) \cdot \left(x + 1\right), 0 - \ell \cdot \ell\right)}}} \]

   5. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 86.2

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

   7. Simplified 86.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. times-frac N/A

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

      6. *-inverses N/A

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

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

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

   9. Applied egg-rr 87.0%

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

   if 0.0195 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 90.5

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

   5. Simplified 90.5%

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

   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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 90.5

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

   8. Simplified 90.5%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.9 \cdot 10^{-225}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \left(\sqrt{2} \cdot \ell\right)}{t \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 0.0195:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-195}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot l\_m\right)}\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{-158}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 0.0022:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e-195)
    (/ (* t_m (sqrt 2.0)) (* (sqrt (/ 1.0 x)) (* (sqrt 2.0) l_m)))
    (if (<= t_m 1.8e-158)
      (+ 1.0 (/ -1.0 x))
      (if (<= t_m 0.0022)
        (/ t_m (sqrt (fma t_m t_m (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x))))
        (sqrt (/ (+ x -1.0) (+ 1.0 x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.25e-195) {
		tmp = (t_m * sqrt(2.0)) / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	} else if (t_m <= 1.8e-158) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 0.0022) {
		tmp = t_m / sqrt(fma(t_m, t_m, (fma(2.0, (t_m * t_m), (l_m * l_m)) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.25e-195)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(sqrt(Float64(1.0 / x)) * Float64(sqrt(2.0) * l_m)));
	elseif (t_m <= 1.8e-158)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 0.0022)
		tmp = Float64(t_m / sqrt(fma(t_m, t_m, Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.25e-195], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e-158], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.0022], N[(t$95$m / N[Sqrt[N[(t$95$m * t$95$m + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 1.25000000000000002e-195

   1. Initial program 30.1%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 2.2

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

   5. Simplified 2.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 24.4

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

   8. Simplified 24.4%

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

   if 1.25000000000000002e-195 < t < 1.79999999999999995e-158

   1. Initial program 12.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 81.4

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

   8. Simplified 81.4%

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

   if 1.79999999999999995e-158 < t < 0.00220000000000000013

   1. Initial program 45.5%

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

      1. sub-neg N/A

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

   4. Applied egg-rr 7.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}, \left(x + 1\right) \cdot \left(x + 1\right), 0 - \ell \cdot \ell\right)}}} \]

   5. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 86.2

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

   7. Simplified 86.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. times-frac N/A

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

      6. *-inverses N/A

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

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

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

   9. Applied egg-rr 87.0%

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

   if 0.00220000000000000013 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 90.5

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

   5. Simplified 90.5%

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

   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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 90.5

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

   8. Simplified 90.5%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-195}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-158}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 0.0022:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-194}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{-159}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 0.0195:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e-194)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 9e-159)
      (+ 1.0 (/ -1.0 x))
      (if (<= t_m 0.0195)
        (/ t_m (sqrt (fma t_m t_m (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x))))
        (sqrt (/ (+ x -1.0) (+ 1.0 x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.1e-194) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 9e-159) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 0.0195) {
		tmp = t_m / sqrt(fma(t_m, t_m, (fma(2.0, (t_m * t_m), (l_m * l_m)) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 3.1e-194)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 9e-159)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 0.0195)
		tmp = Float64(t_m / sqrt(fma(t_m, t_m, Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-194], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 9e-159], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.0195], N[(t$95$m / N[Sqrt[N[(t$95$m * t$95$m + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.1000000000000001e-194

   1. Initial program 30.1%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 2.2

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

   5. Simplified 2.2%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 24.2

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

   8. Simplified 24.2%

      \[\leadsto \color{blue}{\frac{\left(\left(t \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. sqrt-lowering-sqrt.f64 24.4

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

   10. Applied egg-rr 24.4%

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

   if 3.1000000000000001e-194 < t < 8.99999999999999977e-159

   1. Initial program 12.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 81.4

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

   8. Simplified 81.4%

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

   if 8.99999999999999977e-159 < t < 0.0195

   1. Initial program 45.5%

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

      1. sub-neg N/A

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

   4. Applied egg-rr 7.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}, \left(x + 1\right) \cdot \left(x + 1\right), 0 - \ell \cdot \ell\right)}}} \]

   5. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 86.2

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

   7. Simplified 86.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. times-frac N/A

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

      6. *-inverses N/A

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

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

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

   9. Applied egg-rr 87.0%

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

   if 0.0195 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 90.5

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

   5. Simplified 90.5%

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

   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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 90.5

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

   8. Simplified 90.5%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-194}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-159}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 0.0195:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 2e+273)
    (sqrt (/ (+ x -1.0) (+ 1.0 x)))
    (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+273) {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 2d+273) then
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+273) {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 2e+273:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 2e+273)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 2e+273)
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+273], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999989e273

   1. Initial program 41.3%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 45.7

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

   5. Simplified 45.7%

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

   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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 45.7

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

   8. Simplified 45.7%

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

   if 1.99999999999999989e273 < (*.f64 l l)

   1. Initial program 2.7%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 45.6

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

   8. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{\left(\left(t \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 46.0

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

   10. Applied egg-rr 46.0%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+273}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 2e+273) (+ 1.0 (/ -1.0 x)) (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+273) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 2d+273) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+273) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 2e+273:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 2e+273)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 2e+273)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+273], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999989e273

   1. Initial program 41.3%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 45.7

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

   5. Simplified 45.7%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 45.7

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

   8. Simplified 45.7%

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

   if 1.99999999999999989e273 < (*.f64 l l)

   1. Initial program 2.7%

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

   3. Taylor expanded in l around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 45.6

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

   8. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{\left(\left(t \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 46.0

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

   10. Applied egg-rr 46.0%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+273}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+273}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 2e+273) (+ 1.0 (/ -1.0 x)) (* (sqrt x) (/ t_m l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+273) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = sqrt(x) * (t_m / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 2d+273) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = sqrt(x) * (t_m / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+273) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = Math.sqrt(x) * (t_m / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 2e+273:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = math.sqrt(x) * (t_m / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 2e+273)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(sqrt(x) * Float64(t_m / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 2e+273)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = sqrt(x) * (t_m / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+273], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999989e273

   1. Initial program 41.3%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 45.7

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

   5. Simplified 45.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{x}} \]

      4. metadata-eval N/A

         \[\leadsto 1 + \frac{\color{blue}{-1}}{x} \]

      5. /-lowering-/.f64 45.7

         \[\leadsto 1 + \color{blue}{\frac{-1}{x}} \]

   8. Simplified 45.7%

      \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]

   if 1.99999999999999989e273 < (*.f64 l l)

   1. Initial program 2.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \color{blue}{\sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\color{blue}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) + \left(\mathsf{neg}\left(1\right)\right)}}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\color{blue}{\left(\frac{x}{x - 1} + \frac{1}{x - 1}\right)} + \left(\mathsf{neg}\left(1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\left(\frac{x}{x - 1} + \frac{1}{x - 1}\right) + \color{blue}{-1}}} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\color{blue}{\frac{x}{x - 1} + \left(\frac{1}{x - 1} + -1\right)}}} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\color{blue}{\frac{x}{x - 1} + \left(\frac{1}{x - 1} + -1\right)}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\color{blue}{\frac{x}{x - 1}} + \left(\frac{1}{x - 1} + -1\right)}} \]

      9. sub-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{x}{\color{blue}{x + \left(\mathsf{neg}\left(1\right)\right)}} + \left(\frac{1}{x - 1} + -1\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{x}{x + \color{blue}{-1}} + \left(\frac{1}{x - 1} + -1\right)}} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{x}{\color{blue}{x + -1}} + \left(\frac{1}{x - 1} + -1\right)}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{x}{x + -1} + \color{blue}{\left(\frac{1}{x - 1} + -1\right)}}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{x}{x + -1} + \left(\color{blue}{\frac{1}{x - 1}} + -1\right)}} \]

      14. sub-neg N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{x}{x + -1} + \left(\frac{1}{\color{blue}{x + \left(\mathsf{neg}\left(1\right)\right)}} + -1\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{x}{x + -1} + \left(\frac{1}{x + \color{blue}{-1}} + -1\right)}} \]

      16. +-lowering-+.f64 3.1

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{x}{x + -1} + \left(\frac{1}{\color{blue}{x + -1}} + -1\right)}} \]

   5. Simplified 3.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\frac{x}{x + -1} + \left(\frac{1}{x + -1} + -1\right)}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{x}}{\ell}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{x}}{\ell}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{x}}}{\ell} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(t \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{2}\right)} \cdot \sqrt{x}}{\ell} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(t \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{2}\right)} \cdot \sqrt{x}}{\ell} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(t \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\left(\left(t \cdot \color{blue}{\sqrt{\frac{1}{2}}}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell} \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\left(\left(t \cdot \sqrt{\frac{1}{2}}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{x}}{\ell} \]

      9. sqrt-lowering-sqrt.f64 45.6

         \[\leadsto \frac{\left(\left(t \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{x}}}{\ell} \]

   8. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{\left(\left(t \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \left(\left(t \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{2}\right)}}{\ell} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{\left(t \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{2}}{\ell}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{\left(t \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{2}}{\ell}} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{x}} \cdot \frac{\left(t \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{2}}{\ell} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \frac{\color{blue}{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}}{\ell} \]

      6. sqrt-unprod N/A

         \[\leadsto \sqrt{x} \cdot \frac{t \cdot \color{blue}{\sqrt{\frac{1}{2} \cdot 2}}}{\ell} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{x} \cdot \frac{t \cdot \sqrt{\color{blue}{1}}}{\ell} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{x} \cdot \frac{t \cdot \color{blue}{1}}{\ell} \]

      9. *-rgt-identity N/A

         \[\leadsto \sqrt{x} \cdot \frac{\color{blue}{t}}{\ell} \]

      10. /-lowering-/.f64 39.6

          \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{t}{\ell}} \]

   10. Applied egg-rr 39.6%

       \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{t}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((-1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + (-1.0 / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + (-1.0 / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.9%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left(t \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1 + x}{x - 1}}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{1 + x}{x - 1}}}} \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{1 + x}}{x - 1}}} \]

   7. sub-neg N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(\mathsf{neg}\left(1\right)\right)}}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \]

   9. +-lowering-+.f64 40.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + -1}}}} \]

5. Simplified 40.7%

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

   3. distribute-neg-frac N/A

      \[\leadsto 1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{x}} \]

   4. metadata-eval N/A

      \[\leadsto 1 + \frac{\color{blue}{-1}}{x} \]

   5. /-lowering-/.f64 40.7

      \[\leadsto 1 + \color{blue}{\frac{-1}{x}} \]

8. Simplified 40.7%

   \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]
9. Add Preprocessing

Alternative 10: 75.0% accurate, 85.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 34.9%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{2}}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{2}}} \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{1}{2}} \]

   4. sqrt-lowering-sqrt.f64 40.1

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{0.5}} \]

5. Simplified 40.1%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{1}{2}}} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{\color{blue}{1}} \]

   3. metadata-eval 40.7

      \[\leadsto \color{blue}{1} \]

7. Applied egg-rr 40.7%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))))