Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 84.6%

Time: 23.6s

Alternatives: 9

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.9% 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: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{t\_m}{\frac{l\_m}{\sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_2 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 2.3e-180)
      (/ t_m (/ l_m (sqrt x)))
      (if (<= t_m 2.8e-150)
        1.0
        (if (<= t_m 9.5e+27)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
              (/ (+ t_2 (pow l_m 2.0)) x)))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.3e-180) {
		tmp = t_m / (l_m / sqrt(x));
	} else if (t_m <= 2.8e-150) {
		tmp = 1.0;
	} else if (t_m <= 9.5e+27) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 2.3d-180) then
        tmp = t_m / (l_m / sqrt(x))
    else if (t_m <= 2.8d-150) then
        tmp = 1.0d0
    else if (t_m <= 9.5d+27) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.3e-180) {
		tmp = t_m / (l_m / Math.sqrt(x));
	} else if (t_m <= 2.8e-150) {
		tmp = 1.0;
	} else if (t_m <= 9.5e+27) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((t_2 + Math.pow(l_m, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 2.3e-180:
		tmp = t_m / (l_m / math.sqrt(x))
	elif t_m <= 2.8e-150:
		tmp = 1.0
	elif t_m <= 9.5e+27:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((t_2 + math.pow(l_m, 2.0)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.3e-180)
		tmp = Float64(t_m / Float64(l_m / sqrt(x)));
	elseif (t_m <= 2.8e-150)
		tmp = 1.0;
	elseif (t_m <= 9.5e+27)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.3e-180)
		tmp = t_m / (l_m / sqrt(x));
	elseif (t_m <= 2.8e-150)
		tmp = 1.0;
	elseif (t_m <= 9.5e+27)
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.3e-180], N[(t$95$m / N[(l$95$m / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.8e-150], 1.0, If[LessEqual[t$95$m, 9.5e+27], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 2.29999999999999996e-180

   1. Initial program 31.2%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in x around inf 57.5%

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

   6. Taylor expanded in l around inf 23.4%

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

      1. clear-num 23.0%

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

      2. un-div-inv 23.0%

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

      3. associate-*l* 23.0%

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

      4. sqrt-div 23.1%

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

      5. metadata-eval 23.1%

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

      6. un-div-inv 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. associate-/r/ 23.4%

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

      2. associate-*r/ 23.4%

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

      3. *-commutative 23.4%

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

      4. associate-/l* 23.4%

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

   10. Simplified 23.4%

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

       1. clear-num 23.4%

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

       2. associate-*l/ 23.4%

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

       3. *-un-lft-identity 23.4%

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

       4. *-commutative 23.4%

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

       5. associate-/l* 23.4%

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

       6. *-inverses 23.4%

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

       7. *-commutative 23.4%

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

       8. *-un-lft-identity 23.4%

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

   12. Applied egg-rr 23.4%

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

   if 2.29999999999999996e-180 < t < 2.79999999999999996e-150

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if 2.79999999999999996e-150 < t < 9.4999999999999997e27

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in x around inf 86.9%

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

   if 9.4999999999999997e27 < t

   1. Initial program 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in l around 0 95.2%

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

   6. Taylor expanded in t around 0 95.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{l\_m}^{2}}{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{t\_m}{\frac{l\_m}{\sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(2 \cdot {t\_m}^{2} + t\_2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (pow l_m 2.0) x)))
   (*
    t_s
    (if (<= t_m 2.3e-180)
      (/ t_m (/ l_m (sqrt x)))
      (if (<= t_m 2.8e-150)
        1.0
        (if (<= t_m 2.3e-59)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              t_2
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ (* 2.0 (pow t_m 2.0)) t_2))))))
          (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 2.3e-180) {
		tmp = t_m / (l_m / sqrt(x));
	} else if (t_m <= 2.8e-150) {
		tmp = 1.0;
	} else if (t_m <= 2.3e-59) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((2.0 * (pow(t_m, 2.0) / x)) + ((2.0 * pow(t_m, 2.0)) + t_2)))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l_m ** 2.0d0) / x
    if (t_m <= 2.3d-180) then
        tmp = t_m / (l_m / sqrt(x))
    else if (t_m <= 2.8d-150) then
        tmp = 1.0d0
    else if (t_m <= 2.3d-59) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((t_2 + ((2.0d0 * ((t_m ** 2.0d0) / x)) + ((2.0d0 * (t_m ** 2.0d0)) + t_2)))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 2.3e-180) {
		tmp = t_m / (l_m / Math.sqrt(x));
	} else if (t_m <= 2.8e-150) {
		tmp = 1.0;
	} else if (t_m <= 2.3e-59) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + ((2.0 * (Math.pow(t_m, 2.0) / x)) + ((2.0 * Math.pow(t_m, 2.0)) + t_2)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.pow(l_m, 2.0) / x
	tmp = 0
	if t_m <= 2.3e-180:
		tmp = t_m / (l_m / math.sqrt(x))
	elif t_m <= 2.8e-150:
		tmp = 1.0
	elif t_m <= 2.3e-59:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + ((2.0 * (math.pow(t_m, 2.0) / x)) + ((2.0 * math.pow(t_m, 2.0)) + t_2)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64((l_m ^ 2.0) / x)
	tmp = 0.0
	if (t_m <= 2.3e-180)
		tmp = Float64(t_m / Float64(l_m / sqrt(x)));
	elseif (t_m <= 2.8e-150)
		tmp = 1.0;
	elseif (t_m <= 2.3e-59)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(Float64(2.0 * (t_m ^ 2.0)) + t_2))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (l_m ^ 2.0) / x;
	tmp = 0.0;
	if (t_m <= 2.3e-180)
		tmp = t_m / (l_m / sqrt(x));
	elseif (t_m <= 2.8e-150)
		tmp = 1.0;
	elseif (t_m <= 2.3e-59)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((2.0 * ((t_m ^ 2.0) / x)) + ((2.0 * (t_m ^ 2.0)) + t_2)))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.3e-180], N[(t$95$m / N[(l$95$m / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.8e-150], 1.0, If[LessEqual[t$95$m, 2.3e-59], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 2.29999999999999996e-180

   1. Initial program 31.2%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in x around inf 57.5%

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

   6. Taylor expanded in l around inf 23.4%

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

      1. clear-num 23.0%

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

      2. un-div-inv 23.0%

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

      3. associate-*l* 23.0%

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

      4. sqrt-div 23.1%

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

      5. metadata-eval 23.1%

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

      6. un-div-inv 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. associate-/r/ 23.4%

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

      2. associate-*r/ 23.4%

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

      3. *-commutative 23.4%

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

      4. associate-/l* 23.4%

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

   10. Simplified 23.4%

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

       1. clear-num 23.4%

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

       2. associate-*l/ 23.4%

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

       3. *-un-lft-identity 23.4%

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

       4. *-commutative 23.4%

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

       5. associate-/l* 23.4%

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

       6. *-inverses 23.4%

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

       7. *-commutative 23.4%

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

       8. *-un-lft-identity 23.4%

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

   12. Applied egg-rr 23.4%

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

   if 2.29999999999999996e-180 < t < 2.79999999999999996e-150

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if 2.79999999999999996e-150 < t < 2.29999999999999979e-59

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around inf 89.1%

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

   6. Taylor expanded in t around 0 89.1%

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

   if 2.29999999999999979e-59 < t

   1. Initial program 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in l around 0 93.6%

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

   6. Taylor expanded in t around 0 93.8%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{t\_m}{\frac{l\_m}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-180)
    (/ t_m (/ l_m (sqrt x)))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.3e-180) {
		tmp = t_m / (l_m / sqrt(x));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.3d-180) then
        tmp = t_m / (l_m / sqrt(x))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.3e-180) {
		tmp = t_m / (l_m / Math.sqrt(x));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.3e-180:
		tmp = t_m / (l_m / math.sqrt(x))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.3e-180)
		tmp = Float64(t_m / Float64(l_m / sqrt(x)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.3e-180)
		tmp = t_m / (l_m / sqrt(x));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-180], N[(t$95$m / N[(l$95$m / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999996e-180

   1. Initial program 31.2%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in x around inf 57.5%

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

   6. Taylor expanded in l around inf 23.4%

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

      1. clear-num 23.0%

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

      2. un-div-inv 23.0%

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

      3. associate-*l* 23.0%

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

      4. sqrt-div 23.1%

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

      5. metadata-eval 23.1%

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

      6. un-div-inv 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. associate-/r/ 23.4%

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

      2. associate-*r/ 23.4%

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

      3. *-commutative 23.4%

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

      4. associate-/l* 23.4%

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

   10. Simplified 23.4%

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

       1. clear-num 23.4%

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

       2. associate-*l/ 23.4%

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

       3. *-un-lft-identity 23.4%

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

       4. *-commutative 23.4%

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

       5. associate-/l* 23.4%

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

       6. *-inverses 23.4%

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

       7. *-commutative 23.4%

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

       8. *-un-lft-identity 23.4%

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

   12. Applied egg-rr 23.4%

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

   if 2.29999999999999996e-180 < t

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in l around 0 89.3%

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

   6. Taylor expanded in t around 0 89.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-180)
    (* (sqrt x) (/ t_m l_m))
    (- 1.0 (/ (- 1.0 (/ 0.5 x)) x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-180) {
		tmp = sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.4d-180) then
        tmp = sqrt(x) * (t_m / l_m)
    else
        tmp = 1.0d0 - ((1.0d0 - (0.5d0 / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-180) {
		tmp = Math.sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.4e-180:
		tmp = math.sqrt(x) * (t_m / l_m)
	else:
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.4e-180)
		tmp = Float64(sqrt(x) * Float64(t_m / l_m));
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.4e-180)
		tmp = sqrt(x) * (t_m / l_m);
	else
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-180], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.39999999999999979e-180

   1. Initial program 31.2%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in x around inf 57.5%

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

   6. Taylor expanded in l around inf 23.4%

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

   7. Taylor expanded in t around 0 22.2%

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

   if 2.39999999999999979e-180 < t

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in l around 0 89.3%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{0.5 \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x \cdot {\left(\sqrt{-1}\right)}^{2}} - \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   8. Simplified 89.1%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-180}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-180)
    (* t_m (/ (sqrt x) l_m))
    (- 1.0 (/ (- 1.0 (/ 0.5 x)) x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-180) {
		tmp = t_m * (sqrt(x) / l_m);
	} else {
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.4d-180) then
        tmp = t_m * (sqrt(x) / l_m)
    else
        tmp = 1.0d0 - ((1.0d0 - (0.5d0 / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-180) {
		tmp = t_m * (Math.sqrt(x) / l_m);
	} else {
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.4e-180:
		tmp = t_m * (math.sqrt(x) / l_m)
	else:
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.4e-180)
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.4e-180)
		tmp = t_m * (sqrt(x) / l_m);
	else
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-180], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.39999999999999979e-180

   1. Initial program 31.2%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in x around inf 57.5%

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

   6. Taylor expanded in l around inf 23.4%

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

      1. clear-num 23.0%

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

      2. un-div-inv 23.0%

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

      3. associate-*l* 23.0%

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

      4. sqrt-div 23.1%

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

      5. metadata-eval 23.1%

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

      6. un-div-inv 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. associate-/r/ 23.4%

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

      2. associate-*r/ 23.4%

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

      3. *-commutative 23.4%

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

      4. associate-/l* 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in l around 0 23.4%

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

       1. associate-*l/ 23.4%

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

       2. *-lft-identity 23.4%

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

   13. Simplified 23.4%

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

   if 2.39999999999999979e-180 < t

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in l around 0 89.3%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{0.5 \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x \cdot {\left(\sqrt{-1}\right)}^{2}} - \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   8. Simplified 89.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-180}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{t\_m}{\frac{l\_m}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-180)
    (/ t_m (/ l_m (sqrt x)))
    (- 1.0 (/ (- 1.0 (/ 0.5 x)) x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-180) {
		tmp = t_m / (l_m / sqrt(x));
	} else {
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.4d-180) then
        tmp = t_m / (l_m / sqrt(x))
    else
        tmp = 1.0d0 - ((1.0d0 - (0.5d0 / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.4e-180) {
		tmp = t_m / (l_m / Math.sqrt(x));
	} else {
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 2.4e-180:
		tmp = t_m / (l_m / math.sqrt(x))
	else:
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.4e-180)
		tmp = Float64(t_m / Float64(l_m / sqrt(x)));
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 2.4e-180)
		tmp = t_m / (l_m / sqrt(x));
	else
		tmp = 1.0 - ((1.0 - (0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-180], N[(t$95$m / N[(l$95$m / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.39999999999999979e-180

   1. Initial program 31.2%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in x around inf 57.5%

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

   6. Taylor expanded in l around inf 23.4%

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

      1. clear-num 23.0%

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

      2. un-div-inv 23.0%

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

      3. associate-*l* 23.0%

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

      4. sqrt-div 23.1%

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

      5. metadata-eval 23.1%

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

      6. un-div-inv 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. associate-/r/ 23.4%

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

      2. associate-*r/ 23.4%

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

      3. *-commutative 23.4%

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

      4. associate-/l* 23.4%

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

   10. Simplified 23.4%

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

       1. clear-num 23.4%

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

       2. associate-*l/ 23.4%

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

       3. *-un-lft-identity 23.4%

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

       4. *-commutative 23.4%

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

       5. associate-/l* 23.4%

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

       6. *-inverses 23.4%

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

       7. *-commutative 23.4%

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

       8. *-un-lft-identity 23.4%

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

   12. Applied egg-rr 23.4%

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

   if 2.39999999999999979e-180 < t

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in l around 0 89.3%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{0.5 \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x \cdot {\left(\sqrt{-1}\right)}^{2}} - \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   8. Simplified 89.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

3. Simplified 33.5%

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

5. Taylor expanded in l around 0 38.4%

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

6. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{0.5 \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x \cdot {\left(\sqrt{-1}\right)}^{2}} - \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
7. Step-by-step derivation

   1. mul-1-neg 0.0%

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

   2. unsub-neg 0.0%

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

8. Simplified 38.4%

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

9. Final simplification 38.4%

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

Alternative 8: 75.9% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

3. Simplified 33.5%

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

5. Taylor expanded in l around 0 38.4%

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

6. Taylor expanded in x around inf 38.3%

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

7. Final simplification 38.3%

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

Alternative 9: 75.2% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

3. Simplified 33.5%

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

5. Taylor expanded in l around 0 38.4%

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

6. Taylor expanded in x around inf 37.9%

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

7. Final simplification 37.9%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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)))))