Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 82.9%

Time: 26.8s

Alternatives: 10

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: 33.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {\ell}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.8 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{0.5 \cdot \frac{2 \cdot \left({t_m}^{2} + {t_m}^{2}\right) + t_2}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\mathsf{fma}\left(2, \frac{{t_m}^{2}}{\frac{-1 + x}{x + 1}}, \frac{t_2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow l 2.0))))
   (*
    t_s
    (if (<= t_m 5.8e-218)
      (*
       (sqrt 2.0)
       (/
        t_m
        (+
         (*
          0.5
          (/
           (+ (* 2.0 (+ (pow t_m 2.0) (pow t_m 2.0))) t_2)
           (* t_m (* (sqrt 2.0) x))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 5.6e-10)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (fma 2.0 (/ (pow t_m 2.0) (/ (+ -1.0 x) (+ x 1.0))) (/ t_2 x)))))
        (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(l, 2.0);
	double tmp;
	if (t_m <= 5.8e-218) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((2.0 * (pow(t_m, 2.0) + pow(t_m, 2.0))) + t_2) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 5.6e-10) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) / ((-1.0 + x) / (x + 1.0))), (t_2 / x))));
	} else {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * (l ^ 2.0))
	tmp = 0.0
	if (t_m <= 5.8e-218)
		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))) + t_2) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 5.6e-10)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) / Float64(Float64(-1.0 + x) / Float64(x + 1.0))), Float64(t_2 / x)))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e-218], 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] + t$95$2), $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, 5.6e-10], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.8000000000000004e-218

   1. Initial program 32.7%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in l around 0 28.2%

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

      1. fma-def 28.2%

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

      2. associate-/l* 40.7%

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

      3. +-commutative 40.7%

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

      4. sub-neg 40.7%

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

      5. metadata-eval 40.7%

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

      6. +-commutative 40.7%

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

      7. associate--l+ 51.3%

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

      8. sub-neg 51.3%

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

      9. metadata-eval 51.3%

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

      10. +-commutative 51.3%

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

      11. sub-neg 51.3%

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

      12. metadata-eval 51.3%

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

      13. +-commutative 51.3%

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

   6. Simplified 51.3%

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

   7. Taylor expanded in x around inf 14.1%

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

   if 5.8000000000000004e-218 < t < 5.60000000000000031e-10

   1. Initial program 35.7%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in l around 0 47.6%

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

      1. fma-def 47.6%

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

      2. associate-/l* 47.6%

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

      3. +-commutative 47.6%

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

      4. sub-neg 47.6%

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

      5. metadata-eval 47.6%

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

      6. +-commutative 47.6%

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

      7. associate--l+ 60.3%

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

      8. sub-neg 60.3%

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

      9. metadata-eval 60.3%

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

      10. +-commutative 60.3%

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

      11. sub-neg 60.3%

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

      12. metadata-eval 60.3%

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

      13. +-commutative 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in x around inf 85.0%

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

      1. associate-*r/ 85.0%

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

   9. Simplified 85.0%

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

   if 5.60000000000000031e-10 < t

   1. Initial program 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in l around 0 98.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

      4. +-commutative 98.9%

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

   6. Simplified 98.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 98.9%

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

      2. times-frac 98.9%

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

      3. +-commutative 98.9%

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

   8. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 98.9%

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

      2. lft-mult-inverse 98.9%

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

      3. +-commutative 98.9%

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

   10. Simplified 98.9%

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

       1. expm1-log1p-u 98.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)\right)} \]

       2. expm1-udef 98.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1} \]

       3. inv-pow 98.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{x + 1}{-1 + x}}\right)}^{-1}}\right)} - 1 \]

       4. sqrt-pow2 98.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

       5. +-commutative 98.9%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{1 + x}}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       6. +-commutative 98.9%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{\color{blue}{x + -1}}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       7. metadata-eval 98.9%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

   12. Applied egg-rr 98.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 98.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)\right)} \]

       2. expm1-log1p 98.9%

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

       3. +-commutative 98.9%

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

       4. +-commutative 98.9%

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

   14. Simplified 98.9%

       \[\leadsto \color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{2 \cdot \left({t}^{2} + {t}^{2}\right) + 2 \cdot {\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{-1 + x}{x + 1}}, \frac{2 \cdot {\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \end{array} \]

Alternative 2: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t_m \cdot \sqrt{2}\\ t_3 := \frac{x + 1}{-1 + x}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+148}:\\ \;\;\;\;{t_3}^{-0.5}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\frac{2 \cdot {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t_2\right) \cdot \sqrt{t_3}, \ell\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))) (t_3 (/ (+ x 1.0) (+ -1.0 x))))
   (*
    t_s
    (if (<= (* l l) 2e+148)
      (pow t_3 -0.5)
      (if (<= (* l l) 5e+307)
        (* (sqrt 2.0) (/ t_m (sqrt (/ (* 2.0 (pow l 2.0)) x))))
        (/ t_2 (hypot (* (hypot l t_2) (sqrt t_3)) l)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = (x + 1.0) / (-1.0 + x);
	double tmp;
	if ((l * l) <= 2e+148) {
		tmp = pow(t_3, -0.5);
	} else if ((l * l) <= 5e+307) {
		tmp = sqrt(2.0) * (t_m / sqrt(((2.0 * pow(l, 2.0)) / x)));
	} else {
		tmp = t_2 / hypot((hypot(l, t_2) * sqrt(t_3)), l);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double t_3 = (x + 1.0) / (-1.0 + x);
	double tmp;
	if ((l * l) <= 2e+148) {
		tmp = Math.pow(t_3, -0.5);
	} else if ((l * l) <= 5e+307) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt(((2.0 * Math.pow(l, 2.0)) / x)));
	} else {
		tmp = t_2 / Math.hypot((Math.hypot(l, t_2) * Math.sqrt(t_3)), l);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = t_m * math.sqrt(2.0)
	t_3 = (x + 1.0) / (-1.0 + x)
	tmp = 0
	if (l * l) <= 2e+148:
		tmp = math.pow(t_3, -0.5)
	elif (l * l) <= 5e+307:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt(((2.0 * math.pow(l, 2.0)) / x)))
	else:
		tmp = t_2 / math.hypot((math.hypot(l, t_2) * math.sqrt(t_3)), l)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(Float64(x + 1.0) / Float64(-1.0 + x))
	tmp = 0.0
	if (Float64(l * l) <= 2e+148)
		tmp = t_3 ^ -0.5;
	elseif (Float64(l * l) <= 5e+307)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(2.0 * (l ^ 2.0)) / x))));
	else
		tmp = Float64(t_2 / hypot(Float64(hypot(l, t_2) * sqrt(t_3)), l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = t_m * sqrt(2.0);
	t_3 = (x + 1.0) / (-1.0 + x);
	tmp = 0.0;
	if ((l * l) <= 2e+148)
		tmp = t_3 ^ -0.5;
	elseif ((l * l) <= 5e+307)
		tmp = sqrt(2.0) * (t_m / sqrt(((2.0 * (l ^ 2.0)) / x)));
	else
		tmp = t_2 / hypot((hypot(l, t_2) * sqrt(t_3)), l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e+148], N[Power[t$95$3, -0.5], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+307], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[Sqrt[N[(N[Sqrt[l ^ 2 + t$95$2 ^ 2], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] ^ 2 + l ^ 2], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 2.0000000000000001e148

   1. Initial program 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in l around 0 49.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 49.6%

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

      2. sub-neg 49.6%

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

      3. metadata-eval 49.6%

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

      4. +-commutative 49.6%

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

   6. Simplified 49.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 49.6%

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

      2. times-frac 49.6%

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

      3. +-commutative 49.6%

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

   8. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 49.6%

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

      2. lft-mult-inverse 49.6%

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

      3. +-commutative 49.6%

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

   10. Simplified 49.6%

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

       1. expm1-log1p-u 49.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)\right)} \]

       2. expm1-udef 49.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1} \]

       3. inv-pow 49.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{x + 1}{-1 + x}}\right)}^{-1}}\right)} - 1 \]

       4. sqrt-pow2 49.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

       5. +-commutative 49.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{1 + x}}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       6. +-commutative 49.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{\color{blue}{x + -1}}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       7. metadata-eval 49.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

   12. Applied egg-rr 49.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 49.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)\right)} \]

       2. expm1-log1p 49.6%

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

       3. +-commutative 49.6%

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

       4. +-commutative 49.6%

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

   14. Simplified 49.6%

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

   if 2.0000000000000001e148 < (*.f64 l l) < 5e307

   1. Initial program 12.9%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in l around inf 2.6%

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

      1. associate--l+ 16.1%

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

      2. sub-neg 16.1%

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

      3. metadata-eval 16.1%

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

      4. +-commutative 16.1%

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

      5. sub-neg 16.1%

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

      6. metadata-eval 16.1%

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

      7. +-commutative 16.1%

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

   6. Simplified 16.1%

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

   7. Taylor expanded in x around inf 61.2%

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

      1. associate-*r/ 85.5%

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

   9. Simplified 61.2%

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

   if 5e307 < (*.f64 l l)

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Applied egg-rr 61.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+148}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}, \ell\right)}\\ \end{array} \]

Alternative 3: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\mathsf{fma}\left(2, \frac{{t_m}^{2}}{\frac{-1 + x}{x + 1}}, \frac{2 \cdot {\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.5e-10)
    (*
     (sqrt 2.0)
     (/
      t_m
      (sqrt
       (fma
        2.0
        (/ (pow t_m 2.0) (/ (+ -1.0 x) (+ x 1.0)))
        (/ (* 2.0 (pow l 2.0)) x)))))
    (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 6.5e-10) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) / ((-1.0 + x) / (x + 1.0))), ((2.0 * pow(l, 2.0)) / x))));
	} else {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 6.5e-10)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) / Float64(Float64(-1.0 + x) / Float64(x + 1.0))), Float64(Float64(2.0 * (l ^ 2.0)) / x)))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.5e-10], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.5000000000000003e-10

   1. Initial program 33.5%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in l around 0 33.2%

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

      1. fma-def 33.2%

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

      2. associate-/l* 42.5%

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

      3. +-commutative 42.5%

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

      4. sub-neg 42.5%

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

      5. metadata-eval 42.5%

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

      6. +-commutative 42.5%

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

      7. associate--l+ 53.6%

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

      8. sub-neg 53.6%

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

      9. metadata-eval 53.6%

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

      10. +-commutative 53.6%

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

      11. sub-neg 53.6%

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

      12. metadata-eval 53.6%

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

      13. +-commutative 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in x around inf 66.5%

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

      1. associate-*r/ 66.5%

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

   9. Simplified 66.5%

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

   if 6.5000000000000003e-10 < t

   1. Initial program 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in l around 0 98.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

      4. +-commutative 98.9%

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

   6. Simplified 98.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 98.9%

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

      2. times-frac 98.9%

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

      3. +-commutative 98.9%

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

   8. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 98.9%

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

      2. lft-mult-inverse 98.9%

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

      3. +-commutative 98.9%

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

   10. Simplified 98.9%

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

       1. expm1-log1p-u 98.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)\right)} \]

       2. expm1-udef 98.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1} \]

       3. inv-pow 98.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{x + 1}{-1 + x}}\right)}^{-1}}\right)} - 1 \]

       4. sqrt-pow2 98.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

       5. +-commutative 98.9%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{1 + x}}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       6. +-commutative 98.9%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{\color{blue}{x + -1}}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       7. metadata-eval 98.9%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

   12. Applied egg-rr 98.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 98.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)\right)} \]

       2. expm1-log1p 98.9%

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

       3. +-commutative 98.9%

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

       4. +-commutative 98.9%

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

   14. Simplified 98.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{-1 + x}{x + 1}}, \frac{2 \cdot {\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \end{array} \]

Alternative 4: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.3 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\frac{2 \cdot {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-156)
    (* (sqrt 2.0) (/ t_m (sqrt (/ (* 2.0 (pow l 2.0)) x))))
    (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 2.3e-156) {
		tmp = sqrt(2.0) * (t_m / sqrt(((2.0 * pow(l, 2.0)) / x)));
	} else {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.3d-156) then
        tmp = sqrt(2.0d0) * (t_m / sqrt(((2.0d0 * (l ** 2.0d0)) / x)))
    else
        tmp = ((x + 1.0d0) / ((-1.0d0) + x)) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 2.3e-156) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt(((2.0 * Math.pow(l, 2.0)) / x)));
	} else {
		tmp = Math.pow(((x + 1.0) / (-1.0 + x)), -0.5);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 2.3e-156:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt(((2.0 * math.pow(l, 2.0)) / x)))
	else:
		tmp = math.pow(((x + 1.0) / (-1.0 + x)), -0.5)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 2.3e-156)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(2.0 * (l ^ 2.0)) / x))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 2.3e-156)
		tmp = sqrt(2.0) * (t_m / sqrt(((2.0 * (l ^ 2.0)) / x)));
	else
		tmp = ((x + 1.0) / (-1.0 + x)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-156], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.3e-156

   1. Initial program 30.5%

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

   3. Simplified 30.5%

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

   4. Taylor expanded in l around inf 2.3%

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

      1. associate--l+ 12.9%

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

      2. sub-neg 12.9%

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

      3. metadata-eval 12.9%

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

      4. +-commutative 12.9%

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

      5. sub-neg 12.9%

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

      6. metadata-eval 12.9%

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

      7. +-commutative 12.9%

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

   6. Simplified 12.9%

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

   7. Taylor expanded in x around inf 23.6%

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

      1. associate-*r/ 60.2%

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

   9. Simplified 23.6%

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

   if 2.3e-156 < t

   1. Initial program 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in l around 0 85.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 85.6%

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

      2. sub-neg 85.6%

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

      3. metadata-eval 85.6%

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

      4. +-commutative 85.6%

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

   6. Simplified 85.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 85.6%

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

      2. times-frac 85.7%

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

      3. +-commutative 85.7%

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

   8. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 85.7%

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

      2. lft-mult-inverse 85.7%

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

      3. +-commutative 85.7%

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

   10. Simplified 85.7%

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

       1. expm1-log1p-u 85.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)\right)} \]

       2. expm1-udef 85.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1} \]

       3. inv-pow 85.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{x + 1}{-1 + x}}\right)}^{-1}}\right)} - 1 \]

       4. sqrt-pow2 85.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

       5. +-commutative 85.7%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{1 + x}}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       6. +-commutative 85.7%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{\color{blue}{x + -1}}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       7. metadata-eval 85.7%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

   12. Applied egg-rr 85.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 85.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)\right)} \]

       2. expm1-log1p 85.7%

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

       3. +-commutative 85.7%

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

       4. +-commutative 85.7%

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

   14. Simplified 85.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \end{array} \]

Alternative 5: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+77}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+99} \lor \neg \left(\ell \leq 2.1 \cdot 10^{+231}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell}{t_m} \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 5.8e+77)
    (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5)
    (if (or (<= l 2.6e+99) (not (<= l 2.1e+231)))
      (/ (sqrt 2.0) (* (/ l t_m) (sqrt (/ 2.0 x))))
      (sqrt (/ (+ -1.0 x) (+ x 1.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 5.8e+77) {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else if ((l <= 2.6e+99) || !(l <= 2.1e+231)) {
		tmp = sqrt(2.0) / ((l / t_m) * sqrt((2.0 / x)));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 5.8d+77) then
        tmp = ((x + 1.0d0) / ((-1.0d0) + x)) ** (-0.5d0)
    else if ((l <= 2.6d+99) .or. (.not. (l <= 2.1d+231))) then
        tmp = sqrt(2.0d0) / ((l / t_m) * sqrt((2.0d0 / x)))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 5.8e+77) {
		tmp = Math.pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else if ((l <= 2.6e+99) || !(l <= 2.1e+231)) {
		tmp = Math.sqrt(2.0) / ((l / t_m) * Math.sqrt((2.0 / x)));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 5.8e+77:
		tmp = math.pow(((x + 1.0) / (-1.0 + x)), -0.5)
	elif (l <= 2.6e+99) or not (l <= 2.1e+231):
		tmp = math.sqrt(2.0) / ((l / t_m) * math.sqrt((2.0 / x)))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 5.8e+77)
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	elseif ((l <= 2.6e+99) || !(l <= 2.1e+231))
		tmp = Float64(sqrt(2.0) / Float64(Float64(l / t_m) * sqrt(Float64(2.0 / x))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 5.8e+77)
		tmp = ((x + 1.0) / (-1.0 + x)) ^ -0.5;
	elseif ((l <= 2.6e+99) || ~((l <= 2.1e+231)))
		tmp = sqrt(2.0) / ((l / t_m) * sqrt((2.0 / x)));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 5.8e+77], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], If[Or[LessEqual[l, 2.6e+99], N[Not[LessEqual[l, 2.1e+231]], $MachinePrecision]], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 5.8000000000000003e77

   1. Initial program 43.7%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in l around 0 47.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 47.6%

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

      2. sub-neg 47.6%

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

      3. metadata-eval 47.6%

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

      4. +-commutative 47.6%

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

   6. Simplified 47.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 47.6%

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

      2. times-frac 47.6%

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

      3. +-commutative 47.6%

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

   8. Applied egg-rr 47.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 47.6%

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

      2. lft-mult-inverse 47.6%

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

      3. +-commutative 47.6%

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

   10. Simplified 47.6%

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

       1. expm1-log1p-u 47.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)\right)} \]

       2. expm1-udef 47.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1} \]

       3. inv-pow 47.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{x + 1}{-1 + x}}\right)}^{-1}}\right)} - 1 \]

       4. sqrt-pow2 47.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

       5. +-commutative 47.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{1 + x}}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       6. +-commutative 47.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{\color{blue}{x + -1}}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       7. metadata-eval 47.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

   12. Applied egg-rr 47.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 47.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)\right)} \]

       2. expm1-log1p 47.6%

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

       3. +-commutative 47.6%

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

       4. +-commutative 47.6%

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

   14. Simplified 47.6%

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

   if 5.8000000000000003e77 < l < 2.6e99 or 2.09999999999999984e231 < l

   1. Initial program 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 31.7%

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

      3. sub-neg 31.7%

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

      4. metadata-eval 31.7%

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

      5. +-commutative 31.7%

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

      6. sub-neg 31.7%

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

      7. metadata-eval 31.7%

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

      8. +-commutative 31.7%

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

   6. Simplified 31.7%

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

   7. Taylor expanded in l around 0 1.5%

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

      1. associate--l+ 31.7%

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

      2. sub-neg 31.7%

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

      3. metadata-eval 31.7%

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

      4. sub-neg 31.7%

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

      5. metadata-eval 31.7%

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

      6. sub-neg 31.7%

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

      7. metadata-eval 31.7%

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

      8. +-commutative 31.7%

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

      9. associate-+l+ 1.5%

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

      10. +-commutative 1.5%

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

      11. +-commutative 1.5%

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

   9. Simplified 1.5%

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

   10. Taylor expanded in x around inf 69.4%

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

   if 2.6e99 < l < 2.09999999999999984e231

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   4. Taylor expanded in l around 0 27.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 27.8%

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

      2. sub-neg 27.8%

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

      3. metadata-eval 27.8%

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

      4. +-commutative 27.8%

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

   6. Simplified 27.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
   7. Step-by-step derivation

      1. associate-/r* 27.8%

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

      2. sqrt-undiv 27.8%

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

      3. metadata-eval 27.8%

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

      4. metadata-eval 27.8%

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

      5. metadata-eval 27.8%

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

      6. sqrt-div 27.8%

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

      7. clear-num 27.8%

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

      8. +-commutative 27.8%

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

   8. Applied egg-rr 27.8%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+77}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+99} \lor \neg \left(\ell \leq 2.1 \cdot 10^{+231}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell}{t} \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 6: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.66 \cdot 10^{+233}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{2}{\frac{1}{-1 + x} + \left(-1 + \frac{x}{-1 + x}\right)}}}{\frac{\ell}{t_m}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 1.66e+233)
    (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5)
    (/
     (sqrt (/ 2.0 (+ (/ 1.0 (+ -1.0 x)) (+ -1.0 (/ x (+ -1.0 x))))))
     (/ l t_m)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.66e+233) {
		tmp = pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = sqrt((2.0 / ((1.0 / (-1.0 + x)) + (-1.0 + (x / (-1.0 + x)))))) / (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 1.66d+233) then
        tmp = ((x + 1.0d0) / ((-1.0d0) + x)) ** (-0.5d0)
    else
        tmp = sqrt((2.0d0 / ((1.0d0 / ((-1.0d0) + x)) + ((-1.0d0) + (x / ((-1.0d0) + x)))))) / (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.66e+233) {
		tmp = Math.pow(((x + 1.0) / (-1.0 + x)), -0.5);
	} else {
		tmp = Math.sqrt((2.0 / ((1.0 / (-1.0 + x)) + (-1.0 + (x / (-1.0 + x)))))) / (l / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 1.66e+233:
		tmp = math.pow(((x + 1.0) / (-1.0 + x)), -0.5)
	else:
		tmp = math.sqrt((2.0 / ((1.0 / (-1.0 + x)) + (-1.0 + (x / (-1.0 + x)))))) / (l / t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 1.66e+233)
		tmp = Float64(Float64(x + 1.0) / Float64(-1.0 + x)) ^ -0.5;
	else
		tmp = Float64(sqrt(Float64(2.0 / Float64(Float64(1.0 / Float64(-1.0 + x)) + Float64(-1.0 + Float64(x / Float64(-1.0 + x)))))) / Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 1.66e+233)
		tmp = ((x + 1.0) / (-1.0 + x)) ^ -0.5;
	else
		tmp = sqrt((2.0 / ((1.0 / (-1.0 + x)) + (-1.0 + (x / (-1.0 + x)))))) / (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 1.66e+233], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(N[Sqrt[N[(2.0 / N[(N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(x / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.66e233

   1. Initial program 38.2%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in l around 0 44.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   5. Step-by-step derivation

      1. +-commutative 44.5%

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

      2. sub-neg 44.5%

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

      3. metadata-eval 44.5%

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

      4. +-commutative 44.5%

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

   6. Simplified 44.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 44.5%

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

      2. times-frac 44.5%

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

      3. +-commutative 44.5%

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

   8. Applied egg-rr 44.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 44.5%

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

      2. lft-mult-inverse 44.5%

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

      3. +-commutative 44.5%

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

   10. Simplified 44.5%

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

       1. expm1-log1p-u 44.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)\right)} \]

       2. expm1-udef 44.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1} \]

       3. inv-pow 44.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{x + 1}{-1 + x}}\right)}^{-1}}\right)} - 1 \]

       4. sqrt-pow2 44.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

       5. +-commutative 44.5%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{1 + x}}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       6. +-commutative 44.5%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{\color{blue}{x + -1}}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

       7. metadata-eval 44.5%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

   12. Applied egg-rr 44.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 44.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)\right)} \]

       2. expm1-log1p 44.5%

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

       3. +-commutative 44.5%

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

       4. +-commutative 44.5%

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

   14. Simplified 44.5%

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

   if 1.66e233 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in l around inf 1.1%

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

      1. *-commutative 1.1%

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

      2. associate--l+ 38.9%

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

      3. sub-neg 38.9%

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

      4. metadata-eval 38.9%

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

      5. +-commutative 38.9%

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

      6. sub-neg 38.9%

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

      7. metadata-eval 38.9%

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

      8. +-commutative 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in l around 0 1.1%

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

      1. associate--l+ 38.9%

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

      2. sub-neg 38.9%

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

      3. metadata-eval 38.9%

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

      4. sub-neg 38.9%

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

      5. metadata-eval 38.9%

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

      6. sub-neg 38.9%

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

      7. metadata-eval 38.9%

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

      8. +-commutative 38.9%

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

      9. associate-+l+ 1.1%

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

      10. +-commutative 1.1%

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

      11. +-commutative 1.1%

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

   9. Simplified 1.1%

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

   11. Applied egg-rr 34.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{\frac{2}{\frac{1}{x + -1} + \left(-1 + \frac{x}{x + -1}\right)}}}{\frac{\ell}{t}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 38.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\frac{2}{\frac{1}{x + -1} + \left(-1 + \frac{x}{x + -1}\right)}}}{\frac{\ell}{t}}\right)\right)} \]

       2. expm1-log1p 38.9%

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

       3. +-commutative 38.9%

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

       4. +-commutative 38.9%

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

   13. Simplified 38.9%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.66 \cdot 10^{+233}:\\ \;\;\;\;{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{2}{\frac{1}{-1 + x} + \left(-1 + \frac{x}{-1 + x}\right)}}}{\frac{\ell}{t}}\\ \end{array} \]

Alternative 7: 76.8% accurate, 2.1× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (pow (/ (+ x 1.0) (+ -1.0 x)) -0.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

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

3. Simplified 36.1%

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

4. Taylor expanded in l around 0 43.0%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
5. Step-by-step derivation

   1. +-commutative 43.0%

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

   2. sub-neg 43.0%

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

   3. metadata-eval 43.0%

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

   4. +-commutative 43.0%

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

6. Simplified 43.0%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 43.0%

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

   2. times-frac 43.0%

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

   3. +-commutative 43.0%

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

8. Applied egg-rr 43.0%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}}}} \]
9. Step-by-step derivation

   1. associate-*r/ 43.0%

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

   2. lft-mult-inverse 43.0%

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

   3. +-commutative 43.0%

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

10. Simplified 43.0%

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

    1. expm1-log1p-u 43.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)\right)} \]

    2. expm1-udef 43.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{x + 1}{-1 + x}}}\right)} - 1} \]

    3. inv-pow 43.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{x + 1}{-1 + x}}\right)}^{-1}}\right)} - 1 \]

    4. sqrt-pow2 43.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

    5. +-commutative 43.0%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{1 + x}}{-1 + x}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

    6. +-commutative 43.0%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{\color{blue}{x + -1}}\right)}^{\left(\frac{-1}{2}\right)}\right)} - 1 \]

    7. metadata-eval 43.0%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

12. Applied egg-rr 43.0%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 43.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\right)\right)} \]

    2. expm1-log1p 43.0%

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

    3. +-commutative 43.0%

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

    4. +-commutative 43.0%

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

14. Simplified 43.0%

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

15. Final simplification 43.0%

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

Alternative 8: 76.8% accurate, 2.1× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (sqrt (/ (+ -1.0 x) (+ x 1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Simplified 36.1%

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

4. Taylor expanded in l around 0 43.0%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
5. Step-by-step derivation

   1. +-commutative 43.0%

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

   2. sub-neg 43.0%

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

   3. metadata-eval 43.0%

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

   4. +-commutative 43.0%

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

6. Simplified 43.0%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
7. Step-by-step derivation

   1. associate-/r* 43.0%

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

   2. sqrt-undiv 43.0%

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

   3. metadata-eval 43.0%

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

   4. metadata-eval 43.0%

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

   5. metadata-eval 43.0%

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

   6. sqrt-div 43.0%

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

   7. clear-num 43.0%

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

   8. +-commutative 43.0%

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

8. Applied egg-rr 43.0%

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

9. Final simplification 43.0%

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

Alternative 9: 76.1% accurate, 45.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Simplified 36.1%

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

4. Taylor expanded in l around 0 43.0%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
5. Step-by-step derivation

   1. +-commutative 43.0%

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

   2. sub-neg 43.0%

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

   3. metadata-eval 43.0%

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

   4. +-commutative 43.0%

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

6. Simplified 43.0%

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

7. Taylor expanded in x around inf 43.0%

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

8. Final simplification 43.0%

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

Alternative 10: 75.4% accurate, 225.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m) :precision binary64 (* t_s 1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 36.1%

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

3. Simplified 36.1%

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

4. Taylor expanded in l around 0 43.0%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
5. Step-by-step derivation

   1. +-commutative 43.0%

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

   2. sub-neg 43.0%

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

   3. metadata-eval 43.0%

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

   4. +-commutative 43.0%

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

6. Simplified 43.0%

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

7. Taylor expanded in x around inf 42.9%

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

8. Final simplification 42.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023321 
(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)))))