Toniolo and Linder, Equation (7)

Percentage Accurate: 33.0% → 85.2%

Time: 22.6s

Alternatives: 12

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot l\_m}{t\_m \cdot \sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{2 \cdot \left({t\_m}^{2} + {t\_m}^{2}\right) + 2 \cdot {l\_m}^{2}}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{2 \cdot {t\_m}^{2} - \frac{-2 \cdot \left({l\_m}^{2} + \mathsf{fma}\left(t\_m, t\_m, {t\_m}^{2}\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-266)
    (/ (sqrt 2.0) (/ (* (sqrt 2.0) l_m) (* t_m (sqrt x))))
    (if (<= t_m 4.6e-162)
      (*
       (sqrt 2.0)
       (/
        t_m
        (+
         (*
          0.5
          (/
           (+ (* 2.0 (+ (pow t_m 2.0) (pow t_m 2.0))) (* 2.0 (pow l_m 2.0)))
           (* t_m (* (sqrt 2.0) x))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 2.5e+72)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (-
            (* 2.0 (pow t_m 2.0))
            (/ (* -2.0 (+ (pow l_m 2.0) (fma t_m t_m (pow t_m 2.0)))) x)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.6e-266) {
		tmp = sqrt(2.0) / ((sqrt(2.0) * l_m) / (t_m * sqrt(x)));
	} else if (t_m <= 4.6e-162) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((2.0 * (pow(t_m, 2.0) + pow(t_m, 2.0))) + (2.0 * pow(l_m, 2.0))) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 2.5e+72) {
		tmp = sqrt(2.0) * (t_m / sqrt(((2.0 * pow(t_m, 2.0)) - ((-2.0 * (pow(l_m, 2.0) + fma(t_m, t_m, pow(t_m, 2.0)))) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.6e-266)
		tmp = Float64(sqrt(2.0) / Float64(Float64(sqrt(2.0) * l_m) / Float64(t_m * sqrt(x))));
	elseif (t_m <= 4.6e-162)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) + (t_m ^ 2.0))) + Float64(2.0 * (l_m ^ 2.0))) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 2.5e+72)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(2.0 * (t_m ^ 2.0)) - Float64(Float64(-2.0 * Float64((l_m ^ 2.0) + fma(t_m, t_m, (t_m ^ 2.0)))) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	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, 2.6e-266], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] / N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e-162], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+72], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] + N[(t$95$m * t$95$m + N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 2.6e-266

   1. Initial program 33.4%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 9.5%

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

      2. sub-neg 9.5%

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

      3. metadata-eval 9.5%

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

      4. +-commutative 9.5%

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

      5. sub-neg 9.5%

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

      6. metadata-eval 9.5%

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

      7. +-commutative 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in x around inf 19.4%

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

      1. clear-num 19.4%

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

      2. un-div-inv 19.5%

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

      3. sqrt-div 19.4%

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

      4. metadata-eval 19.4%

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

      5. un-div-inv 19.5%

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

   10. Applied egg-rr 19.5%

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

       1. associate-/l/ 19.4%

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

   12. Simplified 19.4%

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

   if 2.6e-266 < t < 4.5999999999999996e-162

   1. Initial program 2.4%

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

   3. Simplified 1.5%

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

   5. Taylor expanded in l around 0 2.4%

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

      1. +-commutative 2.4%

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

      2. fma-define 2.4%

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

      3. associate--l+ 21.5%

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

      4. sub-neg 21.5%

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

      5. metadata-eval 21.5%

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

      6. +-commutative 21.5%

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

      7. sub-neg 21.5%

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

      8. sub-neg 21.5%

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

      9. metadata-eval 21.5%

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

      10. +-commutative 21.5%

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

      11. metadata-eval 21.5%

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

      12. +-commutative 21.5%

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

      13. associate-*r/ 21.5%

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

      14. sub-neg 21.5%

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

      15. metadata-eval 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in x around inf 76.7%

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

   if 4.5999999999999996e-162 < t < 2.49999999999999996e72

   1. Initial program 69.5%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in l around 0 70.1%

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

      1. +-commutative 70.1%

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

      2. fma-define 70.1%

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

      3. associate--l+ 72.1%

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

      4. sub-neg 72.1%

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

      5. metadata-eval 72.1%

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

      6. +-commutative 72.1%

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

      7. sub-neg 72.1%

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

      8. sub-neg 72.1%

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

      9. metadata-eval 72.1%

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

      10. +-commutative 72.1%

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

      11. metadata-eval 72.1%

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

      12. +-commutative 72.1%

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

      13. associate-*r/ 78.3%

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

      14. sub-neg 78.3%

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

      15. metadata-eval 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in x around -inf 85.8%

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

      1. +-commutative 85.8%

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

      2. mul-1-neg 85.8%

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

      3. unsub-neg 85.8%

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

      4. distribute-lft-out 85.8%

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

      5. unpow2 85.8%

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

      6. fma-neg 85.8%

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

      7. mul-1-neg 85.8%

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

      8. remove-double-neg 85.8%

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

   10. Simplified 85.8%

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

   if 2.49999999999999996e72 < t

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in t around inf 97.0%

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

   6. Taylor expanded in t around 0 97.3%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot \ell}{t \cdot \sqrt{x}}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{2 \cdot \left({t}^{2} + {t}^{2}\right) + 2 \cdot {\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} - \frac{-2 \cdot \left({\ell}^{2} + \mathsf{fma}\left(t, t, {t}^{2}\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t\_2}{t\_m \cdot \sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{t\_2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t\_m \leq 10^{+72}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{2 \cdot {t\_m}^{2} - \frac{-2 \cdot \left({l\_m}^{2} + \mathsf{fma}\left(t\_m, t\_m, {t\_m}^{2}\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) l_m)))
   (*
    t_s
    (if (<= t_m 4.8e-227)
      (/ (sqrt 2.0) (/ t_2 (* t_m (sqrt x))))
      (if (<= t_m 1.4e-188)
        1.0
        (if (<= t_m 3.6e-162)
          (* (sqrt 2.0) (/ t_m (* t_2 (sqrt (/ 1.0 x)))))
          (if (<= t_m 1e+72)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (-
                (* 2.0 (pow t_m 2.0))
                (/
                 (* -2.0 (+ (pow l_m 2.0) (fma t_m t_m (pow t_m 2.0))))
                 x)))))
            (sqrt (/ (+ x -1.0) (+ x 1.0))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * l_m;
	double tmp;
	if (t_m <= 4.8e-227) {
		tmp = sqrt(2.0) / (t_2 / (t_m * sqrt(x)));
	} else if (t_m <= 1.4e-188) {
		tmp = 1.0;
	} else if (t_m <= 3.6e-162) {
		tmp = sqrt(2.0) * (t_m / (t_2 * sqrt((1.0 / x))));
	} else if (t_m <= 1e+72) {
		tmp = sqrt(2.0) * (t_m / sqrt(((2.0 * pow(t_m, 2.0)) - ((-2.0 * (pow(l_m, 2.0) + fma(t_m, t_m, pow(t_m, 2.0)))) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * l_m)
	tmp = 0.0
	if (t_m <= 4.8e-227)
		tmp = Float64(sqrt(2.0) / Float64(t_2 / Float64(t_m * sqrt(x))));
	elseif (t_m <= 1.4e-188)
		tmp = 1.0;
	elseif (t_m <= 3.6e-162)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(t_2 * sqrt(Float64(1.0 / x)))));
	elseif (t_m <= 1e+72)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(2.0 * (t_m ^ 2.0)) - Float64(Float64(-2.0 * Float64((l_m ^ 2.0) + fma(t_m, t_m, (t_m ^ 2.0)))) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	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, 4.8e-227], N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e-188], 1.0, If[LessEqual[t$95$m, 3.6e-162], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(t$95$2 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+72], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] + N[(t$95$m * t$95$m + N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 4.7999999999999999e-227

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. clear-num 20.1%

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

      2. un-div-inv 20.2%

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

      3. sqrt-div 20.2%

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

      4. metadata-eval 20.2%

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

      5. un-div-inv 20.2%

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

   10. Applied egg-rr 20.2%

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

       1. associate-/l/ 20.1%

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

   12. Simplified 20.1%

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

   if 4.7999999999999999e-227 < t < 1.4000000000000001e-188

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 1.4000000000000001e-188 < t < 3.5999999999999998e-162

   1. Initial program 2.3%

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

   3. Simplified 1.7%

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

   5. Taylor expanded in l around inf 1.3%

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

      1. associate--l+ 23.5%

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

      2. sub-neg 23.5%

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

      3. metadata-eval 23.5%

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

      4. +-commutative 23.5%

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

      5. sub-neg 23.5%

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

      6. metadata-eval 23.5%

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

      7. +-commutative 23.5%

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

   7. Simplified 23.5%

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

   8. Taylor expanded in x around inf 34.6%

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

   if 3.5999999999999998e-162 < t < 9.99999999999999944e71

   1. Initial program 69.5%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in l around 0 70.1%

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

      1. +-commutative 70.1%

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

      2. fma-define 70.1%

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

      3. associate--l+ 72.1%

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

      4. sub-neg 72.1%

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

      5. metadata-eval 72.1%

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

      6. +-commutative 72.1%

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

      7. sub-neg 72.1%

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

      8. sub-neg 72.1%

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

      9. metadata-eval 72.1%

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

      10. +-commutative 72.1%

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

      11. metadata-eval 72.1%

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

      12. +-commutative 72.1%

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

      13. associate-*r/ 78.3%

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

      14. sub-neg 78.3%

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

      15. metadata-eval 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in x around -inf 85.8%

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

      1. +-commutative 85.8%

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

      2. mul-1-neg 85.8%

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

      3. unsub-neg 85.8%

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

      4. distribute-lft-out 85.8%

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

      5. unpow2 85.8%

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

      6. fma-neg 85.8%

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

      7. mul-1-neg 85.8%

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

      8. remove-double-neg 85.8%

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

   10. Simplified 85.8%

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

   if 9.99999999999999944e71 < t

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in t around inf 97.0%

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

   6. Taylor expanded in t around 0 97.3%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot \ell}{t \cdot \sqrt{x}}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 10^{+72}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} - \frac{-2 \cdot \left({\ell}^{2} + \mathsf{fma}\left(t, t, {t}^{2}\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t\_2}{t\_m \cdot \sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 8.8 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{t\_2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) l_m)))
   (*
    t_s
    (if (<= t_m 7e-228)
      (/ (sqrt 2.0) (/ t_2 (* t_m (sqrt x))))
      (if (<= t_m 8.6e-189)
        1.0
        (if (<= t_m 8.8e-179)
          (* (sqrt 2.0) (/ t_m (* t_2 (sqrt (/ 1.0 x)))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * l_m;
	double tmp;
	if (t_m <= 7e-228) {
		tmp = sqrt(2.0) / (t_2 / (t_m * sqrt(x)));
	} else if (t_m <= 8.6e-189) {
		tmp = 1.0;
	} else if (t_m <= 8.8e-179) {
		tmp = sqrt(2.0) * (t_m / (t_2 * sqrt((1.0 / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * l_m
    if (t_m <= 7d-228) then
        tmp = sqrt(2.0d0) / (t_2 / (t_m * sqrt(x)))
    else if (t_m <= 8.6d-189) then
        tmp = 1.0d0
    else if (t_m <= 8.8d-179) then
        tmp = sqrt(2.0d0) * (t_m / (t_2 * sqrt((1.0d0 / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) * l_m;
	double tmp;
	if (t_m <= 7e-228) {
		tmp = Math.sqrt(2.0) / (t_2 / (t_m * Math.sqrt(x)));
	} else if (t_m <= 8.6e-189) {
		tmp = 1.0;
	} else if (t_m <= 8.8e-179) {
		tmp = Math.sqrt(2.0) * (t_m / (t_2 * Math.sqrt((1.0 / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) * l_m
	tmp = 0
	if t_m <= 7e-228:
		tmp = math.sqrt(2.0) / (t_2 / (t_m * math.sqrt(x)))
	elif t_m <= 8.6e-189:
		tmp = 1.0
	elif t_m <= 8.8e-179:
		tmp = math.sqrt(2.0) * (t_m / (t_2 * math.sqrt((1.0 / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * l_m)
	tmp = 0.0
	if (t_m <= 7e-228)
		tmp = Float64(sqrt(2.0) / Float64(t_2 / Float64(t_m * sqrt(x))));
	elseif (t_m <= 8.6e-189)
		tmp = 1.0;
	elseif (t_m <= 8.8e-179)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(t_2 * sqrt(Float64(1.0 / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) * l_m;
	tmp = 0.0;
	if (t_m <= 7e-228)
		tmp = sqrt(2.0) / (t_2 / (t_m * sqrt(x)));
	elseif (t_m <= 8.6e-189)
		tmp = 1.0;
	elseif (t_m <= 8.8e-179)
		tmp = sqrt(2.0) * (t_m / (t_2 * sqrt((1.0 / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7e-228], N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$2 / N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e-189], 1.0, If[LessEqual[t$95$m, 8.8e-179], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(t$95$2 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 6.9999999999999995e-228

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. clear-num 20.1%

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

      2. un-div-inv 20.2%

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

      3. sqrt-div 20.2%

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

      4. metadata-eval 20.2%

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

      5. un-div-inv 20.2%

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

   10. Applied egg-rr 20.2%

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

       1. associate-/l/ 20.1%

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

   12. Simplified 20.1%

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

   if 6.9999999999999995e-228 < t < 8.60000000000000071e-189

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 8.60000000000000071e-189 < t < 8.80000000000000018e-179

   1. Initial program 0.7%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 65.5%

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

      2. sub-neg 65.5%

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

      3. metadata-eval 65.5%

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

      4. +-commutative 65.5%

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

      5. sub-neg 65.5%

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

      6. metadata-eval 65.5%

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

      7. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 99.0%

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

   if 8.80000000000000018e-179 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot \ell}{t \cdot \sqrt{x}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot l\_m}{t\_m \cdot \sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 6.9 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 10^{-178}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{l\_m \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.6e-228)
    (/ (sqrt 2.0) (/ (* (sqrt 2.0) l_m) (* t_m (sqrt x))))
    (if (<= t_m 6.9e-187)
      1.0
      (if (<= t_m 1e-178)
        (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (/ 2.0 x))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.6e-228) {
		tmp = sqrt(2.0) / ((sqrt(2.0) * l_m) / (t_m * sqrt(x)));
	} else if (t_m <= 6.9e-187) {
		tmp = 1.0;
	} else if (t_m <= 1e-178) {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt((2.0 / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.6d-228) then
        tmp = sqrt(2.0d0) / ((sqrt(2.0d0) * l_m) / (t_m * sqrt(x)))
    else if (t_m <= 6.9d-187) then
        tmp = 1.0d0
    else if (t_m <= 1d-178) then
        tmp = (t_m * sqrt(2.0d0)) / (l_m * sqrt((2.0d0 / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.6e-228) {
		tmp = Math.sqrt(2.0) / ((Math.sqrt(2.0) * l_m) / (t_m * Math.sqrt(x)));
	} else if (t_m <= 6.9e-187) {
		tmp = 1.0;
	} else if (t_m <= 1e-178) {
		tmp = (t_m * Math.sqrt(2.0)) / (l_m * Math.sqrt((2.0 / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 3.6e-228:
		tmp = math.sqrt(2.0) / ((math.sqrt(2.0) * l_m) / (t_m * math.sqrt(x)))
	elif t_m <= 6.9e-187:
		tmp = 1.0
	elif t_m <= 1e-178:
		tmp = (t_m * math.sqrt(2.0)) / (l_m * math.sqrt((2.0 / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 3.6e-228)
		tmp = Float64(sqrt(2.0) / Float64(Float64(sqrt(2.0) * l_m) / Float64(t_m * sqrt(x))));
	elseif (t_m <= 6.9e-187)
		tmp = 1.0;
	elseif (t_m <= 1e-178)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(Float64(2.0 / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 3.6e-228)
		tmp = sqrt(2.0) / ((sqrt(2.0) * l_m) / (t_m * sqrt(x)));
	elseif (t_m <= 6.9e-187)
		tmp = 1.0;
	elseif (t_m <= 1e-178)
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt((2.0 / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-228], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] / N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.9e-187], 1.0, If[LessEqual[t$95$m, 1e-178], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.6000000000000002e-228

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. clear-num 20.1%

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

      2. un-div-inv 20.2%

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

      3. sqrt-div 20.2%

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

      4. metadata-eval 20.2%

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

      5. un-div-inv 20.2%

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

   10. Applied egg-rr 20.2%

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

       1. associate-/l/ 20.1%

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

   12. Simplified 20.1%

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

   if 3.6000000000000002e-228 < t < 6.90000000000000045e-187

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 6.90000000000000045e-187 < t < 9.9999999999999995e-179

   1. Initial program 0.7%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 65.5%

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

      2. sub-neg 65.5%

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

      3. metadata-eval 65.5%

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

      4. +-commutative 65.5%

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

      5. sub-neg 65.5%

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

      6. metadata-eval 65.5%

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

      7. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. add-log-exp 59.6%

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

      2. sqrt-div 59.6%

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

      3. metadata-eval 59.6%

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

      4. un-div-inv 59.6%

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

   10. Applied egg-rr 59.6%

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

       1. rem-log-exp 99.0%

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

       2. associate-*r/ 99.5%

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

       3. associate-/l* 99.5%

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

       4. sqrt-undiv 99.5%

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

   12. Applied egg-rr 99.5%

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

   if 9.9999999999999995e-179 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot \ell}{t \cdot \sqrt{x}}}\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 10^{-178}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{l\_m} \cdot \frac{t\_m}{\sqrt{\frac{2}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-224}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 9.6 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{-178}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (/ (sqrt 2.0) l_m) (/ t_m (sqrt (/ 2.0 x))))))
   (*
    t_s
    (if (<= t_m 7.8e-224)
      t_2
      (if (<= t_m 9.6e-188)
        1.0
        (if (<= t_m 1.45e-178) t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (sqrt(2.0) / l_m) * (t_m / sqrt((2.0 / x)));
	double tmp;
	if (t_m <= 7.8e-224) {
		tmp = t_2;
	} else if (t_m <= 9.6e-188) {
		tmp = 1.0;
	} else if (t_m <= 1.45e-178) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (sqrt(2.0d0) / l_m) * (t_m / sqrt((2.0d0 / x)))
    if (t_m <= 7.8d-224) then
        tmp = t_2
    else if (t_m <= 9.6d-188) then
        tmp = 1.0d0
    else if (t_m <= 1.45d-178) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (Math.sqrt(2.0) / l_m) * (t_m / Math.sqrt((2.0 / x)));
	double tmp;
	if (t_m <= 7.8e-224) {
		tmp = t_2;
	} else if (t_m <= 9.6e-188) {
		tmp = 1.0;
	} else if (t_m <= 1.45e-178) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (math.sqrt(2.0) / l_m) * (t_m / math.sqrt((2.0 / x)))
	tmp = 0
	if t_m <= 7.8e-224:
		tmp = t_2
	elif t_m <= 9.6e-188:
		tmp = 1.0
	elif t_m <= 1.45e-178:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(sqrt(2.0) / l_m) * Float64(t_m / sqrt(Float64(2.0 / x))))
	tmp = 0.0
	if (t_m <= 7.8e-224)
		tmp = t_2;
	elseif (t_m <= 9.6e-188)
		tmp = 1.0;
	elseif (t_m <= 1.45e-178)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (sqrt(2.0) / l_m) * (t_m / sqrt((2.0 / x)));
	tmp = 0.0;
	if (t_m <= 7.8e-224)
		tmp = t_2;
	elseif (t_m <= 9.6e-188)
		tmp = 1.0;
	elseif (t_m <= 1.45e-178)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.8e-224], t$95$2, If[LessEqual[t$95$m, 9.6e-188], 1.0, If[LessEqual[t$95$m, 1.45e-178], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.7999999999999996e-224 or 9.6e-188 < t < 1.4499999999999999e-178

   1. Initial program 31.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 11.9%

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

      2. sub-neg 11.9%

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

      3. metadata-eval 11.9%

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

      4. +-commutative 11.9%

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

      5. sub-neg 11.9%

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

      6. metadata-eval 11.9%

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

      7. +-commutative 11.9%

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

   7. Simplified 11.9%

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

   8. Taylor expanded in x around inf 22.0%

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

      1. associate-*r/ 22.0%

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

      2. *-commutative 22.0%

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

      3. sqrt-div 22.0%

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

      4. metadata-eval 22.0%

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

      5. un-div-inv 22.0%

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

   10. Applied egg-rr 22.0%

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

       1. *-commutative 22.0%

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

       2. associate-/l* 22.0%

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

       3. times-frac 22.0%

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

       4. sqrt-undiv 22.0%

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

   12. Applied egg-rr 22.0%

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

   if 7.7999999999999996e-224 < t < 9.6e-188

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 1.4499999999999999e-178 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-188}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\sqrt{2}}{l\_m \cdot \sqrt{\frac{2}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.4 \cdot 10^{-223}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.62 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{-179}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ (sqrt 2.0) (* l_m (sqrt (/ 2.0 x)))))))
   (*
    t_s
    (if (<= t_m 6.4e-223)
      t_2
      (if (<= t_m 1.62e-187)
        1.0
        (if (<= t_m 5.6e-179) t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * (sqrt(2.0) / (l_m * sqrt((2.0 / x))));
	double tmp;
	if (t_m <= 6.4e-223) {
		tmp = t_2;
	} else if (t_m <= 1.62e-187) {
		tmp = 1.0;
	} else if (t_m <= 5.6e-179) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (sqrt(2.0d0) / (l_m * sqrt((2.0d0 / x))))
    if (t_m <= 6.4d-223) then
        tmp = t_2
    else if (t_m <= 1.62d-187) then
        tmp = 1.0d0
    else if (t_m <= 5.6d-179) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * (Math.sqrt(2.0) / (l_m * Math.sqrt((2.0 / x))));
	double tmp;
	if (t_m <= 6.4e-223) {
		tmp = t_2;
	} else if (t_m <= 1.62e-187) {
		tmp = 1.0;
	} else if (t_m <= 5.6e-179) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * (math.sqrt(2.0) / (l_m * math.sqrt((2.0 / x))))
	tmp = 0
	if t_m <= 6.4e-223:
		tmp = t_2
	elif t_m <= 1.62e-187:
		tmp = 1.0
	elif t_m <= 5.6e-179:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * Float64(sqrt(2.0) / Float64(l_m * sqrt(Float64(2.0 / x)))))
	tmp = 0.0
	if (t_m <= 6.4e-223)
		tmp = t_2;
	elseif (t_m <= 1.62e-187)
		tmp = 1.0;
	elseif (t_m <= 5.6e-179)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * (sqrt(2.0) / (l_m * sqrt((2.0 / x))));
	tmp = 0.0;
	if (t_m <= 6.4e-223)
		tmp = t_2;
	elseif (t_m <= 1.62e-187)
		tmp = 1.0;
	elseif (t_m <= 5.6e-179)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.4e-223], t$95$2, If[LessEqual[t$95$m, 1.62e-187], 1.0, If[LessEqual[t$95$m, 5.6e-179], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.4000000000000001e-223 or 1.6200000000000001e-187 < t < 5.6000000000000001e-179

   1. Initial program 31.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 11.9%

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

      2. sub-neg 11.9%

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

      3. metadata-eval 11.9%

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

      4. +-commutative 11.9%

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

      5. sub-neg 11.9%

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

      6. metadata-eval 11.9%

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

      7. +-commutative 11.9%

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

   7. Simplified 11.9%

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

   8. Taylor expanded in x around inf 22.0%

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

      1. associate-*r/ 22.0%

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

      2. *-commutative 22.0%

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

      3. sqrt-div 22.0%

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

      4. metadata-eval 22.0%

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

      5. un-div-inv 22.0%

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

   10. Applied egg-rr 22.0%

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

       1. associate-/l* 22.1%

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

       2. *-commutative 22.1%

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

       3. associate-/l* 22.0%

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

       4. sqrt-undiv 22.1%

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

   12. Applied egg-rr 22.1%

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

   if 6.4000000000000001e-223 < t < 1.6200000000000001e-187

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 5.6000000000000001e-179 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-223}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := l\_m \cdot \sqrt{\frac{2}{x}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-226}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{t\_2}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.35 \cdot 10^{-178}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* l_m (sqrt (/ 2.0 x)))))
   (*
    t_s
    (if (<= t_m 9.5e-226)
      (* t_m (/ (sqrt 2.0) t_2))
      (if (<= t_m 1.2e-186)
        1.0
        (if (<= t_m 2.35e-178)
          (/ (* t_m (sqrt 2.0)) t_2)
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = l_m * sqrt((2.0 / x));
	double tmp;
	if (t_m <= 9.5e-226) {
		tmp = t_m * (sqrt(2.0) / t_2);
	} else if (t_m <= 1.2e-186) {
		tmp = 1.0;
	} else if (t_m <= 2.35e-178) {
		tmp = (t_m * sqrt(2.0)) / t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l_m * sqrt((2.0d0 / x))
    if (t_m <= 9.5d-226) then
        tmp = t_m * (sqrt(2.0d0) / t_2)
    else if (t_m <= 1.2d-186) then
        tmp = 1.0d0
    else if (t_m <= 2.35d-178) then
        tmp = (t_m * sqrt(2.0d0)) / t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = l_m * Math.sqrt((2.0 / x));
	double tmp;
	if (t_m <= 9.5e-226) {
		tmp = t_m * (Math.sqrt(2.0) / t_2);
	} else if (t_m <= 1.2e-186) {
		tmp = 1.0;
	} else if (t_m <= 2.35e-178) {
		tmp = (t_m * Math.sqrt(2.0)) / t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = l_m * math.sqrt((2.0 / x))
	tmp = 0
	if t_m <= 9.5e-226:
		tmp = t_m * (math.sqrt(2.0) / t_2)
	elif t_m <= 1.2e-186:
		tmp = 1.0
	elif t_m <= 2.35e-178:
		tmp = (t_m * math.sqrt(2.0)) / t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(l_m * sqrt(Float64(2.0 / x)))
	tmp = 0.0
	if (t_m <= 9.5e-226)
		tmp = Float64(t_m * Float64(sqrt(2.0) / t_2));
	elseif (t_m <= 1.2e-186)
		tmp = 1.0;
	elseif (t_m <= 2.35e-178)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / t_2);
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = l_m * sqrt((2.0 / x));
	tmp = 0.0;
	if (t_m <= 9.5e-226)
		tmp = t_m * (sqrt(2.0) / t_2);
	elseif (t_m <= 1.2e-186)
		tmp = 1.0;
	elseif (t_m <= 2.35e-178)
		tmp = (t_m * sqrt(2.0)) / t_2;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-226], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e-186], 1.0, If[LessEqual[t$95$m, 2.35e-178], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 9.5000000000000007e-226

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. associate-*r/ 20.2%

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

      2. *-commutative 20.2%

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

      3. sqrt-div 20.1%

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

      4. metadata-eval 20.1%

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

      5. un-div-inv 20.2%

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

   10. Applied egg-rr 20.2%

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

       1. associate-/l* 20.2%

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

       2. *-commutative 20.2%

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

       3. associate-/l* 20.2%

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

       4. sqrt-undiv 20.2%

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

   12. Applied egg-rr 20.2%

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

   if 9.5000000000000007e-226 < t < 1.20000000000000002e-186

   1. Initial program 3.0%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in t around inf 79.0%

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

   6. Taylor expanded in x around inf 79.0%

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

   if 1.20000000000000002e-186 < t < 2.35e-178

   1. Initial program 0.7%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 65.5%

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

      2. sub-neg 65.5%

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

      3. metadata-eval 65.5%

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

      4. +-commutative 65.5%

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

      5. sub-neg 65.5%

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

      6. metadata-eval 65.5%

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

      7. +-commutative 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. add-log-exp 59.6%

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

      2. sqrt-div 59.6%

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

      3. metadata-eval 59.6%

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

      4. un-div-inv 59.6%

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

   10. Applied egg-rr 59.6%

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

       1. rem-log-exp 99.0%

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

       2. associate-*r/ 99.5%

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

       3. associate-/l* 99.5%

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

       4. sqrt-undiv 99.5%

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

   12. Applied egg-rr 99.5%

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

   if 2.35e-178 < t

   1. Initial program 46.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in t around inf 88.3%

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

   6. Taylor expanded in t around 0 88.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-178}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\frac{t\_m}{l\_m}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.7e-228)
    (* (sqrt 2.0) (/ (/ t_m l_m) (sqrt (/ 2.0 x))))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.7e-228) {
		tmp = sqrt(2.0) * ((t_m / l_m) / sqrt((2.0 / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.7d-228) then
        tmp = sqrt(2.0d0) * ((t_m / l_m) / sqrt((2.0d0 / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.7e-228) {
		tmp = Math.sqrt(2.0) * ((t_m / l_m) / Math.sqrt((2.0 / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 3.7e-228:
		tmp = math.sqrt(2.0) * ((t_m / l_m) / math.sqrt((2.0 / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 3.7e-228)
		tmp = Float64(sqrt(2.0) * Float64(Float64(t_m / l_m) / sqrt(Float64(2.0 / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 3.7e-228)
		tmp = sqrt(2.0) * ((t_m / l_m) / sqrt((2.0 / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.7e-228], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.7e-228

   1. Initial program 31.9%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in l around inf 1.9%

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

      1. associate--l+ 10.6%

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

      2. sub-neg 10.6%

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

      3. metadata-eval 10.6%

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

      4. +-commutative 10.6%

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

      5. sub-neg 10.6%

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

      6. metadata-eval 10.6%

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

      7. +-commutative 10.6%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in x around inf 20.1%

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

      1. add-log-exp 8.9%

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

      2. sqrt-div 8.9%

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

      3. metadata-eval 8.9%

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

      4. un-div-inv 8.9%

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

   10. Applied egg-rr 8.9%

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

       1. rem-log-exp 20.2%

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

       2. associate-/l* 20.1%

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

       3. associate-/r* 18.0%

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

       4. sqrt-undiv 18.1%

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

   12. Applied egg-rr 18.1%

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

   if 3.7e-228 < t

   1. Initial program 42.3%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in t around inf 85.8%

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

   6. Taylor expanded in t around 0 85.9%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\frac{t}{\ell}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * sqrt(((x + -1.0) / (x + 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 * sqrt(((x + (-1.0d0)) / (x + 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 * Math.sqrt(((x + -1.0) / (x + 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 * math.sqrt(((x + -1.0) / (x + 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 * sqrt(Float64(Float64(x + -1.0) / Float64(x + 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 * sqrt(((x + -1.0) / (x + 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 * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 46.3%

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

6. Taylor expanded in t around 0 46.4%

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

7. Final simplification 46.4%

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

Alternative 10: 76.0% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 46.3%

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

6. Taylor expanded in x around -inf 0.0%

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

8. Simplified 46.2%

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

9. Final simplification 46.2%

   \[\leadsto 1 + \frac{-1 - \frac{-0.5}{x}}{x} \]
10. Add Preprocessing

Alternative 11: 75.7% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 46.3%

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

6. Taylor expanded in x around inf 46.2%

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

7. Final simplification 46.2%

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

Alternative 12: 75.0% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

3. Simplified 31.4%

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

5. Taylor expanded in t around inf 46.3%

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

6. Taylor expanded in x around inf 45.8%

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

7. Final simplification 45.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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)))))