Toniolo and Linder, Equation (7)

Percentage Accurate: 33.1% → 85.4%

Time: 14.3s

Alternatives: 8

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{t\_2}{l\_m \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(2, \frac{t\_m}{\sqrt{2} \cdot x}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{l\_m \cdot l\_m}{\sqrt{2} \cdot \left(t\_m \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \frac{l\_m \cdot l\_m}{x}\right) + \left(\frac{t\_3}{x} - t\_3 \cdot -2\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_2 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))) (t_3 (fma 2.0 (* t_m t_m) (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 8.6e-231)
      (/ t_2 (* l_m (sqrt (/ (+ 2.0 (/ 2.0 x)) x))))
      (if (<= t_m 4.2e-160)
        (/
         t_2
         (fma
          2.0
          (/ t_m (* (sqrt 2.0) x))
          (fma t_m (sqrt 2.0) (/ (* l_m l_m) (* (sqrt 2.0) (* t_m x))))))
        (if (<= t_m 9.6e+14)
          (/
           t_2
           (sqrt
            (fma
             2.0
             (* t_m t_m)
             (/
              (+
               (fma 2.0 (/ (* t_m t_m) x) (/ (* l_m l_m) x))
               (- (/ t_3 x) (* t_3 -2.0)))
              x))))
          (/ t_2 (* t_2 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = fma(2.0, (t_m * t_m), (l_m * l_m));
	double tmp;
	if (t_m <= 8.6e-231) {
		tmp = t_2 / (l_m * sqrt(((2.0 + (2.0 / x)) / x)));
	} else if (t_m <= 4.2e-160) {
		tmp = t_2 / fma(2.0, (t_m / (sqrt(2.0) * x)), fma(t_m, sqrt(2.0), ((l_m * l_m) / (sqrt(2.0) * (t_m * x)))));
	} else if (t_m <= 9.6e+14) {
		tmp = t_2 / sqrt(fma(2.0, (t_m * t_m), ((fma(2.0, ((t_m * t_m) / x), ((l_m * l_m) / x)) + ((t_3 / x) - (t_3 * -2.0))) / x)));
	} else {
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 8.6e-231)
		tmp = Float64(t_2 / Float64(l_m * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / x))));
	elseif (t_m <= 4.2e-160)
		tmp = Float64(t_2 / fma(2.0, Float64(t_m / Float64(sqrt(2.0) * x)), fma(t_m, sqrt(2.0), Float64(Float64(l_m * l_m) / Float64(sqrt(2.0) * Float64(t_m * x))))));
	elseif (t_m <= 9.6e+14)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(t_m * t_m), Float64(Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), Float64(Float64(l_m * l_m) / x)) + Float64(Float64(t_3 / x) - Float64(t_3 * -2.0))) / x))));
	else
		tmp = Float64(t_2 / Float64(t_2 * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.6e-231], N[(t$95$2 / N[(l$95$m * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e-160], N[(t$95$2 / N[(2.0 * N[(t$95$m / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.6e+14], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / x), $MachinePrecision] - N[(t$95$3 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$2 * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 8.59999999999999996e-231

   1. Initial program 35.1%

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

   3. Taylor expanded in l around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 3.4

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

   5. Simplified 3.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 22.1%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 23.7

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

   11. Simplified 23.7%

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

   if 8.59999999999999996e-231 < t < 4.2000000000000001e-160

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   5. Simplified 67.3%

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

   6. Taylor expanded in l around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 68.0

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

   8. Simplified 68.0%

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

   if 4.2000000000000001e-160 < t < 9.6e14

   1. Initial program 57.4%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

   5. Simplified 78.9%

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

   if 9.6e14 < t

   1. Initial program 39.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 95.7

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

   5. Simplified 95.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} - \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot -2\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-231}:\\ \;\;\;\;\frac{t\_2}{l\_m \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{elif}\;t\_m \leq 1.22 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(2, \frac{t\_m}{\sqrt{2} \cdot x}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{l\_m \cdot l\_m}{\sqrt{2} \cdot \left(t\_m \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x} + \mathsf{fma}\left(2, t\_m \cdot t\_m + \frac{t\_m \cdot t\_m}{x}, \frac{l\_m \cdot l\_m}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_2 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 2.45e-231)
      (/ t_2 (* l_m (sqrt (/ (+ 2.0 (/ 2.0 x)) x))))
      (if (<= t_m 1.22e-160)
        (/
         t_2
         (fma
          2.0
          (/ t_m (* (sqrt 2.0) x))
          (fma t_m (sqrt 2.0) (/ (* l_m l_m) (* (sqrt 2.0) (* t_m x))))))
        (if (<= t_m 4e+14)
          (/
           t_2
           (sqrt
            (+
             (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x)
             (fma 2.0 (+ (* t_m t_m) (/ (* t_m t_m) x)) (/ (* l_m l_m) x)))))
          (/ t_2 (* t_2 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 2.45e-231) {
		tmp = t_2 / (l_m * sqrt(((2.0 + (2.0 / x)) / x)));
	} else if (t_m <= 1.22e-160) {
		tmp = t_2 / fma(2.0, (t_m / (sqrt(2.0) * x)), fma(t_m, sqrt(2.0), ((l_m * l_m) / (sqrt(2.0) * (t_m * x)))));
	} else if (t_m <= 4e+14) {
		tmp = t_2 / sqrt(((fma(2.0, (t_m * t_m), (l_m * l_m)) / x) + fma(2.0, ((t_m * t_m) + ((t_m * t_m) / x)), ((l_m * l_m) / x))));
	} else {
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 2.45e-231)
		tmp = Float64(t_2 / Float64(l_m * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / x))));
	elseif (t_m <= 1.22e-160)
		tmp = Float64(t_2 / fma(2.0, Float64(t_m / Float64(sqrt(2.0) * x)), fma(t_m, sqrt(2.0), Float64(Float64(l_m * l_m) / Float64(sqrt(2.0) * Float64(t_m * x))))));
	elseif (t_m <= 4e+14)
		tmp = Float64(t_2 / sqrt(Float64(Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x) + fma(2.0, Float64(Float64(t_m * t_m) + Float64(Float64(t_m * t_m) / x)), Float64(Float64(l_m * l_m) / x)))));
	else
		tmp = Float64(t_2 / Float64(t_2 * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.45e-231], N[(t$95$2 / N[(l$95$m * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.22e-160], N[(t$95$2 / N[(2.0 * N[(t$95$m / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+14], N[(t$95$2 / N[Sqrt[N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$2 * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 2.45000000000000002e-231

   1. Initial program 35.1%

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

   3. Taylor expanded in l around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 3.4

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

   5. Simplified 3.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 22.1%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 23.7

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

   11. Simplified 23.7%

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

   if 2.45000000000000002e-231 < t < 1.22000000000000003e-160

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   5. Simplified 67.3%

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

   6. Taylor expanded in l around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 68.0

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

   8. Simplified 68.0%

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

   if 1.22000000000000003e-160 < t < 4e14

   1. Initial program 57.4%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 78.6%

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

   if 4e14 < t

   1. Initial program 39.5%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 95.7

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

   5. Simplified 95.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-231}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(2, t \cdot t + \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.05 \cdot 10^{+163}:\\ \;\;\;\;\frac{t\_2}{t\_2 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= l_m 1.05e+163)
      (/ t_2 (* t_2 (sqrt (/ (+ x 1.0) (+ x -1.0)))))
      (* t_m (/ (sqrt x) l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (l_m <= 1.05e+163) {
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    if (l_m <= 1.05d+163) then
        tmp = t_2 / (t_2 * sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double tmp;
	if (l_m <= 1.05e+163) {
		tmp = t_2 / (t_2 * Math.sqrt(((x + 1.0) / (x + -1.0))));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	tmp = 0
	if l_m <= 1.05e+163:
		tmp = t_2 / (t_2 * math.sqrt(((x + 1.0) / (x + -1.0))))
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (l_m <= 1.05e+163)
		tmp = Float64(t_2 / Float64(t_2 * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (l_m <= 1.05e+163)
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 1.05e+163], N[(t$95$2 / N[(t$95$2 * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.05e163

   1. Initial program 42.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 42.8

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

   5. Simplified 42.8%

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

   if 1.05e163 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 18.7%

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

      1. lift-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

   10. Applied egg-rr 18.7%

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

   11. Taylor expanded in x around inf

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 79.7

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

   13. Simplified 79.7%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+163}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.15 \cdot 10^{+163}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.15e+163)
    (/ 1.0 (sqrt (/ (+ x 1.0) (+ x -1.0))))
    (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.15e+163) {
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 1.15d+163) then
        tmp = 1.0d0 / sqrt(((x + 1.0d0) / (x + (-1.0d0))))
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.15e+163) {
		tmp = 1.0 / Math.sqrt(((x + 1.0) / (x + -1.0)));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 1.15e+163:
		tmp = 1.0 / math.sqrt(((x + 1.0) / (x + -1.0)))
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 1.15e+163)
		tmp = Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 1.15e+163)
		tmp = 1.0 / sqrt(((x + 1.0) / (x + -1.0)));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.15e+163], N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.15000000000000001e163

   1. Initial program 42.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 42.8

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

   5. Simplified 42.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-inverses N/A

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

      14. lower-/.f64 42.8

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

   7. Applied egg-rr 42.8%

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

   if 1.15000000000000001e163 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 18.7%

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

      1. lift-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

   10. Applied egg-rr 18.7%

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

   11. Taylor expanded in x around inf

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 79.7

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

   13. Simplified 79.7%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+163}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.5 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 3.5e+160)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3.5e+160) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 3.5d+160) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3.5e+160) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 3.5e+160:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 3.5e+160)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 3.5e+160)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 3.5e+160], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.50000000000000026e160

   1. Initial program 42.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 42.8

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

   5. Simplified 42.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 42.8

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

   8. Simplified 42.8%

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

   if 3.50000000000000026e160 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 18.7%

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

      1. lift-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

   10. Applied egg-rr 18.7%

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

   11. Taylor expanded in x around inf

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 79.7

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

   13. Simplified 79.7%

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

Alternative 6: 80.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 6.4 \cdot 10^{+164}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (if (<= l_m 6.4e+164) (+ (/ -1.0 x) 1.0) (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 6.4e+164) {
		tmp = (-1.0 / x) + 1.0;
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 6.4d+164) then
        tmp = ((-1.0d0) / x) + 1.0d0
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 6.4e+164) {
		tmp = (-1.0 / x) + 1.0;
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 6.4e+164:
		tmp = (-1.0 / x) + 1.0
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 6.4e+164)
		tmp = Float64(Float64(-1.0 / x) + 1.0);
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 6.4e+164)
		tmp = (-1.0 / x) + 1.0;
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 6.4e+164], N[(N[(-1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 6.3999999999999996e164

   1. Initial program 42.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 42.8

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 42.6

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

   8. Simplified 42.6%

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

   if 6.3999999999999996e164 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 0.0

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 18.7%

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

      1. lift-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

   10. Applied egg-rr 18.7%

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

   11. Taylor expanded in x around inf

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

       1. associate-*l/ N/A

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

       2. *-lft-identity N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 79.7

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

   13. Simplified 79.7%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{+164}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f64 40.4

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

5. Simplified 40.4%

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

6. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 40.2

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

8. Simplified 40.2%

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

9. Final simplification 40.2%

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

Alternative 8: 75.5% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 39.2

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{0.5}} \]

5. Simplified 39.2%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval 39.7

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

7. Applied egg-rr 39.7%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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)))))