Toniolo and Linder, Equation (7)

Percentage Accurate: 32.5% → 80.3%

Time: 10.8s

Alternatives: 7

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: 32.5% 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: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+297}:\\ \;\;\;\;{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\frac{\frac{\frac{2}{x} + 2}{x} + 2}{x}} \cdot l\_m} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 2e+297)
    (pow (sqrt (/ (+ 1.0 x) (- x 1.0))) -1.0)
    (*
     (/ t_m (* (sqrt (/ (+ (/ (+ (/ 2.0 x) 2.0) x) 2.0) x)) l_m))
     (sqrt 2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+297) {
		tmp = pow(sqrt(((1.0 + x) / (x - 1.0))), -1.0);
	} else {
		tmp = (t_m / (sqrt((((((2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m)) * sqrt(2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 2d+297) then
        tmp = sqrt(((1.0d0 + x) / (x - 1.0d0))) ** (-1.0d0)
    else
        tmp = (t_m / (sqrt((((((2.0d0 / x) + 2.0d0) / x) + 2.0d0) / x)) * l_m)) * sqrt(2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+297) {
		tmp = Math.pow(Math.sqrt(((1.0 + x) / (x - 1.0))), -1.0);
	} else {
		tmp = (t_m / (Math.sqrt((((((2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m)) * Math.sqrt(2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 2e+297:
		tmp = math.pow(math.sqrt(((1.0 + x) / (x - 1.0))), -1.0)
	else:
		tmp = (t_m / (math.sqrt((((((2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m)) * math.sqrt(2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 2e+297)
		tmp = sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0))) ^ -1.0;
	else
		tmp = Float64(Float64(t_m / Float64(sqrt(Float64(Float64(Float64(Float64(Float64(2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m)) * sqrt(2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 2e+297)
		tmp = sqrt(((1.0 + x) / (x - 1.0))) ^ -1.0;
	else
		tmp = (t_m / (sqrt((((((2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m)) * sqrt(2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+297], N[Power[N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], N[(N[(t$95$m / N[(N[Sqrt[N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e297

   1. Initial program 44.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 43.7

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

   5. Applied rewrites 43.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 43.7

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower--.f64 43.7

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

   10. Applied rewrites 43.7%

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

   if 2e297 < (*.f64 l l)

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 24.0

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 46.4%

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

   12. Applied rewrites 46.3%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+297}:\\ \;\;\;\;{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\frac{2}{x} + 2}{x} + 2}{x}} \cdot \ell} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.6% accurate, 0.4× speedup

Math

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

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 (pow (+ (pow x -1.0) 1.0) -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 * pow((pow(x, -1.0) + 1.0), -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 * (((x ** (-1.0d0)) + 1.0d0) ** (-1.0d0))
end function

Java

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

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * math.pow((math.pow(x, -1.0) + 1.0), -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((x ^ -1.0) + 1.0) ^ -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 * (((x ^ -1.0) + 1.0) ^ -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[Power[N[(N[Power[x, -1.0], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 39.9

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

5. Applied rewrites 39.9%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 39.9

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

7. Applied rewrites 39.9%

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

8. Taylor expanded in l around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower--.f64 39.9

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

10. Applied rewrites 39.9%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 39.5%

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

14. Final simplification 39.5%

    \[\leadsto {\left({x}^{-1} + 1\right)}^{-1} \]
15. Add Preprocessing

Alternative 3: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+297}:\\ \;\;\;\;{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{\frac{\frac{\frac{2}{x} + 2}{x} + 2}{x}} \cdot l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 2e+297)
    (pow (sqrt (/ (+ 1.0 x) (- x 1.0))) -1.0)
    (*
     t_m
     (/ (sqrt 2.0) (* (sqrt (/ (+ (/ (+ (/ 2.0 x) 2.0) x) 2.0) x)) l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+297) {
		tmp = pow(sqrt(((1.0 + x) / (x - 1.0))), -1.0);
	} else {
		tmp = t_m * (sqrt(2.0) / (sqrt((((((2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 2d+297) then
        tmp = sqrt(((1.0d0 + x) / (x - 1.0d0))) ** (-1.0d0)
    else
        tmp = t_m * (sqrt(2.0d0) / (sqrt((((((2.0d0 / x) + 2.0d0) / x) + 2.0d0) / x)) * l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+297) {
		tmp = Math.pow(Math.sqrt(((1.0 + x) / (x - 1.0))), -1.0);
	} else {
		tmp = t_m * (Math.sqrt(2.0) / (Math.sqrt((((((2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 2e+297:
		tmp = math.pow(math.sqrt(((1.0 + x) / (x - 1.0))), -1.0)
	else:
		tmp = t_m * (math.sqrt(2.0) / (math.sqrt((((((2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 2e+297)
		tmp = sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0))) ^ -1.0;
	else
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(Float64(Float64(Float64(2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 2e+297)
		tmp = sqrt(((1.0 + x) / (x - 1.0))) ^ -1.0;
	else
		tmp = t_m * (sqrt(2.0) / (sqrt((((((2.0 / x) + 2.0) / x) + 2.0) / x)) * l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+297], N[Power[N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e297

   1. Initial program 44.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 43.7

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

   5. Applied rewrites 43.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 43.7

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower--.f64 43.7

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

   10. Applied rewrites 43.7%

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

   if 2e297 < (*.f64 l l)

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 24.0

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 46.4%

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

   12. Applied rewrites 46.4%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+297}:\\ \;\;\;\;{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\frac{\frac{2}{x} + 2}{x} + 2}{x}} \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+297}:\\ \;\;\;\;{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{\frac{2}{x} + 2}{x}} \cdot l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 2e+297)
    (pow (sqrt (/ (+ 1.0 x) (- x 1.0))) -1.0)
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ (+ (/ 2.0 x) 2.0) x)) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+297) {
		tmp = pow(sqrt(((1.0 + x) / (x - 1.0))), -1.0);
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 2d+297) then
        tmp = sqrt(((1.0d0 + x) / (x - 1.0d0))) ** (-1.0d0)
    else
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((((2.0d0 / x) + 2.0d0) / x)) * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+297) {
		tmp = Math.pow(Math.sqrt(((1.0 + x) / (x - 1.0))), -1.0);
	} else {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 2e+297:
		tmp = math.pow(math.sqrt(((1.0 + x) / (x - 1.0))), -1.0)
	else:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((((2.0 / x) + 2.0) / x)) * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 2e+297)
		tmp = sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0))) ^ -1.0;
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(Float64(Float64(2.0 / x) + 2.0) / x)) * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 2e+297)
		tmp = sqrt(((1.0 + x) / (x - 1.0))) ^ -1.0;
	else
		tmp = (sqrt(2.0) * t_m) / (sqrt((((2.0 / x) + 2.0) / x)) * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+297], N[Power[N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e297

   1. Initial program 44.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 43.7

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

   5. Applied rewrites 43.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 43.7

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower--.f64 43.7

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

   10. Applied rewrites 43.7%

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

   if 2e297 < (*.f64 l l)

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 24.0

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 45.9%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+297}:\\ \;\;\;\;{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\frac{2}{x} + 2}{x}} \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+297}:\\ \;\;\;\;{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{2}{x}} \cdot l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 2e+297)
    (pow (sqrt (/ (+ 1.0 x) (- x 1.0))) -1.0)
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 2.0 x)) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+297) {
		tmp = pow(sqrt(((1.0 + x) / (x - 1.0))), -1.0);
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 2d+297) then
        tmp = sqrt(((1.0d0 + x) / (x - 1.0d0))) ** (-1.0d0)
    else
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((2.0d0 / x)) * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 2e+297) {
		tmp = Math.pow(Math.sqrt(((1.0 + x) / (x - 1.0))), -1.0);
	} else {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((2.0 / x)) * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 2e+297:
		tmp = math.pow(math.sqrt(((1.0 + x) / (x - 1.0))), -1.0)
	else:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((2.0 / x)) * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 2e+297)
		tmp = sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0))) ^ -1.0;
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(2.0 / x)) * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 2e+297)
		tmp = sqrt(((1.0 + x) / (x - 1.0))) ^ -1.0;
	else
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+297], N[Power[N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e297

   1. Initial program 44.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 43.7

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

   5. Applied rewrites 43.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 43.7

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower--.f64 43.7

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

   10. Applied rewrites 43.7%

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

   if 2e297 < (*.f64 l l)

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 24.0

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 45.3%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+297}:\\ \;\;\;\;{\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 0.7× speedup

Math

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

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 (pow (sqrt (/ (+ 1.0 x) (- x 1.0))) -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 * pow(sqrt(((1.0 + x) / (x - 1.0))), -1.0);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (sqrt(((1.0 + x) / (x - 1.0))) ^ -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[Power[N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 39.9

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

5. Applied rewrites 39.9%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 39.9

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

7. Applied rewrites 39.9%

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

8. Taylor expanded in l around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower--.f64 39.9

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

10. Applied rewrites 39.9%

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

11. Final simplification 39.9%

    \[\leadsto {\left(\sqrt{\frac{1 + x}{x - 1}}\right)}^{-1} \]
12. Add Preprocessing

Alternative 7: 74.9% 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 36.2%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 38.6

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

5. Applied rewrites 38.6%

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

7. Applied rewrites 39.2%

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

Reproduce

herbie shell --seed 2024296 
(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)))))