Toniolo and Linder, Equation (7)

Percentage Accurate: 33.6% → 81.8%

Time: 15.8s

Alternatives: 13

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: 33.6% 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: 81.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot {t}^{2}\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.12 \cdot 10^{-148}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(t_1 + \frac{{\ell}^{2}}{x}\right)\right) + \frac{t_1 + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (pow t 2.0))))
   (if (<= t -6.4e+66)
     (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
     (if (<= t -1.12e-148)
       (/
        t
        (/
         (sqrt
          (+
           (+ (* 2.0 (/ (pow t 2.0) x)) (+ t_1 (/ (pow l 2.0) x)))
           (/ (+ t_1 (pow l 2.0)) x)))
         (sqrt 2.0)))
       (if (<= t 2.9e-140) (/ t (* l (sqrt (/ 1.0 x)))) (+ 1.0 (/ -1.0 x)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 2.0 * pow(t, 2.0);
	double tmp;
	if (t <= -6.4e+66) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= -1.12e-148) {
		tmp = t / (sqrt((((2.0 * (pow(t, 2.0) / x)) + (t_1 + (pow(l, 2.0) / x))) + ((t_1 + pow(l, 2.0)) / x))) / sqrt(2.0));
	} else if (t <= 2.9e-140) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (t ** 2.0d0)
    if (t <= (-6.4d+66)) then
        tmp = t / (t * -sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    else if (t <= (-1.12d-148)) then
        tmp = t / (sqrt((((2.0d0 * ((t ** 2.0d0) / x)) + (t_1 + ((l ** 2.0d0) / x))) + ((t_1 + (l ** 2.0d0)) / x))) / sqrt(2.0d0))
    else if (t <= 2.9d-140) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = 2.0 * Math.pow(t, 2.0);
	double tmp;
	if (t <= -6.4e+66) {
		tmp = t / (t * -Math.sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= -1.12e-148) {
		tmp = t / (Math.sqrt((((2.0 * (Math.pow(t, 2.0) / x)) + (t_1 + (Math.pow(l, 2.0) / x))) + ((t_1 + Math.pow(l, 2.0)) / x))) / Math.sqrt(2.0));
	} else if (t <= 2.9e-140) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 2.0 * math.pow(t, 2.0)
	tmp = 0
	if t <= -6.4e+66:
		tmp = t / (t * -math.sqrt(((x + 1.0) / (x + -1.0))))
	elif t <= -1.12e-148:
		tmp = t / (math.sqrt((((2.0 * (math.pow(t, 2.0) / x)) + (t_1 + (math.pow(l, 2.0) / x))) + ((t_1 + math.pow(l, 2.0)) / x))) / math.sqrt(2.0))
	elif t <= 2.9e-140:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(2.0 * (t ^ 2.0))
	tmp = 0.0
	if (t <= -6.4e+66)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= -1.12e-148)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(t_1 + Float64((l ^ 2.0) / x))) + Float64(Float64(t_1 + (l ^ 2.0)) / x))) / sqrt(2.0)));
	elseif (t <= 2.9e-140)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 2.0 * (t ^ 2.0);
	tmp = 0.0;
	if (t <= -6.4e+66)
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	elseif (t <= -1.12e-148)
		tmp = t / (sqrt((((2.0 * ((t ^ 2.0) / x)) + (t_1 + ((l ^ 2.0) / x))) + ((t_1 + (l ^ 2.0)) / x))) / sqrt(2.0));
	elseif (t <= 2.9e-140)
		tmp = t / (l * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.4e+66], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.12e-148], N[(t / N[(N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.9e-140], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.3999999999999999e66

   1. Initial program 31.6%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in t around -inf 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

      3. +-commutative 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if -6.3999999999999999e66 < t < -1.1199999999999999e-148

   1. Initial program 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in x around inf 87.6%

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

   if -1.1199999999999999e-148 < t < 2.89999999999999997e-140

   1. Initial program 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in x around inf 52.7%

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

   5. Taylor expanded in t around 0 52.2%

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

      1. cancel-sign-sub-inv 52.2%

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

      2. metadata-eval 52.2%

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

      3. distribute-rgt1-in 52.2%

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

      4. metadata-eval 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in l around 0 51.2%

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

   if 2.89999999999999997e-140 < t

   1. Initial program 38.7%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in t around -inf 1.6%

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

      1. associate-*r* 1.6%

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

      2. neg-mul-1 1.6%

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

      3. +-commutative 1.6%

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

      4. sub-neg 1.6%

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

      5. metadata-eval 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in t around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. +-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.12 \cdot 10^{-148}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2: 79.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-146}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -4e-146)
   (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
   (if (<= t 2.9e-140) (/ t (* l (sqrt (/ 1.0 x)))) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -4e-146) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 2.9e-140) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4d-146)) then
        tmp = t / (t * -sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    else if (t <= 2.9d-140) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -4e-146) {
		tmp = t / (t * -Math.sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 2.9e-140) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -4e-146:
		tmp = t / (t * -math.sqrt(((x + 1.0) / (x + -1.0))))
	elif t <= 2.9e-140:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -4e-146)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= 2.9e-140)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -4e-146)
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	elseif (t <= 2.9e-140)
		tmp = t / (l * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -4e-146], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.9e-140], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.0000000000000001e-146

   1. Initial program 41.4%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around -inf 89.9%

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

      1. associate-*r* 89.9%

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

      2. neg-mul-1 89.9%

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

      3. +-commutative 89.9%

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

      4. sub-neg 89.9%

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

      5. metadata-eval 89.9%

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

      6. +-commutative 89.9%

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

   6. Simplified 89.9%

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

   if -4.0000000000000001e-146 < t < 2.89999999999999997e-140

   1. Initial program 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in x around inf 52.7%

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

   5. Taylor expanded in t around 0 52.2%

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

      1. cancel-sign-sub-inv 52.2%

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

      2. metadata-eval 52.2%

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

      3. distribute-rgt1-in 52.2%

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

      4. metadata-eval 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in l around 0 51.2%

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

   if 2.89999999999999997e-140 < t

   1. Initial program 38.7%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in t around -inf 1.6%

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

      1. associate-*r* 1.6%

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

      2. neg-mul-1 1.6%

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

      3. +-commutative 1.6%

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

      4. sub-neg 1.6%

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

      5. metadata-eval 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in t around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. +-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-146}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 79.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.45 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -3.45e-143)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (<= t 4e-140) (/ t (* l (sqrt (/ 1.0 x)))) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.45e-143) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 4e-140) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.45d-143)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if (t <= 4d-140) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.45e-143) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 4e-140) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -3.45e-143:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif t <= 4e-140:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -3.45e-143)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif (t <= 4e-140)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.45e-143)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif (t <= 4e-140)
		tmp = t / (l * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -3.45e-143], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-140], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.44999999999999994e-143

   1. Initial program 41.4%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around -inf 89.9%

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

      1. associate-*r* 89.9%

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

      2. neg-mul-1 89.9%

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

      3. +-commutative 89.9%

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

      4. sub-neg 89.9%

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

      5. metadata-eval 89.9%

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

      6. +-commutative 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in x around inf 89.0%

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

      1. associate-*r/ 89.0%

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

      2. metadata-eval 89.0%

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

   9. Simplified 89.0%

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

       1. unpow2 89.0%

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

   11. Applied egg-rr 89.0%

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

   if -3.44999999999999994e-143 < t < 3.9999999999999999e-140

   1. Initial program 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in x around inf 52.7%

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

   5. Taylor expanded in t around 0 52.2%

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

      1. cancel-sign-sub-inv 52.2%

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

      2. metadata-eval 52.2%

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

      3. distribute-rgt1-in 52.2%

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

      4. metadata-eval 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in l around 0 51.2%

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

   if 3.9999999999999999e-140 < t

   1. Initial program 38.7%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in t around -inf 1.6%

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

      1. associate-*r* 1.6%

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

      2. neg-mul-1 1.6%

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

      3. +-commutative 1.6%

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

      4. sub-neg 1.6%

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

      5. metadata-eval 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in t around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. +-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.45 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.6e-141)
   (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t 2.9e-140) (/ t (* l (sqrt (/ 1.0 x)))) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.6e-141) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.9e-140) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.6d-141)) then
        tmp = -sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else if (t <= 2.9d-140) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.6e-141) {
		tmp = -Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= 2.9e-140) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.6e-141:
		tmp = -math.sqrt(((x + -1.0) / (x + 1.0)))
	elif t <= 2.9e-140:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.6e-141)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= 2.9e-140)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.6e-141)
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	elseif (t <= 2.9e-140)
		tmp = t / (l * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.6e-141], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 2.9e-140], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.6000000000000001e-141

   1. Initial program 41.4%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around -inf 89.9%

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

      1. associate-*r* 89.9%

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

      2. neg-mul-1 89.9%

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

      3. +-commutative 89.9%

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

      4. sub-neg 89.9%

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

      5. metadata-eval 89.9%

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

      6. +-commutative 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in t around 0 89.9%

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

      1. mul-1-neg 89.9%

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

      2. sub-neg 89.9%

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

      3. metadata-eval 89.9%

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

      4. +-commutative 89.9%

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

   9. Simplified 89.9%

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

   if -1.6000000000000001e-141 < t < 2.89999999999999997e-140

   1. Initial program 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in x around inf 52.7%

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

   5. Taylor expanded in t around 0 52.2%

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

      1. cancel-sign-sub-inv 52.2%

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

      2. metadata-eval 52.2%

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

      3. distribute-rgt1-in 52.2%

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

      4. metadata-eval 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in l around 0 51.2%

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

   if 2.89999999999999997e-140 < t

   1. Initial program 38.7%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in t around -inf 1.6%

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

      1. associate-*r* 1.6%

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

      2. neg-mul-1 1.6%

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

      3. +-commutative 1.6%

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

      4. sub-neg 1.6%

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

      5. metadata-eval 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in t around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. +-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-141}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 5: 78.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -3.9e-205)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (<= t 3.7e-140) (* (sqrt x) (/ t l)) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.9e-205) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 3.7e-140) {
		tmp = sqrt(x) * (t / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.9d-205)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if (t <= 3.7d-140) then
        tmp = sqrt(x) * (t / l)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.9e-205) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 3.7e-140) {
		tmp = Math.sqrt(x) * (t / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -3.9e-205:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif t <= 3.7e-140:
		tmp = math.sqrt(x) * (t / l)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -3.9e-205)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif (t <= 3.7e-140)
		tmp = Float64(sqrt(x) * Float64(t / l));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.9e-205)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif (t <= 3.7e-140)
		tmp = sqrt(x) * (t / l);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -3.9e-205], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.7e-140], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.90000000000000018e-205

   1. Initial program 36.4%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around -inf 84.7%

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

      1. associate-*r* 84.7%

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

      2. neg-mul-1 84.7%

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

      3. +-commutative 84.7%

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

      4. sub-neg 84.7%

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

      5. metadata-eval 84.7%

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

      6. +-commutative 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in x around inf 83.9%

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

      1. associate-*r/ 83.9%

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

      2. metadata-eval 83.9%

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

   9. Simplified 83.9%

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

       1. unpow2 83.9%

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

   11. Applied egg-rr 83.9%

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

   if -3.90000000000000018e-205 < t < 3.69999999999999977e-140

   1. Initial program 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in x around inf 57.9%

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

   5. Taylor expanded in t around 0 57.4%

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

      1. cancel-sign-sub-inv 57.4%

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

      2. metadata-eval 57.4%

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

      3. distribute-rgt1-in 57.4%

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

      4. metadata-eval 57.4%

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

   7. Simplified 57.4%

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

   8. Taylor expanded in t around 0 51.2%

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

   if 3.69999999999999977e-140 < t

   1. Initial program 38.7%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in t around -inf 1.6%

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

      1. associate-*r* 1.6%

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

      2. neg-mul-1 1.6%

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

      3. +-commutative 1.6%

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

      4. sub-neg 1.6%

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

      5. metadata-eval 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in t around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. +-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 6: 79.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-140}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -9.2e-146)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (<= t 3e-140) (* t (/ (sqrt x) l)) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -9.2e-146) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 3e-140) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9.2d-146)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if (t <= 3d-140) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -9.2e-146) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 3e-140) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -9.2e-146:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif t <= 3e-140:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -9.2e-146)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif (t <= 3e-140)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -9.2e-146)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif (t <= 3e-140)
		tmp = t * (sqrt(x) / l);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -9.2e-146], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e-140], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.2000000000000003e-146

   1. Initial program 41.4%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around -inf 89.9%

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

      1. associate-*r* 89.9%

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

      2. neg-mul-1 89.9%

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

      3. +-commutative 89.9%

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

      4. sub-neg 89.9%

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

      5. metadata-eval 89.9%

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

      6. +-commutative 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in x around inf 89.0%

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

      1. associate-*r/ 89.0%

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

      2. metadata-eval 89.0%

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

   9. Simplified 89.0%

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

       1. unpow2 89.0%

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

   11. Applied egg-rr 89.0%

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

   if -9.2000000000000003e-146 < t < 3.00000000000000018e-140

   1. Initial program 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in x around inf 52.7%

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

   5. Taylor expanded in t around 0 52.2%

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

      1. cancel-sign-sub-inv 52.2%

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

      2. metadata-eval 52.2%

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

      3. distribute-rgt1-in 52.2%

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

      4. metadata-eval 52.2%

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

   7. Simplified 52.2%

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

      1. clear-num 51.2%

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

      2. associate-/r/ 52.3%

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

      3. clear-num 52.3%

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

      4. sqrt-undiv 52.4%

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

      5. associate-*r/ 52.4%

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

   9. Applied egg-rr 52.4%

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

   10. Taylor expanded in l around 0 51.0%

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

       1. associate-*l/ 51.0%

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

       2. *-lft-identity 51.0%

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

   12. Simplified 51.0%

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

   if 3.00000000000000018e-140 < t

   1. Initial program 38.7%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in t around -inf 1.6%

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

      1. associate-*r* 1.6%

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

      2. neg-mul-1 1.6%

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

      3. +-commutative 1.6%

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

      4. sub-neg 1.6%

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

      5. metadata-eval 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in t around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. +-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-140}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 7: 79.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.65e-141)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (<= t 3.3e-140) (/ t (/ l (sqrt x))) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.65e-141) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 3.3e-140) {
		tmp = t / (l / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.65d-141)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if (t <= 3.3d-140) then
        tmp = t / (l / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.65e-141) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if (t <= 3.3e-140) {
		tmp = t / (l / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.65e-141:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif t <= 3.3e-140:
		tmp = t / (l / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.65e-141)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif (t <= 3.3e-140)
		tmp = Float64(t / Float64(l / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.65e-141)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif (t <= 3.3e-140)
		tmp = t / (l / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.65e-141], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e-140], N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.65e-141

   1. Initial program 41.4%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around -inf 89.9%

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

      1. associate-*r* 89.9%

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

      2. neg-mul-1 89.9%

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

      3. +-commutative 89.9%

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

      4. sub-neg 89.9%

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

      5. metadata-eval 89.9%

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

      6. +-commutative 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in x around inf 89.0%

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

      1. associate-*r/ 89.0%

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

      2. metadata-eval 89.0%

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

   9. Simplified 89.0%

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

       1. unpow2 89.0%

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

   11. Applied egg-rr 89.0%

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

   if -1.65e-141 < t < 3.29999999999999987e-140

   1. Initial program 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in x around inf 52.7%

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

   5. Taylor expanded in t around 0 52.2%

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

      1. cancel-sign-sub-inv 52.2%

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

      2. metadata-eval 52.2%

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

      3. distribute-rgt1-in 52.2%

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

      4. metadata-eval 52.2%

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

   7. Simplified 52.2%

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

      1. expm1-log1p-u 51.8%

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

      2. expm1-udef 24.7%

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

      3. sqrt-undiv 24.7%

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

      4. *-commutative 24.7%

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

      5. associate-/l* 24.7%

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

      6. metadata-eval 24.7%

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

   9. Applied egg-rr 24.7%

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

       1. expm1-def 52.0%

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

       2. expm1-log1p 52.5%

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

       3. /-rgt-identity 52.5%

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

   11. Simplified 52.5%

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

       1. expm1-log1p-u 52.1%

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

       2. expm1-udef 35.0%

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

       3. sqrt-div 34.4%

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

       4. unpow2 34.4%

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

       5. sqrt-prod 24.4%

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

       6. add-sqr-sqrt 24.8%

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

   13. Applied egg-rr 24.8%

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

       1. expm1-def 40.8%

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

       2. expm1-log1p 51.2%

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

   15. Simplified 51.2%

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

   if 3.29999999999999987e-140 < t

   1. Initial program 38.7%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in t around -inf 1.6%

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

      1. associate-*r* 1.6%

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

      2. neg-mul-1 1.6%

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

      3. +-commutative 1.6%

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

      4. sub-neg 1.6%

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

      5. metadata-eval 1.6%

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

      6. +-commutative 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in t around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. +-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 87.6%

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

       3. +-commutative 87.6%

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

   12. Simplified 87.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 8: 76.9% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x)))) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 33.2%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around -inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. neg-mul-1 77.7%

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

      3. +-commutative 77.7%

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

      4. sub-neg 77.7%

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

      5. metadata-eval 77.7%

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

      6. +-commutative 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in x around inf 77.0%

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

      1. associate-*r/ 77.0%

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

      2. metadata-eval 77.0%

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

   9. Simplified 77.0%

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

       1. unpow2 77.0%

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

   11. Applied egg-rr 77.0%

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

   if -4.999999999999985e-310 < t

   1. Initial program 30.5%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. associate-*r* 1.7%

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

      2. neg-mul-1 1.7%

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

      3. +-commutative 1.7%

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

      4. sub-neg 1.7%

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

      5. metadata-eval 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 76.0%

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

       3. +-commutative 76.0%

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

   12. Simplified 76.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 9: 76.8% accurate, 22.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (/ t (- (- t) (/ t x))) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = t / (-t - (t / x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = t / (-t - (t / x))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = t / (-t - (t / x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = t / (-t - (t / x))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(t / Float64(Float64(-t) - Float64(t / x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = t / (-t - (t / x));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(t / N[((-t) - N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 33.2%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around -inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. neg-mul-1 77.7%

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

      3. +-commutative 77.7%

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

      4. sub-neg 77.7%

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

      5. metadata-eval 77.7%

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

      6. +-commutative 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in x around inf 76.8%

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

      1. neg-mul-1 76.8%

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

      2. +-commutative 76.8%

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

      3. unsub-neg 76.8%

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

      4. associate-*r/ 76.8%

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

      5. neg-mul-1 76.8%

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

   9. Simplified 76.8%

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

   if -4.999999999999985e-310 < t

   1. Initial program 30.5%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. associate-*r* 1.7%

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

      2. neg-mul-1 1.7%

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

      3. +-commutative 1.7%

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

      4. sub-neg 1.7%

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

      5. metadata-eval 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 76.0%

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

       3. +-commutative 76.0%

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

   12. Simplified 76.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 10: 76.5% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-310) (+ -1.0 (/ 1.0 x)) 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 33.2%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around -inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. neg-mul-1 77.7%

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

      3. +-commutative 77.7%

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

      4. sub-neg 77.7%

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

      5. metadata-eval 77.7%

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

      6. +-commutative 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in x around inf 76.8%

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

   if -4.999999999999985e-310 < t

   1. Initial program 30.5%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. associate-*r* 1.7%

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

      2. neg-mul-1 1.7%

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

      3. +-commutative 1.7%

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

      4. sub-neg 1.7%

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

      5. metadata-eval 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

       \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 0.0%

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

       2. rem-square-sqrt 75.1%

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

   12. Simplified 75.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 76.8% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ -1.0 (/ 1.0 x)) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 33.2%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around -inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. neg-mul-1 77.7%

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

      3. +-commutative 77.7%

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

      4. sub-neg 77.7%

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

      5. metadata-eval 77.7%

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

      6. +-commutative 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in x around inf 76.8%

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

   if -4.999999999999985e-310 < t

   1. Initial program 30.5%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. associate-*r* 1.7%

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

      2. neg-mul-1 1.7%

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

      3. +-commutative 1.7%

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

      4. sub-neg 1.7%

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

      5. metadata-eval 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 76.0%

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

       3. +-commutative 76.0%

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

   12. Simplified 76.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12: 76.1% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-310) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 33.2%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around -inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. neg-mul-1 77.7%

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

      3. +-commutative 77.7%

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

      4. sub-neg 77.7%

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

      5. metadata-eval 77.7%

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

      6. +-commutative 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in x around inf 76.3%

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

   if -4.999999999999985e-310 < t

   1. Initial program 30.5%

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

   3. Simplified 30.4%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. associate-*r* 1.7%

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

      2. neg-mul-1 1.7%

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

      3. +-commutative 1.7%

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

      4. sub-neg 1.7%

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

      5. metadata-eval 1.7%

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

      6. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

       \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 0.0%

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

       2. rem-square-sqrt 75.1%

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

   12. Simplified 75.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 38.7% accurate, 225.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ -1 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 -1.0)

C

l = abs(l);
double code(double x, double l, double t) {
	return -1.0;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return -1.0;
}

Python

l = abs(l)
def code(x, l, t):
	return -1.0

Julia

l = abs(l)
function code(x, l, t)
	return -1.0
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := -1.0

Derivation

1. Initial program 31.9%

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

3. Simplified 31.9%

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

4. Taylor expanded in t around -inf 41.2%

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

   1. associate-*r* 41.2%

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

   2. neg-mul-1 41.2%

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

   3. +-commutative 41.2%

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

   4. sub-neg 41.2%

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

   5. metadata-eval 41.2%

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

   6. +-commutative 41.2%

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

6. Simplified 41.2%

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

7. Taylor expanded in x around inf 40.5%

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

8. Final simplification 40.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023299 
(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)))))