Toniolo and Linder, Equation (7)

Percentage Accurate: 32.4% → 89.1%

Time: 27.0s

Alternatives: 9

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 32.4% 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: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t\_2 \cdot \sqrt{\frac{1}{x}}}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 2.05 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(2, {t\_m}^{2} \cdot \frac{1 + x}{x + -1}, t\_2 \cdot \frac{t\_2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) l_m)))
   (*
    t_s
    (if (<= t_m 1.2e-198)
      (/ (sqrt 2.0) (/ (* t_2 (sqrt (/ 1.0 x))) t_m))
      (if (<= t_m 2.05e-167)
        1.0
        (if (<= t_m 2.7e+147)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (fma
              2.0
              (* (pow t_m 2.0) (/ (+ 1.0 x) (+ x -1.0)))
              (* t_2 (/ t_2 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 tmp;
	if (t_m <= 1.2e-198) {
		tmp = sqrt(2.0) / ((t_2 * sqrt((1.0 / x))) / t_m);
	} else if (t_m <= 2.05e-167) {
		tmp = 1.0;
	} else if (t_m <= 2.7e+147) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) * ((1.0 + x) / (x + -1.0))), (t_2 * (t_2 / 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)
	tmp = 0.0
	if (t_m <= 1.2e-198)
		tmp = Float64(sqrt(2.0) / Float64(Float64(t_2 * sqrt(Float64(1.0 / x))) / t_m));
	elseif (t_m <= 2.05e-167)
		tmp = 1.0;
	elseif (t_m <= 2.7e+147)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) * Float64(Float64(1.0 + x) / Float64(x + -1.0))), Float64(t_2 * Float64(t_2 / 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]}, N[(t$95$s * If[LessEqual[t$95$m, 1.2e-198], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t$95$2 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.05e-167], 1.0, If[LessEqual[t$95$m, 2.7e+147], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $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 < 1.19999999999999993e-198

   1. Initial program 33.6%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in l around 0 25.9%

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

      1. fma-define 25.9%

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

      2. +-commutative 25.9%

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

      3. associate-*r/ 35.1%

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

      4. sub-neg 35.1%

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

      5. metadata-eval 35.1%

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

      6. +-commutative 35.1%

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

      7. associate--l+ 41.8%

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

      8. sub-neg 41.8%

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

      9. metadata-eval 41.8%

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

      10. +-commutative 41.8%

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

      11. sub-neg 41.8%

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

      12. metadata-eval 41.8%

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

      13. +-commutative 41.8%

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

   7. Simplified 41.8%

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

   8. Taylor expanded in x around inf 50.3%

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

      1. associate-*r/ 50.3%

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

   10. Simplified 50.3%

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

       1. clear-num 50.4%

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

       2. un-div-inv 50.4%

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

       3. fma-undefine 50.4%

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

       4. associate-/l* 50.4%

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

       5. distribute-lft-out 50.4%

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

       6. +-commutative 50.4%

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

   12. Applied egg-rr 50.4%

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

   13. Taylor expanded in t around 0 15.4%

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

       1. *-commutative 15.4%

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

   15. Simplified 15.4%

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

   if 1.19999999999999993e-198 < t < 2.05000000000000009e-167

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in l around 0 71.5%

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

      1. +-commutative 71.5%

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

      2. sub-neg 71.5%

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

      3. metadata-eval 71.5%

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

      4. +-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in x around inf 71.6%

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

   if 2.05000000000000009e-167 < t < 2.69999999999999998e147

   1. Initial program 62.5%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in l around 0 46.7%

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

      1. fma-define 46.7%

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

      2. +-commutative 46.7%

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

      3. associate-*r/ 71.8%

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

      4. sub-neg 71.8%

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

      5. metadata-eval 71.8%

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

      6. +-commutative 71.8%

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

      7. associate--l+ 76.6%

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

      8. sub-neg 76.6%

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

      9. metadata-eval 76.6%

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

      10. +-commutative 76.6%

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

      11. sub-neg 76.6%

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

      12. metadata-eval 76.6%

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

      13. +-commutative 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in x around inf 82.9%

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

      1. associate-*r/ 82.9%

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

   10. Simplified 82.9%

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

       1. add-sqr-sqrt 82.9%

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

       2. *-un-lft-identity 82.9%

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

       3. times-frac 82.9%

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

       4. sqrt-prod 82.9%

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

       5. unpow2 82.9%

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

       6. sqrt-prod 42.3%

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

       7. add-sqr-sqrt 74.6%

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

       8. sqrt-prod 74.6%

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

       9. unpow2 74.6%

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

       10. sqrt-prod 45.0%

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

       11. add-sqr-sqrt 92.6%

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

   12. Applied egg-rr 92.6%

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

   if 2.69999999999999998e147 < t

   1. Initial program 4.8%

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

   3. Simplified 4.8%

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

   5. Taylor expanded in l around 0 92.9%

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

      1. +-commutative 92.9%

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

      2. sub-neg 92.9%

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

      3. metadata-eval 92.9%

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

      4. +-commutative 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in t around 0 93.2%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}{t}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{1 + x}{x + -1}, \left(\sqrt{2} \cdot \ell\right) \cdot \frac{\sqrt{2} \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\frac{1}{x}}}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{\sqrt{2 \cdot \mathsf{fma}\left({t\_m}^{2}, \frac{1 + x}{x + -1}, \frac{{l\_m}^{2}}{x}\right)}} \cdot \left(t\_m \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.2e-198)
    (/ (sqrt 2.0) (/ (* (* (sqrt 2.0) l_m) (sqrt (/ 1.0 x))) t_m))
    (if (<= t_m 5.2e-166)
      1.0
      (if (<= t_m 5e+94)
        (*
         (/
          1.0
          (sqrt
           (*
            2.0
            (fma (pow t_m 2.0) (/ (+ 1.0 x) (+ x -1.0)) (/ (pow l_m 2.0) x)))))
         (* t_m (sqrt 2.0)))
        (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 <= 3.2e-198) {
		tmp = sqrt(2.0) / (((sqrt(2.0) * l_m) * sqrt((1.0 / x))) / t_m);
	} else if (t_m <= 5.2e-166) {
		tmp = 1.0;
	} else if (t_m <= 5e+94) {
		tmp = (1.0 / sqrt((2.0 * fma(pow(t_m, 2.0), ((1.0 + x) / (x + -1.0)), (pow(l_m, 2.0) / x))))) * (t_m * sqrt(2.0));
	} 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 <= 3.2e-198)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(1.0 / x))) / t_m));
	elseif (t_m <= 5.2e-166)
		tmp = 1.0;
	elseif (t_m <= 5e+94)
		tmp = Float64(Float64(1.0 / sqrt(Float64(2.0 * fma((t_m ^ 2.0), Float64(Float64(1.0 + x) / Float64(x + -1.0)), Float64((l_m ^ 2.0) / x))))) * Float64(t_m * sqrt(2.0)));
	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, 3.2e-198], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.2e-166], 1.0, If[LessEqual[t$95$m, 5e+94], N[(N[(1.0 / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Sqrt[2.0], $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 < 3.19999999999999994e-198

   1. Initial program 33.6%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in l around 0 25.9%

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

      1. fma-define 25.9%

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

      2. +-commutative 25.9%

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

      3. associate-*r/ 35.1%

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

      4. sub-neg 35.1%

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

      5. metadata-eval 35.1%

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

      6. +-commutative 35.1%

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

      7. associate--l+ 41.8%

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

      8. sub-neg 41.8%

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

      9. metadata-eval 41.8%

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

      10. +-commutative 41.8%

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

      11. sub-neg 41.8%

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

      12. metadata-eval 41.8%

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

      13. +-commutative 41.8%

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

   7. Simplified 41.8%

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

   8. Taylor expanded in x around inf 50.3%

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

      1. associate-*r/ 50.3%

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

   10. Simplified 50.3%

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

       1. clear-num 50.4%

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

       2. un-div-inv 50.4%

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

       3. fma-undefine 50.4%

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

       4. associate-/l* 50.4%

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

       5. distribute-lft-out 50.4%

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

       6. +-commutative 50.4%

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

   12. Applied egg-rr 50.4%

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

   13. Taylor expanded in t around 0 15.4%

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

       1. *-commutative 15.4%

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

   15. Simplified 15.4%

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

   if 3.19999999999999994e-198 < t < 5.19999999999999979e-166

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in l around 0 71.5%

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

      1. +-commutative 71.5%

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

      2. sub-neg 71.5%

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

      3. metadata-eval 71.5%

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

      4. +-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in x around inf 71.6%

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

   if 5.19999999999999979e-166 < t < 5.0000000000000001e94

   1. Initial program 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in l around 0 56.6%

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

      1. fma-define 56.6%

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

      2. +-commutative 56.6%

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

      3. associate-*r/ 65.9%

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

      4. sub-neg 65.9%

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

      5. metadata-eval 65.9%

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

      6. +-commutative 65.9%

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

      7. associate--l+ 72.3%

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

      8. sub-neg 72.3%

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

      9. metadata-eval 72.3%

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

      10. +-commutative 72.3%

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

      11. sub-neg 72.3%

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

      12. metadata-eval 72.3%

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

      13. +-commutative 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in x around inf 80.9%

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

      1. associate-*r/ 80.9%

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

   10. Simplified 80.9%

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

       1. associate-*r/ 80.9%

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

       2. clear-num 81.0%

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

       3. fma-undefine 81.0%

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

       4. associate-/l* 81.0%

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

       5. distribute-lft-out 81.0%

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

       6. +-commutative 81.0%

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

   12. Applied egg-rr 81.0%

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

       1. associate-/r/ 81.0%

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

       2. fma-define 81.0%

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

       3. +-commutative 81.0%

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

   14. Simplified 81.0%

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

   if 5.0000000000000001e94 < t

   1. Initial program 30.4%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in l around 0 92.9%

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

      1. +-commutative 92.9%

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

      2. sub-neg 92.9%

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

      3. metadata-eval 92.9%

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

      4. +-commutative 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in t around 0 93.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}{t}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{\sqrt{2 \cdot \mathsf{fma}\left({t}^{2}, \frac{1 + x}{x + -1}, \frac{{\ell}^{2}}{x}\right)}} \cdot \left(t \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\frac{1}{x}}}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 3.7 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \left(\frac{{l\_m}^{2}}{x} + {t\_m}^{2} \cdot \frac{1 + x}{x + -1}\right)}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.25e-206)
    (/ (sqrt 2.0) (/ (* (* (sqrt 2.0) l_m) (sqrt (/ 1.0 x))) t_m))
    (if (<= t_m 3.7e-167)
      1.0
      (if (<= t_m 3.5e+94)
        (/
         (sqrt 2.0)
         (/
          (sqrt
           (*
            2.0
            (+
             (/ (pow l_m 2.0) x)
             (* (pow t_m 2.0) (/ (+ 1.0 x) (+ x -1.0))))))
          t_m))
        (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 <= 2.25e-206) {
		tmp = sqrt(2.0) / (((sqrt(2.0) * l_m) * sqrt((1.0 / x))) / t_m);
	} else if (t_m <= 3.7e-167) {
		tmp = 1.0;
	} else if (t_m <= 3.5e+94) {
		tmp = sqrt(2.0) / (sqrt((2.0 * ((pow(l_m, 2.0) / x) + (pow(t_m, 2.0) * ((1.0 + x) / (x + -1.0)))))) / t_m);
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.25d-206) then
        tmp = sqrt(2.0d0) / (((sqrt(2.0d0) * l_m) * sqrt((1.0d0 / x))) / t_m)
    else if (t_m <= 3.7d-167) then
        tmp = 1.0d0
    else if (t_m <= 3.5d+94) then
        tmp = sqrt(2.0d0) / (sqrt((2.0d0 * (((l_m ** 2.0d0) / x) + ((t_m ** 2.0d0) * ((1.0d0 + x) / (x + (-1.0d0))))))) / t_m)
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.25e-206) {
		tmp = Math.sqrt(2.0) / (((Math.sqrt(2.0) * l_m) * Math.sqrt((1.0 / x))) / t_m);
	} else if (t_m <= 3.7e-167) {
		tmp = 1.0;
	} else if (t_m <= 3.5e+94) {
		tmp = Math.sqrt(2.0) / (Math.sqrt((2.0 * ((Math.pow(l_m, 2.0) / x) + (Math.pow(t_m, 2.0) * ((1.0 + x) / (x + -1.0)))))) / t_m);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.25e-206:
		tmp = math.sqrt(2.0) / (((math.sqrt(2.0) * l_m) * math.sqrt((1.0 / x))) / t_m)
	elif t_m <= 3.7e-167:
		tmp = 1.0
	elif t_m <= 3.5e+94:
		tmp = math.sqrt(2.0) / (math.sqrt((2.0 * ((math.pow(l_m, 2.0) / x) + (math.pow(t_m, 2.0) * ((1.0 + x) / (x + -1.0)))))) / t_m)
	else:
		tmp = math.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 <= 2.25e-206)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(1.0 / x))) / t_m));
	elseif (t_m <= 3.7e-167)
		tmp = 1.0;
	elseif (t_m <= 3.5e+94)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(2.0 * Float64(Float64((l_m ^ 2.0) / x) + Float64((t_m ^ 2.0) * Float64(Float64(1.0 + x) / Float64(x + -1.0)))))) / t_m));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.25e-206)
		tmp = sqrt(2.0) / (((sqrt(2.0) * l_m) * sqrt((1.0 / x))) / t_m);
	elseif (t_m <= 3.7e-167)
		tmp = 1.0;
	elseif (t_m <= 3.5e+94)
		tmp = sqrt(2.0) / (sqrt((2.0 * (((l_m ^ 2.0) / x) + ((t_m ^ 2.0) * ((1.0 + x) / (x + -1.0)))))) / t_m);
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.25e-206], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.7e-167], 1.0, If[LessEqual[t$95$m, 3.5e+94], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$m), $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 < 2.2499999999999999e-206

   1. Initial program 33.6%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in l around 0 25.9%

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

      1. fma-define 25.9%

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

      2. +-commutative 25.9%

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

      3. associate-*r/ 35.1%

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

      4. sub-neg 35.1%

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

      5. metadata-eval 35.1%

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

      6. +-commutative 35.1%

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

      7. associate--l+ 41.8%

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

      8. sub-neg 41.8%

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

      9. metadata-eval 41.8%

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

      10. +-commutative 41.8%

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

      11. sub-neg 41.8%

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

      12. metadata-eval 41.8%

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

      13. +-commutative 41.8%

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

   7. Simplified 41.8%

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

   8. Taylor expanded in x around inf 50.3%

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

      1. associate-*r/ 50.3%

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

   10. Simplified 50.3%

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

       1. clear-num 50.4%

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

       2. un-div-inv 50.4%

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

       3. fma-undefine 50.4%

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

       4. associate-/l* 50.4%

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

       5. distribute-lft-out 50.4%

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

       6. +-commutative 50.4%

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

   12. Applied egg-rr 50.4%

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

   13. Taylor expanded in t around 0 15.4%

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

       1. *-commutative 15.4%

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

   15. Simplified 15.4%

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

   if 2.2499999999999999e-206 < t < 3.7000000000000003e-167

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in l around 0 71.5%

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

      1. +-commutative 71.5%

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

      2. sub-neg 71.5%

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

      3. metadata-eval 71.5%

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

      4. +-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in x around inf 71.6%

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

   if 3.7000000000000003e-167 < t < 3.4999999999999997e94

   1. Initial program 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in l around 0 56.6%

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

      1. fma-define 56.6%

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

      2. +-commutative 56.6%

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

      3. associate-*r/ 65.9%

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

      4. sub-neg 65.9%

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

      5. metadata-eval 65.9%

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

      6. +-commutative 65.9%

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

      7. associate--l+ 72.3%

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

      8. sub-neg 72.3%

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

      9. metadata-eval 72.3%

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

      10. +-commutative 72.3%

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

      11. sub-neg 72.3%

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

      12. metadata-eval 72.3%

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

      13. +-commutative 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in x around inf 80.9%

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

      1. associate-*r/ 80.9%

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

   10. Simplified 80.9%

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

       1. clear-num 81.0%

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

       2. un-div-inv 80.9%

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

       3. fma-undefine 80.9%

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

       4. associate-/l* 80.9%

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

       5. distribute-lft-out 80.9%

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

       6. +-commutative 80.9%

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

   12. Applied egg-rr 80.9%

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

   if 3.4999999999999997e94 < t

   1. Initial program 30.4%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in l around 0 92.9%

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

      1. +-commutative 92.9%

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

      2. sub-neg 92.9%

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

      3. metadata-eval 92.9%

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

      4. +-commutative 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in t around 0 93.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}{t}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \frac{1 + x}{x + -1}\right)}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;l\_m \leq 1.26 \cdot 10^{+151}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{x}\\ \mathbf{elif}\;l\_m \leq 2.3 \cdot 10^{+226}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 0.5 - 0.5} \cdot \left(t\_m \cdot \frac{\sqrt{2}}{l\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 3e+134)
    (sqrt (/ (+ x -1.0) (+ 1.0 x)))
    (if (<= l_m 1.26e+151)
      (* (/ t_m l_m) (sqrt x))
      (if (<= l_m 2.3e+226)
        (+ 1.0 (/ -1.0 x))
        (* (sqrt (- (* x 0.5) 0.5)) (* t_m (/ (sqrt 2.0) l_m))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3e+134) {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	} else if (l_m <= 1.26e+151) {
		tmp = (t_m / l_m) * sqrt(x);
	} else if (l_m <= 2.3e+226) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = sqrt(((x * 0.5) - 0.5)) * (t_m * (sqrt(2.0) / l_m));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 3d+134) then
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    else if (l_m <= 1.26d+151) then
        tmp = (t_m / l_m) * sqrt(x)
    else if (l_m <= 2.3d+226) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = sqrt(((x * 0.5d0) - 0.5d0)) * (t_m * (sqrt(2.0d0) / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3e+134) {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	} else if (l_m <= 1.26e+151) {
		tmp = (t_m / l_m) * Math.sqrt(x);
	} else if (l_m <= 2.3e+226) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = Math.sqrt(((x * 0.5) - 0.5)) * (t_m * (Math.sqrt(2.0) / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 3e+134:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	elif l_m <= 1.26e+151:
		tmp = (t_m / l_m) * math.sqrt(x)
	elif l_m <= 2.3e+226:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = math.sqrt(((x * 0.5) - 0.5)) * (t_m * (math.sqrt(2.0) / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 3e+134)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	elseif (l_m <= 1.26e+151)
		tmp = Float64(Float64(t_m / l_m) * sqrt(x));
	elseif (l_m <= 2.3e+226)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(sqrt(Float64(Float64(x * 0.5) - 0.5)) * Float64(t_m * Float64(sqrt(2.0) / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 3e+134)
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	elseif (l_m <= 1.26e+151)
		tmp = (t_m / l_m) * sqrt(x);
	elseif (l_m <= 2.3e+226)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = sqrt(((x * 0.5) - 0.5)) * (t_m * (sqrt(2.0) / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 3e+134], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.26e+151], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 2.3e+226], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(x * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if l < 2.99999999999999997e134

   1. Initial program 41.2%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in l around 0 43.8%

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

      1. +-commutative 43.8%

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

      2. sub-neg 43.8%

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

      3. metadata-eval 43.8%

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

      4. +-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in t around 0 43.9%

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

   if 2.99999999999999997e134 < l < 1.26000000000000006e151

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   5. Taylor expanded in l around 0 6.1%

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

      1. fma-define 6.1%

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

      2. +-commutative 6.1%

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

      3. associate-*r/ 6.1%

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

      4. sub-neg 6.1%

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

      5. metadata-eval 6.1%

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

      6. +-commutative 6.1%

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

      7. associate--l+ 20.4%

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

      8. sub-neg 20.4%

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

      9. metadata-eval 20.4%

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

      10. +-commutative 20.4%

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

      11. sub-neg 20.4%

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

      12. metadata-eval 20.4%

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

      13. +-commutative 20.4%

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

   7. Simplified 20.4%

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

   8. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

   10. Simplified 99.2%

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

       1. clear-num 98.8%

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

       2. un-div-inv 98.8%

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

       3. fma-undefine 98.8%

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

       4. associate-/l* 98.8%

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

       5. distribute-lft-out 98.8%

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

       6. +-commutative 98.8%

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

   12. Applied egg-rr 98.8%

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

   13. Taylor expanded in t around 0 100.0%

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

   if 1.26000000000000006e151 < l < 2.29999999999999995e226

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around 0 24.7%

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

      1. +-commutative 24.7%

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

      2. sub-neg 24.7%

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

      3. metadata-eval 24.7%

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

      4. +-commutative 24.7%

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

   7. Simplified 24.7%

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

   8. Taylor expanded in x around inf 24.7%

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

   if 2.29999999999999995e226 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 2.2%

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

      1. *-commutative 2.2%

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

      2. associate--l+ 29.8%

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

      3. sub-neg 29.8%

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

      4. metadata-eval 29.8%

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

      5. +-commutative 29.8%

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

      6. sub-neg 29.8%

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

      7. metadata-eval 29.8%

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

      8. +-commutative 29.8%

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

      9. associate-/l* 29.8%

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

   7. Simplified 29.8%

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

   8. Taylor expanded in x around 0 58.9%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot x - 0.5}} \cdot \left(t \cdot \frac{\sqrt{2}}{\ell}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{+151}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+226}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 0.5 - 0.5} \cdot \left(t \cdot \frac{\sqrt{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 10^{+267}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_m \cdot \left(\sqrt{\frac{1}{\frac{1}{x} + \frac{1}{x + -1}}} \cdot \frac{1}{l\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 1e+267)
    (sqrt (/ (+ x -1.0) (+ 1.0 x)))
    (*
     (sqrt 2.0)
     (*
      t_m
      (* (sqrt (/ 1.0 (+ (/ 1.0 x) (/ 1.0 (+ x -1.0))))) (/ 1.0 l_m)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 1e+267) {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	} else {
		tmp = sqrt(2.0) * (t_m * (sqrt((1.0 / ((1.0 / x) + (1.0 / (x + -1.0))))) * (1.0 / l_m)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 1d+267) then
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    else
        tmp = sqrt(2.0d0) * (t_m * (sqrt((1.0d0 / ((1.0d0 / x) + (1.0d0 / (x + (-1.0d0)))))) * (1.0d0 / l_m)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 1e+267) {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	} else {
		tmp = Math.sqrt(2.0) * (t_m * (Math.sqrt((1.0 / ((1.0 / x) + (1.0 / (x + -1.0))))) * (1.0 / l_m)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 1e+267:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	else:
		tmp = math.sqrt(2.0) * (t_m * (math.sqrt((1.0 / ((1.0 / x) + (1.0 / (x + -1.0))))) * (1.0 / l_m)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e+267)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m * Float64(sqrt(Float64(1.0 / Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0))))) * Float64(1.0 / l_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 1e+267)
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	else
		tmp = sqrt(2.0) * (t_m * (sqrt((1.0 / ((1.0 / x) + (1.0 / (x + -1.0))))) * (1.0 / l_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e+267], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999997e266

   1. Initial program 47.6%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in l around 0 48.8%

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

      1. +-commutative 48.8%

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

      2. sub-neg 48.8%

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

      3. metadata-eval 48.8%

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

      4. +-commutative 48.8%

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

   7. Simplified 48.8%

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

   8. Taylor expanded in t around 0 48.9%

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

   if 9.9999999999999997e266 < (*.f64 l l)

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in t around 0 0.3%

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

   6. Taylor expanded in l around 0 2.0%

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

      1. *-commutative 2.0%

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

      2. associate--l+ 19.6%

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

      3. sub-neg 19.6%

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

      4. metadata-eval 19.6%

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

      5. sub-neg 19.6%

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

      6. metadata-eval 19.6%

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

   8. Simplified 19.6%

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

   9. Taylor expanded in x around inf 35.7%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+267}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \left(\sqrt{\frac{1}{\frac{1}{x} + \frac{1}{x + -1}}} \cdot \frac{1}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{+134} \lor \neg \left(l\_m \leq 2.3 \cdot 10^{+151}\right) \land l\_m \leq 2 \cdot 10^{+226}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (or (<= l_m 3e+134) (and (not (<= l_m 2.3e+151)) (<= l_m 2e+226)))
    (+ 1.0 (/ -1.0 x))
    (* (/ t_m l_m) (sqrt 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 ((l_m <= 3e+134) || (!(l_m <= 2.3e+151) && (l_m <= 2e+226))) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = (t_m / l_m) * sqrt(x);
	}
	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 <= 3d+134) .or. (.not. (l_m <= 2.3d+151)) .and. (l_m <= 2d+226)) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = (t_m / l_m) * sqrt(x)
    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 <= 3e+134) || (!(l_m <= 2.3e+151) && (l_m <= 2e+226))) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = (t_m / l_m) * Math.sqrt(x);
	}
	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 <= 3e+134) or (not (l_m <= 2.3e+151) and (l_m <= 2e+226)):
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = (t_m / l_m) * math.sqrt(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 ((l_m <= 3e+134) || (!(l_m <= 2.3e+151) && (l_m <= 2e+226)))
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(t_m / l_m) * sqrt(x));
	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 <= 3e+134) || (~((l_m <= 2.3e+151)) && (l_m <= 2e+226)))
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = (t_m / l_m) * sqrt(x);
	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[Or[LessEqual[l$95$m, 3e+134], And[N[Not[LessEqual[l$95$m, 2.3e+151]], $MachinePrecision], LessEqual[l$95$m, 2e+226]]], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.99999999999999997e134 or 2.3000000000000001e151 < l < 1.99999999999999992e226

   1. Initial program 38.9%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in l around 0 42.7%

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

      1. +-commutative 42.7%

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

      2. sub-neg 42.7%

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

      3. metadata-eval 42.7%

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

      4. +-commutative 42.7%

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

   7. Simplified 42.7%

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

   8. Taylor expanded in x around inf 42.8%

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

   if 2.99999999999999997e134 < l < 2.3000000000000001e151 or 1.99999999999999992e226 < l

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in l around 0 0.6%

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

      1. fma-define 0.6%

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

      2. +-commutative 0.6%

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

      3. associate-*r/ 0.6%

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

      4. sub-neg 0.6%

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

      5. metadata-eval 0.6%

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

      6. +-commutative 0.6%

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

      7. associate--l+ 22.3%

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

      8. sub-neg 22.3%

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

      9. metadata-eval 22.3%

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

      10. +-commutative 22.3%

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

      11. sub-neg 22.3%

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

      12. metadata-eval 22.3%

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

      13. +-commutative 22.3%

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

   7. Simplified 22.3%

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

   8. Taylor expanded in x around inf 29.5%

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

      1. associate-*r/ 29.5%

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

   10. Simplified 29.5%

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

       1. clear-num 29.4%

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

       2. un-div-inv 29.4%

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

       3. fma-undefine 29.4%

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

       4. associate-/l* 29.4%

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

       5. distribute-lft-out 29.4%

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

       6. +-commutative 29.4%

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

   12. Applied egg-rr 29.4%

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

   13. Taylor expanded in t around 0 62.9%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+134} \lor \neg \left(\ell \leq 2.3 \cdot 10^{+151}\right) \land \ell \leq 2 \cdot 10^{+226}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;l\_m \leq 7.6 \cdot 10^{+152} \lor \neg \left(l\_m \leq 2.3 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 3e+134)
    (sqrt (/ (+ x -1.0) (+ 1.0 x)))
    (if (or (<= l_m 7.6e+152) (not (<= l_m 2.3e+226)))
      (* (/ t_m l_m) (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 (l_m <= 3e+134) {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	} else if ((l_m <= 7.6e+152) || !(l_m <= 2.3e+226)) {
		tmp = (t_m / l_m) * sqrt(x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	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 <= 3d+134) then
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    else if ((l_m <= 7.6d+152) .or. (.not. (l_m <= 2.3d+226))) then
        tmp = (t_m / l_m) * sqrt(x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    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 <= 3e+134) {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	} else if ((l_m <= 7.6e+152) || !(l_m <= 2.3e+226)) {
		tmp = (t_m / l_m) * Math.sqrt(x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	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 <= 3e+134:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	elif (l_m <= 7.6e+152) or not (l_m <= 2.3e+226):
		tmp = (t_m / l_m) * math.sqrt(x)
	else:
		tmp = 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 (l_m <= 3e+134)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	elseif ((l_m <= 7.6e+152) || !(l_m <= 2.3e+226))
		tmp = Float64(Float64(t_m / l_m) * sqrt(x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	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 <= 3e+134)
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	elseif ((l_m <= 7.6e+152) || ~((l_m <= 2.3e+226)))
		tmp = (t_m / l_m) * sqrt(x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 3e+134], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l$95$m, 7.6e+152], N[Not[LessEqual[l$95$m, 2.3e+226]], $MachinePrecision]], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 2.99999999999999997e134

   1. Initial program 41.2%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in l around 0 43.8%

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

      1. +-commutative 43.8%

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

      2. sub-neg 43.8%

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

      3. metadata-eval 43.8%

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

      4. +-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in t around 0 43.9%

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

   if 2.99999999999999997e134 < l < 7.6000000000000001e152 or 2.29999999999999995e226 < l

   1. Initial program 0.2%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in l around 0 0.6%

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

      1. fma-define 0.6%

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

      2. +-commutative 0.6%

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

      3. associate-*r/ 0.6%

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

      4. sub-neg 0.6%

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

      5. metadata-eval 0.6%

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

      6. +-commutative 0.6%

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

      7. associate--l+ 22.3%

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

      8. sub-neg 22.3%

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

      9. metadata-eval 22.3%

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

      10. +-commutative 22.3%

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

      11. sub-neg 22.3%

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

      12. metadata-eval 22.3%

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

      13. +-commutative 22.3%

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

   7. Simplified 22.3%

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

   8. Taylor expanded in x around inf 29.5%

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

      1. associate-*r/ 29.5%

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

   10. Simplified 29.5%

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

       1. clear-num 29.4%

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

       2. un-div-inv 29.4%

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

       3. fma-undefine 29.4%

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

       4. associate-/l* 29.4%

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

       5. distribute-lft-out 29.4%

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

       6. +-commutative 29.4%

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

   12. Applied egg-rr 29.4%

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

   13. Taylor expanded in t around 0 62.9%

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

   if 7.6000000000000001e152 < l < 2.29999999999999995e226

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around 0 24.7%

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

      1. +-commutative 24.7%

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

      2. sub-neg 24.7%

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

      3. metadata-eval 24.7%

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

      4. +-commutative 24.7%

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

   7. Simplified 24.7%

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

   8. Taylor expanded in x around inf 24.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+152} \lor \neg \left(\ell \leq 2.3 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 45.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 35.6%

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

3. Simplified 35.5%

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

5. Taylor expanded in l around 0 39.7%

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

   1. +-commutative 39.7%

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

   2. sub-neg 39.7%

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

   3. metadata-eval 39.7%

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

   4. +-commutative 39.7%

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

7. Simplified 39.7%

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

8. Taylor expanded in x around inf 39.8%

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

9. Final simplification 39.8%

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

Alternative 9: 76.1% accurate, 225.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 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 35.6%

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

3. Simplified 35.5%

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

5. Taylor expanded in l around 0 39.7%

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

   1. +-commutative 39.7%

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

   2. sub-neg 39.7%

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

   3. metadata-eval 39.7%

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

   4. +-commutative 39.7%

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

7. Simplified 39.7%

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

8. Taylor expanded in x around inf 39.5%

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

9. Final simplification 39.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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)))))