Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 83.3%

Time: 24.0s

Alternatives: 14

Speedup: 75.0×

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: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot {t}^{2}\\ t_2 := 2 \cdot \frac{{t}^{2}}{x} + \left(t_1 + \frac{{\ell}^{2}}{x}\right)\\ t_3 := t_1 + {\ell}^{2}\\ t_4 := \frac{t_3}{x}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{{x}^{2}}\right)\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-125}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(\frac{t_3 + t_3}{{x}^{2}} + t_2\right) + t_4}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{t_2 + t_4}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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)))
        (t_2 (+ (* 2.0 (/ (pow t 2.0) x)) (+ t_1 (/ (pow l 2.0) x))))
        (t_3 (+ t_1 (pow l 2.0)))
        (t_4 (/ t_3 x)))
   (if (<= t -3.2e+64)
     (+ (/ 1.0 x) (- -1.0 (/ 0.5 (pow x 2.0))))
     (if (<= t -5e-125)
       (/ t (/ (sqrt (+ (+ (/ (+ t_3 t_3) (pow x 2.0)) t_2) t_4)) (sqrt 2.0)))
       (if (<= t 1.62e-198)
         (* (/ 1.0 l) (/ t (pow x -0.5)))
         (if (<= t 4.6e+35)
           (/ t (/ (sqrt (+ t_2 t_4)) (sqrt 2.0)))
           (sqrt (/ (+ 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 t_2 = (2.0 * (pow(t, 2.0) / x)) + (t_1 + (pow(l, 2.0) / x));
	double t_3 = t_1 + pow(l, 2.0);
	double t_4 = t_3 / x;
	double tmp;
	if (t <= -3.2e+64) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / pow(x, 2.0)));
	} else if (t <= -5e-125) {
		tmp = t / (sqrt(((((t_3 + t_3) / pow(x, 2.0)) + t_2) + t_4)) / sqrt(2.0));
	} else if (t <= 1.62e-198) {
		tmp = (1.0 / l) * (t / pow(x, -0.5));
	} else if (t <= 4.6e+35) {
		tmp = t / (sqrt((t_2 + t_4)) / sqrt(2.0));
	} else {
		tmp = sqrt(((x + -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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = 2.0d0 * (t ** 2.0d0)
    t_2 = (2.0d0 * ((t ** 2.0d0) / x)) + (t_1 + ((l ** 2.0d0) / x))
    t_3 = t_1 + (l ** 2.0d0)
    t_4 = t_3 / x
    if (t <= (-3.2d+64)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x ** 2.0d0)))
    else if (t <= (-5d-125)) then
        tmp = t / (sqrt(((((t_3 + t_3) / (x ** 2.0d0)) + t_2) + t_4)) / sqrt(2.0d0))
    else if (t <= 1.62d-198) then
        tmp = (1.0d0 / l) * (t / (x ** (-0.5d0)))
    else if (t <= 4.6d+35) then
        tmp = t / (sqrt((t_2 + t_4)) / sqrt(2.0d0))
    else
        tmp = sqrt(((x + (-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 t_2 = (2.0 * (Math.pow(t, 2.0) / x)) + (t_1 + (Math.pow(l, 2.0) / x));
	double t_3 = t_1 + Math.pow(l, 2.0);
	double t_4 = t_3 / x;
	double tmp;
	if (t <= -3.2e+64) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / Math.pow(x, 2.0)));
	} else if (t <= -5e-125) {
		tmp = t / (Math.sqrt(((((t_3 + t_3) / Math.pow(x, 2.0)) + t_2) + t_4)) / Math.sqrt(2.0));
	} else if (t <= 1.62e-198) {
		tmp = (1.0 / l) * (t / Math.pow(x, -0.5));
	} else if (t <= 4.6e+35) {
		tmp = t / (Math.sqrt((t_2 + t_4)) / Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(((x + -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)
	t_2 = (2.0 * (math.pow(t, 2.0) / x)) + (t_1 + (math.pow(l, 2.0) / x))
	t_3 = t_1 + math.pow(l, 2.0)
	t_4 = t_3 / x
	tmp = 0
	if t <= -3.2e+64:
		tmp = (1.0 / x) + (-1.0 - (0.5 / math.pow(x, 2.0)))
	elif t <= -5e-125:
		tmp = t / (math.sqrt(((((t_3 + t_3) / math.pow(x, 2.0)) + t_2) + t_4)) / math.sqrt(2.0))
	elif t <= 1.62e-198:
		tmp = (1.0 / l) * (t / math.pow(x, -0.5))
	elif t <= 4.6e+35:
		tmp = t / (math.sqrt((t_2 + t_4)) / math.sqrt(2.0))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(2.0 * (t ^ 2.0))
	t_2 = Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(t_1 + Float64((l ^ 2.0) / x)))
	t_3 = Float64(t_1 + (l ^ 2.0))
	t_4 = Float64(t_3 / x)
	tmp = 0.0
	if (t <= -3.2e+64)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / (x ^ 2.0))));
	elseif (t <= -5e-125)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(Float64(Float64(t_3 + t_3) / (x ^ 2.0)) + t_2) + t_4)) / sqrt(2.0)));
	elseif (t <= 1.62e-198)
		tmp = Float64(Float64(1.0 / l) * Float64(t / (x ^ -0.5)));
	elseif (t <= 4.6e+35)
		tmp = Float64(t / Float64(sqrt(Float64(t_2 + t_4)) / sqrt(2.0)));
	else
		tmp = sqrt(Float64(Float64(x + -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);
	t_2 = (2.0 * ((t ^ 2.0) / x)) + (t_1 + ((l ^ 2.0) / x));
	t_3 = t_1 + (l ^ 2.0);
	t_4 = t_3 / x;
	tmp = 0.0;
	if (t <= -3.2e+64)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x ^ 2.0)));
	elseif (t <= -5e-125)
		tmp = t / (sqrt(((((t_3 + t_3) / (x ^ 2.0)) + t_2) + t_4)) / sqrt(2.0));
	elseif (t <= 1.62e-198)
		tmp = (1.0 / l) * (t / (x ^ -0.5));
	elseif (t <= 4.6e+35)
		tmp = t / (sqrt((t_2 + t_4)) / sqrt(2.0));
	else
		tmp = sqrt(((x + -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]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / x), $MachinePrecision]}, If[LessEqual[t, -3.2e+64], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5e-125], N[(t / N[(N[Sqrt[N[(N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.62e-198], N[(N[(1.0 / l), $MachinePrecision] * N[(t / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e+35], N[(t / N[(N[Sqrt[N[(t$95$2 + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -3.20000000000000019e64

   1. Initial program 27.6%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in t around -inf 89.3%

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

      1. associate-*r* 89.3%

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

      2. neg-mul-1 89.3%

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

      3. distribute-rgt-neg-out 89.3%

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

      4. +-commutative 89.3%

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

      5. sub-neg 89.3%

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

      6. metadata-eval 89.3%

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

      7. +-commutative 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in x around inf 89.4%

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

      1. associate-*r/ 89.4%

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

      2. metadata-eval 89.4%

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

   9. Simplified 89.4%

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

   if -3.20000000000000019e64 < t < -4.99999999999999967e-125

   1. Initial program 48.3%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in x around -inf 89.5%

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

   if -4.99999999999999967e-125 < t < 1.62e-198

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 58.4%

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

   5. Taylor expanded in l around inf 47.9%

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

   6. Taylor expanded in l around 0 47.9%

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

      1. *-un-lft-identity 47.9%

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

      2. inv-pow 47.9%

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

      3. sqrt-pow1 47.9%

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

      4. metadata-eval 47.9%

         \[\leadsto \frac{1 \cdot t}{\ell \cdot {x}^{\color{blue}{-0.5}}} \]

      5. times-frac 48.1%

         \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   8. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   if 1.62e-198 < t < 4.5999999999999996e35

   1. Initial program 50.8%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in x around inf 88.4%

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

   if 4.5999999999999996e35 < t

   1. Initial program 41.0%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in t around inf 92.9%

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

      1. +-commutative 92.9%

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

      2. sub-neg 92.9%

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

      3. metadata-eval 92.9%

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

      4. +-commutative 92.9%

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

   6. Simplified 92.9%

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

   7. Taylor expanded in t around 0 92.9%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{{x}^{2}}\right)\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-125}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 2: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot {t}^{2}\\ t_2 := \frac{{\ell}^{2}}{x}\\ \mathbf{if}\;t \leq -1.86 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{{x}^{2}}\right)\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-125}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t_2 + {t}^{2} \cdot \frac{1 + x}{x + -1}\right)}}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(t_1 + t_2\right)\right) + \frac{t_1 + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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))) (t_2 (/ (pow l 2.0) x)))
   (if (<= t -1.86e+65)
     (+ (/ 1.0 x) (- -1.0 (/ 0.5 (pow x 2.0))))
     (if (<= t -5e-125)
       (/
        (* t (sqrt 2.0))
        (sqrt (* 2.0 (+ t_2 (* (pow t 2.0) (/ (+ 1.0 x) (+ x -1.0)))))))
       (if (<= t 1.62e-198)
         (* (/ 1.0 l) (/ t (pow x -0.5)))
         (if (<= t 2.9e+35)
           (/
            t
            (/
             (sqrt
              (+
               (+ (* 2.0 (/ (pow t 2.0) x)) (+ t_1 t_2))
               (/ (+ t_1 (pow l 2.0)) x)))
             (sqrt 2.0)))
           (sqrt (/ (+ 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 t_2 = pow(l, 2.0) / x;
	double tmp;
	if (t <= -1.86e+65) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / pow(x, 2.0)));
	} else if (t <= -5e-125) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * (t_2 + (pow(t, 2.0) * ((1.0 + x) / (x + -1.0))))));
	} else if (t <= 1.62e-198) {
		tmp = (1.0 / l) * (t / pow(x, -0.5));
	} else if (t <= 2.9e+35) {
		tmp = t / (sqrt((((2.0 * (pow(t, 2.0) / x)) + (t_1 + t_2)) + ((t_1 + pow(l, 2.0)) / x))) / sqrt(2.0));
	} else {
		tmp = sqrt(((x + -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) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * (t ** 2.0d0)
    t_2 = (l ** 2.0d0) / x
    if (t <= (-1.86d+65)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x ** 2.0d0)))
    else if (t <= (-5d-125)) then
        tmp = (t * sqrt(2.0d0)) / sqrt((2.0d0 * (t_2 + ((t ** 2.0d0) * ((1.0d0 + x) / (x + (-1.0d0)))))))
    else if (t <= 1.62d-198) then
        tmp = (1.0d0 / l) * (t / (x ** (-0.5d0)))
    else if (t <= 2.9d+35) then
        tmp = t / (sqrt((((2.0d0 * ((t ** 2.0d0) / x)) + (t_1 + t_2)) + ((t_1 + (l ** 2.0d0)) / x))) / sqrt(2.0d0))
    else
        tmp = sqrt(((x + (-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 t_2 = Math.pow(l, 2.0) / x;
	double tmp;
	if (t <= -1.86e+65) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / Math.pow(x, 2.0)));
	} else if (t <= -5e-125) {
		tmp = (t * Math.sqrt(2.0)) / Math.sqrt((2.0 * (t_2 + (Math.pow(t, 2.0) * ((1.0 + x) / (x + -1.0))))));
	} else if (t <= 1.62e-198) {
		tmp = (1.0 / l) * (t / Math.pow(x, -0.5));
	} else if (t <= 2.9e+35) {
		tmp = t / (Math.sqrt((((2.0 * (Math.pow(t, 2.0) / x)) + (t_1 + t_2)) + ((t_1 + Math.pow(l, 2.0)) / x))) / Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(((x + -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)
	t_2 = math.pow(l, 2.0) / x
	tmp = 0
	if t <= -1.86e+65:
		tmp = (1.0 / x) + (-1.0 - (0.5 / math.pow(x, 2.0)))
	elif t <= -5e-125:
		tmp = (t * math.sqrt(2.0)) / math.sqrt((2.0 * (t_2 + (math.pow(t, 2.0) * ((1.0 + x) / (x + -1.0))))))
	elif t <= 1.62e-198:
		tmp = (1.0 / l) * (t / math.pow(x, -0.5))
	elif t <= 2.9e+35:
		tmp = t / (math.sqrt((((2.0 * (math.pow(t, 2.0) / x)) + (t_1 + t_2)) + ((t_1 + math.pow(l, 2.0)) / x))) / math.sqrt(2.0))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(2.0 * (t ^ 2.0))
	t_2 = Float64((l ^ 2.0) / x)
	tmp = 0.0
	if (t <= -1.86e+65)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / (x ^ 2.0))));
	elseif (t <= -5e-125)
		tmp = Float64(Float64(t * sqrt(2.0)) / sqrt(Float64(2.0 * Float64(t_2 + Float64((t ^ 2.0) * Float64(Float64(1.0 + x) / Float64(x + -1.0)))))));
	elseif (t <= 1.62e-198)
		tmp = Float64(Float64(1.0 / l) * Float64(t / (x ^ -0.5)));
	elseif (t <= 2.9e+35)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(t_1 + t_2)) + Float64(Float64(t_1 + (l ^ 2.0)) / x))) / sqrt(2.0)));
	else
		tmp = sqrt(Float64(Float64(x + -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);
	t_2 = (l ^ 2.0) / x;
	tmp = 0.0;
	if (t <= -1.86e+65)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x ^ 2.0)));
	elseif (t <= -5e-125)
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * (t_2 + ((t ^ 2.0) * ((1.0 + x) / (x + -1.0))))));
	elseif (t <= 1.62e-198)
		tmp = (1.0 / l) * (t / (x ^ -0.5));
	elseif (t <= 2.9e+35)
		tmp = t / (sqrt((((2.0 * ((t ^ 2.0) / x)) + (t_1 + t_2)) + ((t_1 + (l ^ 2.0)) / x))) / sqrt(2.0));
	else
		tmp = sqrt(((x + -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]}, Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -1.86e+65], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5e-125], N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(t$95$2 + N[(N[Power[t, 2.0], $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.62e-198], N[(N[(1.0 / l), $MachinePrecision] * N[(t / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.9e+35], N[(t / N[(N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + t$95$2), $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], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.8599999999999999e65

   1. Initial program 27.6%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in t around -inf 89.3%

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

      1. associate-*r* 89.3%

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

      2. neg-mul-1 89.3%

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

      3. distribute-rgt-neg-out 89.3%

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

      4. +-commutative 89.3%

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

      5. sub-neg 89.3%

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

      6. metadata-eval 89.3%

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

      7. +-commutative 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in x around inf 89.4%

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

      1. associate-*r/ 89.4%

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

      2. metadata-eval 89.4%

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

   9. Simplified 89.4%

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

   if -1.8599999999999999e65 < t < -4.99999999999999967e-125

   1. Initial program 48.3%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in l around 0 66.9%

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

      1. fma-def 66.9%

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

      2. +-commutative 66.9%

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

      3. associate-*r/ 66.9%

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

      4. sub-neg 66.9%

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

      5. metadata-eval 66.9%

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

      6. +-commutative 66.9%

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

      7. associate--l+ 70.7%

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

      8. sub-neg 70.7%

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

      9. metadata-eval 70.7%

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

      10. +-commutative 70.7%

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

      11. sub-neg 70.7%

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

      12. metadata-eval 70.7%

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

      13. +-commutative 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in x around inf 89.2%

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

      1. associate-*r/ 89.4%

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

      2. associate-*r/ 89.4%

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

      3. +-commutative 89.4%

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

   9. Applied egg-rr 89.4%

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

       1. expm1-log1p-u 86.7%

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

       2. expm1-udef 36.5%

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

       3. associate-/l* 36.5%

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

   11. Applied egg-rr 36.5%

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

       1. expm1-def 86.7%

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

       2. expm1-log1p 89.4%

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

       3. fma-udef 89.4%

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

       4. distribute-lft-out 89.4%

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

       5. associate-/l* 89.4%

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

       6. associate-*r/ 89.4%

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

       7. +-commutative 89.4%

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

   13. Simplified 89.4%

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

   if -4.99999999999999967e-125 < t < 1.62e-198

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 58.4%

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

   5. Taylor expanded in l around inf 47.9%

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

   6. Taylor expanded in l around 0 47.9%

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

      1. *-un-lft-identity 47.9%

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

      2. inv-pow 47.9%

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

      3. sqrt-pow1 47.9%

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

      4. metadata-eval 47.9%

         \[\leadsto \frac{1 \cdot t}{\ell \cdot {x}^{\color{blue}{-0.5}}} \]

      5. times-frac 48.1%

         \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   8. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   if 1.62e-198 < t < 2.89999999999999995e35

   1. Initial program 50.8%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in x around inf 88.4%

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

   if 2.89999999999999995e35 < t

   1. Initial program 41.0%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in t around inf 92.9%

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

      1. +-commutative 92.9%

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

      2. sub-neg 92.9%

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

      3. metadata-eval 92.9%

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

      4. +-commutative 92.9%

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

   6. Simplified 92.9%

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

   7. Taylor expanded in t around 0 92.9%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.86 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{{x}^{2}}\right)\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-125}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \frac{1 + x}{x + -1}\right)}}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 3: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \frac{1 + x}{x + -1}\right)}}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{{x}^{2}}\right)\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-188}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          (* t (sqrt 2.0))
          (sqrt
           (*
            2.0
            (+ (/ (pow l 2.0) x) (* (pow t 2.0) (/ (+ 1.0 x) (+ x -1.0)))))))))
   (if (<= t -1.35e+63)
     (+ (/ 1.0 x) (- -1.0 (/ 0.5 (pow x 2.0))))
     (if (<= t -4.5e-125)
       t_1
       (if (<= t 3e-188)
         (* (/ 1.0 l) (/ t (pow x -0.5)))
         (if (<= t 4e+35) t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (t * sqrt(2.0)) / sqrt((2.0 * ((pow(l, 2.0) / x) + (pow(t, 2.0) * ((1.0 + x) / (x + -1.0))))));
	double tmp;
	if (t <= -1.35e+63) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / pow(x, 2.0)));
	} else if (t <= -4.5e-125) {
		tmp = t_1;
	} else if (t <= 3e-188) {
		tmp = (1.0 / l) * (t / pow(x, -0.5));
	} else if (t <= 4e+35) {
		tmp = t_1;
	} else {
		tmp = sqrt(((x + -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 = (t * sqrt(2.0d0)) / sqrt((2.0d0 * (((l ** 2.0d0) / x) + ((t ** 2.0d0) * ((1.0d0 + x) / (x + (-1.0d0)))))))
    if (t <= (-1.35d+63)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x ** 2.0d0)))
    else if (t <= (-4.5d-125)) then
        tmp = t_1
    else if (t <= 3d-188) then
        tmp = (1.0d0 / l) * (t / (x ** (-0.5d0)))
    else if (t <= 4d+35) then
        tmp = t_1
    else
        tmp = sqrt(((x + (-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 = (t * Math.sqrt(2.0)) / Math.sqrt((2.0 * ((Math.pow(l, 2.0) / x) + (Math.pow(t, 2.0) * ((1.0 + x) / (x + -1.0))))));
	double tmp;
	if (t <= -1.35e+63) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / Math.pow(x, 2.0)));
	} else if (t <= -4.5e-125) {
		tmp = t_1;
	} else if (t <= 3e-188) {
		tmp = (1.0 / l) * (t / Math.pow(x, -0.5));
	} else if (t <= 4e+35) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (t * math.sqrt(2.0)) / math.sqrt((2.0 * ((math.pow(l, 2.0) / x) + (math.pow(t, 2.0) * ((1.0 + x) / (x + -1.0))))))
	tmp = 0
	if t <= -1.35e+63:
		tmp = (1.0 / x) + (-1.0 - (0.5 / math.pow(x, 2.0)))
	elif t <= -4.5e-125:
		tmp = t_1
	elif t <= 3e-188:
		tmp = (1.0 / l) * (t / math.pow(x, -0.5))
	elif t <= 4e+35:
		tmp = t_1
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(t * sqrt(2.0)) / sqrt(Float64(2.0 * Float64(Float64((l ^ 2.0) / x) + Float64((t ^ 2.0) * Float64(Float64(1.0 + x) / Float64(x + -1.0)))))))
	tmp = 0.0
	if (t <= -1.35e+63)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / (x ^ 2.0))));
	elseif (t <= -4.5e-125)
		tmp = t_1;
	elseif (t <= 3e-188)
		tmp = Float64(Float64(1.0 / l) * Float64(t / (x ^ -0.5)));
	elseif (t <= 4e+35)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (t * sqrt(2.0)) / sqrt((2.0 * (((l ^ 2.0) / x) + ((t ^ 2.0) * ((1.0 + x) / (x + -1.0))))));
	tmp = 0.0;
	if (t <= -1.35e+63)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x ^ 2.0)));
	elseif (t <= -4.5e-125)
		tmp = t_1;
	elseif (t <= 3e-188)
		tmp = (1.0 / l) * (t / (x ^ -0.5));
	elseif (t <= 4e+35)
		tmp = t_1;
	else
		tmp = sqrt(((x + -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[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[t, 2.0], $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.35e+63], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.5e-125], t$95$1, If[LessEqual[t, 3e-188], N[(N[(1.0 / l), $MachinePrecision] * N[(t / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+35], t$95$1, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.35000000000000009e63

   1. Initial program 27.6%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in t around -inf 89.3%

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

      1. associate-*r* 89.3%

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

      2. neg-mul-1 89.3%

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

      3. distribute-rgt-neg-out 89.3%

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

      4. +-commutative 89.3%

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

      5. sub-neg 89.3%

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

      6. metadata-eval 89.3%

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

      7. +-commutative 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in x around inf 89.4%

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

      1. associate-*r/ 89.4%

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

      2. metadata-eval 89.4%

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

   9. Simplified 89.4%

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

   if -1.35000000000000009e63 < t < -4.50000000000000012e-125 or 3.00000000000000017e-188 < t < 3.9999999999999999e35

   1. Initial program 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in l around 0 66.0%

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

      1. fma-def 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-*r/ 67.1%

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

      4. sub-neg 67.1%

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

      5. metadata-eval 67.1%

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

      6. +-commutative 67.1%

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

      7. associate--l+ 71.6%

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

      8. sub-neg 71.6%

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

      9. metadata-eval 71.6%

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

      10. +-commutative 71.6%

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

      11. sub-neg 71.6%

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

      12. metadata-eval 71.6%

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

      13. +-commutative 71.6%

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

   6. Simplified 71.6%

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

   7. Taylor expanded in x around inf 88.5%

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

      1. associate-*r/ 88.6%

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

      2. associate-*r/ 87.5%

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

      3. +-commutative 87.5%

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

   9. Applied egg-rr 87.5%

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

       1. expm1-log1p-u 85.7%

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

       2. expm1-udef 30.5%

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

       3. associate-/l* 31.6%

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

   11. Applied egg-rr 31.6%

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

       1. expm1-def 86.8%

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

       2. expm1-log1p 88.6%

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

       3. fma-udef 88.6%

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

       4. distribute-lft-out 88.6%

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

       5. associate-/l* 87.5%

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

       6. associate-*r/ 88.6%

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

       7. +-commutative 88.6%

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

   13. Simplified 88.6%

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

   if -4.50000000000000012e-125 < t < 3.00000000000000017e-188

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 59.2%

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

   5. Taylor expanded in l around inf 48.8%

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

   6. Taylor expanded in l around 0 48.9%

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

      1. *-un-lft-identity 48.9%

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

      2. inv-pow 48.9%

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

      3. sqrt-pow1 48.9%

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

      4. metadata-eval 48.9%

         \[\leadsto \frac{1 \cdot t}{\ell \cdot {x}^{\color{blue}{-0.5}}} \]

      5. times-frac 49.0%

         \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   8. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   if 3.9999999999999999e35 < t

   1. Initial program 41.0%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in t around inf 92.9%

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

      1. +-commutative 92.9%

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

      2. sub-neg 92.9%

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

      3. metadata-eval 92.9%

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

      4. +-commutative 92.9%

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

   6. Simplified 92.9%

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

   7. Taylor expanded in t around 0 92.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{{x}^{2}}\right)\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \frac{1 + x}{x + -1}\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-188}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+35}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \frac{1 + x}{x + -1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 4: 79.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-125}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-168}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -5.4e-125)
     (- t_1)
     (if (<= t 5e-168) (* (/ 1.0 l) (/ t (pow x -0.5))) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -5.4e-125) {
		tmp = -t_1;
	} else if (t <= 5e-168) {
		tmp = (1.0 / l) * (t / pow(x, -0.5));
	} else {
		tmp = t_1;
	}
	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 = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    if (t <= (-5.4d-125)) then
        tmp = -t_1
    else if (t <= 5d-168) then
        tmp = (1.0d0 / l) * (t / (x ** (-0.5d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -5.4e-125) {
		tmp = -t_1;
	} else if (t <= 5e-168) {
		tmp = (1.0 / l) * (t / Math.pow(x, -0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -5.4e-125:
		tmp = -t_1
	elif t <= 5e-168:
		tmp = (1.0 / l) * (t / math.pow(x, -0.5))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -5.4e-125)
		tmp = Float64(-t_1);
	elseif (t <= 5e-168)
		tmp = Float64(Float64(1.0 / l) * Float64(t / (x ^ -0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -5.4e-125)
		tmp = -t_1;
	elseif (t <= 5e-168)
		tmp = (1.0 / l) * (t / (x ^ -0.5));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -5.4e-125], (-t$95$1), If[LessEqual[t, 5e-168], N[(N[(1.0 / l), $MachinePrecision] * N[(t / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.3999999999999995e-125

   1. Initial program 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in t around -inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

      6. metadata-eval 79.8%

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

      7. +-commutative 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in t around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. sub-neg 79.9%

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

      3. metadata-eval 79.9%

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

      4. +-commutative 79.9%

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

   9. Simplified 79.9%

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

   if -5.3999999999999995e-125 < t < 5.00000000000000001e-168

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in l around inf 48.9%

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

   6. Taylor expanded in l around 0 49.0%

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

      1. *-un-lft-identity 49.0%

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

      2. inv-pow 49.0%

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

      3. sqrt-pow1 49.0%

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

      4. metadata-eval 49.0%

         \[\leadsto \frac{1 \cdot t}{\ell \cdot {x}^{\color{blue}{-0.5}}} \]

      5. times-frac 49.1%

         \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   8. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   if 5.00000000000000001e-168 < t

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in t around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

      4. +-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in t around 0 84.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-125}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-168}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 5: 79.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{{x}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -4.5e-125)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (pow x 2.0))))
   (if (<= t 5.7e-165)
     (* (/ 1.0 l) (/ t (pow x -0.5)))
     (sqrt (/ (+ x -1.0) (+ 1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.5e-125) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / pow(x, 2.0)));
	} else if (t <= 5.7e-165) {
		tmp = (1.0 / l) * (t / pow(x, -0.5));
	} else {
		tmp = sqrt(((x + -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 <= (-4.5d-125)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x ** 2.0d0)))
    else if (t <= 5.7d-165) then
        tmp = (1.0d0 / l) * (t / (x ** (-0.5d0)))
    else
        tmp = sqrt(((x + (-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 <= -4.5e-125) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / Math.pow(x, 2.0)));
	} else if (t <= 5.7e-165) {
		tmp = (1.0 / l) * (t / Math.pow(x, -0.5));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -4.5e-125:
		tmp = (1.0 / x) + (-1.0 - (0.5 / math.pow(x, 2.0)))
	elif t <= 5.7e-165:
		tmp = (1.0 / l) * (t / math.pow(x, -0.5))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -4.5e-125)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / (x ^ 2.0))));
	elseif (t <= 5.7e-165)
		tmp = Float64(Float64(1.0 / l) * Float64(t / (x ^ -0.5)));
	else
		tmp = sqrt(Float64(Float64(x + -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 <= -4.5e-125)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x ^ 2.0)));
	elseif (t <= 5.7e-165)
		tmp = (1.0 / l) * (t / (x ^ -0.5));
	else
		tmp = sqrt(((x + -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, -4.5e-125], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.7e-165], N[(N[(1.0 / l), $MachinePrecision] * N[(t / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.50000000000000012e-125

   1. Initial program 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in t around -inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

      6. metadata-eval 79.8%

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

      7. +-commutative 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in x around inf 79.9%

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

      1. associate-*r/ 79.9%

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

      2. metadata-eval 79.9%

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

   9. Simplified 79.9%

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

   if -4.50000000000000012e-125 < t < 5.70000000000000003e-165

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in l around inf 48.9%

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

   6. Taylor expanded in l around 0 49.0%

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

      1. *-un-lft-identity 49.0%

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

      2. inv-pow 49.0%

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

      3. sqrt-pow1 49.0%

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

      4. metadata-eval 49.0%

         \[\leadsto \frac{1 \cdot t}{\ell \cdot {x}^{\color{blue}{-0.5}}} \]

      5. times-frac 49.1%

         \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   8. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

   if 5.70000000000000003e-165 < t

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in t around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

      4. +-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in t around 0 84.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{{x}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 6: 79.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5.8e-125)
   (+ (/ 1.0 x) -1.0)
   (if (<= t 9.5e-167)
     (/ t (* l (pow x -0.5)))
     (sqrt (/ (+ x -1.0) (+ 1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.8e-125) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 9.5e-167) {
		tmp = t / (l * pow(x, -0.5));
	} else {
		tmp = sqrt(((x + -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 <= (-5.8d-125)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 9.5d-167) then
        tmp = t / (l * (x ** (-0.5d0)))
    else
        tmp = sqrt(((x + (-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 <= -5.8e-125) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 9.5e-167) {
		tmp = t / (l * Math.pow(x, -0.5));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5.8e-125:
		tmp = (1.0 / x) + -1.0
	elif t <= 9.5e-167:
		tmp = t / (l * math.pow(x, -0.5))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5.8e-125)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 9.5e-167)
		tmp = Float64(t / Float64(l * (x ^ -0.5)));
	else
		tmp = sqrt(Float64(Float64(x + -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 <= -5.8e-125)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 9.5e-167)
		tmp = t / (l * (x ^ -0.5));
	else
		tmp = sqrt(((x + -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, -5.8e-125], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 9.5e-167], N[(t / N[(l * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.8000000000000004e-125

   1. Initial program 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in t around -inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

      6. metadata-eval 79.8%

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

      7. +-commutative 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in x around inf 79.8%

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

   if -5.8000000000000004e-125 < t < 9.49999999999999955e-167

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in l around inf 48.9%

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

      1. div-inv 49.0%

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

      2. associate-/l* 48.9%

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

      3. inv-pow 48.9%

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

      4. sqrt-pow1 48.8%

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

      5. metadata-eval 48.8%

         \[\leadsto t \cdot \frac{1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}}}} \]

   7. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{t \cdot \frac{1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.8%

         \[\leadsto \color{blue}{\frac{t \cdot 1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}}} \]

      2. *-rgt-identity 48.8%

         \[\leadsto \frac{\color{blue}{t}}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}} \]

      3. associate-/r/ 48.9%

         \[\leadsto \frac{t}{\color{blue}{\frac{\ell \cdot \sqrt{2}}{\sqrt{2}} \cdot {x}^{-0.5}}} \]

      4. associate-/l* 49.0%

         \[\leadsto \frac{t}{\color{blue}{\frac{\ell}{\frac{\sqrt{2}}{\sqrt{2}}}} \cdot {x}^{-0.5}} \]

      5. *-inverses 49.0%

         \[\leadsto \frac{t}{\frac{\ell}{\color{blue}{1}} \cdot {x}^{-0.5}} \]

      6. /-rgt-identity 49.0%

         \[\leadsto \frac{t}{\color{blue}{\ell} \cdot {x}^{-0.5}} \]

   9. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}} \]

   if 9.49999999999999955e-167 < t

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in t around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

      4. +-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in t around 0 84.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 7: 79.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-124}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-172}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ 1.0 x)))))
   (if (<= t -1.6e-124)
     (- t_1)
     (if (<= t 1.06e-172) (/ t (* l (pow x -0.5))) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -1.6e-124) {
		tmp = -t_1;
	} else if (t <= 1.06e-172) {
		tmp = t / (l * pow(x, -0.5));
	} else {
		tmp = t_1;
	}
	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 = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    if (t <= (-1.6d-124)) then
        tmp = -t_1
    else if (t <= 1.06d-172) then
        tmp = t / (l * (x ** (-0.5d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (1.0 + x)));
	double tmp;
	if (t <= -1.6e-124) {
		tmp = -t_1;
	} else if (t <= 1.06e-172) {
		tmp = t / (l * Math.pow(x, -0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (1.0 + x)))
	tmp = 0
	if t <= -1.6e-124:
		tmp = -t_1
	elif t <= 1.06e-172:
		tmp = t / (l * math.pow(x, -0.5))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)))
	tmp = 0.0
	if (t <= -1.6e-124)
		tmp = Float64(-t_1);
	elseif (t <= 1.06e-172)
		tmp = Float64(t / Float64(l * (x ^ -0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (1.0 + x)));
	tmp = 0.0;
	if (t <= -1.6e-124)
		tmp = -t_1;
	elseif (t <= 1.06e-172)
		tmp = t / (l * (x ^ -0.5));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.6e-124], (-t$95$1), If[LessEqual[t, 1.06e-172], N[(t / N[(l * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.60000000000000002e-124

   1. Initial program 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in t around -inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

      6. metadata-eval 79.8%

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

      7. +-commutative 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in t around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. sub-neg 79.9%

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

      3. metadata-eval 79.9%

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

      4. +-commutative 79.9%

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

   9. Simplified 79.9%

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

   if -1.60000000000000002e-124 < t < 1.05999999999999993e-172

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in l around inf 48.9%

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

      1. div-inv 49.0%

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

      2. associate-/l* 48.9%

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

      3. inv-pow 48.9%

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

      4. sqrt-pow1 48.8%

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

      5. metadata-eval 48.8%

         \[\leadsto t \cdot \frac{1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}}}} \]

   7. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{t \cdot \frac{1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.8%

         \[\leadsto \color{blue}{\frac{t \cdot 1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}}} \]

      2. *-rgt-identity 48.8%

         \[\leadsto \frac{\color{blue}{t}}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}} \]

      3. associate-/r/ 48.9%

         \[\leadsto \frac{t}{\color{blue}{\frac{\ell \cdot \sqrt{2}}{\sqrt{2}} \cdot {x}^{-0.5}}} \]

      4. associate-/l* 49.0%

         \[\leadsto \frac{t}{\color{blue}{\frac{\ell}{\frac{\sqrt{2}}{\sqrt{2}}}} \cdot {x}^{-0.5}} \]

      5. *-inverses 49.0%

         \[\leadsto \frac{t}{\frac{\ell}{\color{blue}{1}} \cdot {x}^{-0.5}} \]

      6. /-rgt-identity 49.0%

         \[\leadsto \frac{t}{\color{blue}{\ell} \cdot {x}^{-0.5}} \]

   9. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}} \]

   if 1.05999999999999993e-172 < t

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in t around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

      4. +-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in t around 0 84.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-124}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-172}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 8: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-165}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.26e-124)
   (+ (/ 1.0 x) -1.0)
   (if (<= t 1.36e-165) (/ t (* l (pow x -0.5))) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.26e-124) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.36e-165) {
		tmp = t / (l * pow(x, -0.5));
	} 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 <= (-1.26d-124)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 1.36d-165) then
        tmp = t / (l * (x ** (-0.5d0)))
    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 <= -1.26e-124) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.36e-165) {
		tmp = t / (l * Math.pow(x, -0.5));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.26e-124:
		tmp = (1.0 / x) + -1.0
	elif t <= 1.36e-165:
		tmp = t / (l * math.pow(x, -0.5))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.26e-124)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 1.36e-165)
		tmp = Float64(t / Float64(l * (x ^ -0.5)));
	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 <= -1.26e-124)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 1.36e-165)
		tmp = t / (l * (x ^ -0.5));
	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, -1.26e-124], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 1.36e-165], N[(t / N[(l * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2600000000000001e-124

   1. Initial program 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in t around -inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

      6. metadata-eval 79.8%

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

      7. +-commutative 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in x around inf 79.8%

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

   if -1.2600000000000001e-124 < t < 1.3599999999999999e-165

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in l around inf 48.9%

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

      1. div-inv 49.0%

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

      2. associate-/l* 48.9%

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

      3. inv-pow 48.9%

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

      4. sqrt-pow1 48.8%

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

      5. metadata-eval 48.8%

         \[\leadsto t \cdot \frac{1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}}}} \]

   7. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{t \cdot \frac{1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 48.8%

         \[\leadsto \color{blue}{\frac{t \cdot 1}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}}} \]

      2. *-rgt-identity 48.8%

         \[\leadsto \frac{\color{blue}{t}}{\frac{\ell \cdot \sqrt{2}}{\frac{\sqrt{2}}{{x}^{-0.5}}}} \]

      3. associate-/r/ 48.9%

         \[\leadsto \frac{t}{\color{blue}{\frac{\ell \cdot \sqrt{2}}{\sqrt{2}} \cdot {x}^{-0.5}}} \]

      4. associate-/l* 49.0%

         \[\leadsto \frac{t}{\color{blue}{\frac{\ell}{\frac{\sqrt{2}}{\sqrt{2}}}} \cdot {x}^{-0.5}} \]

      5. *-inverses 49.0%

         \[\leadsto \frac{t}{\frac{\ell}{\color{blue}{1}} \cdot {x}^{-0.5}} \]

      6. /-rgt-identity 49.0%

         \[\leadsto \frac{t}{\color{blue}{\ell} \cdot {x}^{-0.5}} \]

   9. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}} \]

   if 1.3599999999999999e-165 < t

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in t around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

      4. +-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in x around inf 83.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-165}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 9: 79.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-167}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -6.1e-125)
   (+ (/ 1.0 x) -1.0)
   (if (<= t 2.5e-167) (* 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 <= -6.1e-125) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 2.5e-167) {
		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 <= (-6.1d-125)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 2.5d-167) 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 <= -6.1e-125) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 2.5e-167) {
		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 <= -6.1e-125:
		tmp = (1.0 / x) + -1.0
	elif t <= 2.5e-167:
		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 <= -6.1e-125)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 2.5e-167)
		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 <= -6.1e-125)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 2.5e-167)
		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, -6.1e-125], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 2.5e-167], 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 < -6.10000000000000003e-125

   1. Initial program 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in t around -inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

      6. metadata-eval 79.8%

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

      7. +-commutative 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in x around inf 79.8%

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

   if -6.10000000000000003e-125 < t < 2.5000000000000001e-167

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in l around inf 48.9%

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

   6. Taylor expanded in l around 0 49.0%

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

   7. Taylor expanded in t around 0 43.8%

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

      1. associate-*l/ 49.1%

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

      2. associate-/l* 48.9%

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

      3. *-rgt-identity 48.9%

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

      4. associate-*r/ 48.9%

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

      5. associate-/r/ 48.9%

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

      6. associate-*l/ 48.9%

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

      7. *-lft-identity 48.9%

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

   9. Simplified 48.9%

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

   if 2.5000000000000001e-167 < t

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in t around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

      4. +-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in x around inf 83.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-167}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 10: 79.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5.8e-125)
   (+ (/ 1.0 x) -1.0)
   (if (<= t 1.3e-161) (/ 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 <= -5.8e-125) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.3e-161) {
		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 <= (-5.8d-125)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 1.3d-161) 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 <= -5.8e-125) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.3e-161) {
		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 <= -5.8e-125:
		tmp = (1.0 / x) + -1.0
	elif t <= 1.3e-161:
		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 <= -5.8e-125)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 1.3e-161)
		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 <= -5.8e-125)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 1.3e-161)
		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, -5.8e-125], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 1.3e-161], 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 < -5.8000000000000004e-125

   1. Initial program 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in t around -inf 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. +-commutative 79.8%

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

      5. sub-neg 79.8%

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

      6. metadata-eval 79.8%

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

      7. +-commutative 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in x around inf 79.8%

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

   if -5.8000000000000004e-125 < t < 1.29999999999999998e-161

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in x around inf 60.7%

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

   5. Taylor expanded in l around inf 48.9%

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

   6. Taylor expanded in l around 0 49.0%

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

      1. expm1-log1p-u 30.3%

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

      2. expm1-udef 17.1%

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

      3. add-sqr-sqrt 16.4%

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

      4. sqrt-prod 37.2%

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

      5. unpow2 37.2%

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

      6. sqrt-prod 37.6%

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

      7. div-inv 37.6%

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

      8. sqrt-div 37.2%

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

      9. unpow2 37.2%

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

      10. sqrt-prod 16.4%

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

      11. add-sqr-sqrt 17.1%

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

   8. Applied egg-rr 17.1%

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

      1. expm1-def 30.2%

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

      2. expm1-log1p 48.9%

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

   10. Simplified 48.9%

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

   if 1.29999999999999998e-161 < t

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in t around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

      4. +-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in x around inf 83.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11: 75.3% accurate, 31.9× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (/ t (- t)) (+ 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;
	} 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
    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;
	} 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
	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(-t));
	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;
	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 / (-t)), $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 26.1%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in t around inf 1.8%

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

      1. +-commutative 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

      4. +-commutative 1.8%

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

   6. Simplified 1.8%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 64.7%

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

      4. mul-1-neg 64.7%

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

   9. Simplified 64.7%

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

   if -4.999999999999985e-310 < t

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 76.4%

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

      1. +-commutative 76.4%

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

      2. sub-neg 76.4%

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

      3. metadata-eval 76.4%

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

      4. +-commutative 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in x around inf 76.1%

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

4. Final simplification 70.6%

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

Alternative 12: 75.6% accurate, 31.9× speedup

Math

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

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) (+ 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;
	} 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)
    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;
	} 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
	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) + -1.0);
	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;
	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] + -1.0), $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 26.1%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in t around -inf 65.3%

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

      1. associate-*r* 65.3%

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

      2. neg-mul-1 65.3%

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

      3. distribute-rgt-neg-out 65.3%

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

      4. +-commutative 65.3%

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

      5. sub-neg 65.3%

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

      6. metadata-eval 65.3%

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

      7. +-commutative 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in x around inf 65.3%

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

   if -4.999999999999985e-310 < t

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 76.4%

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

      1. +-commutative 76.4%

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

      2. sub-neg 76.4%

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

      3. metadata-eval 76.4%

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

      4. +-commutative 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in x around inf 76.1%

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

4. Final simplification 70.9%

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

Alternative 13: 74.9% accurate, 37.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(t / Float64(-t));
	else
		tmp = Float64(t / t);
	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;
	else
		tmp = t / t;
	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 / (-t)), $MachinePrecision], N[(t / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 26.1%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in t around inf 1.8%

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

      1. +-commutative 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

      4. +-commutative 1.8%

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

   6. Simplified 1.8%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 64.7%

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

      4. mul-1-neg 64.7%

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

   9. Simplified 64.7%

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

   if -4.999999999999985e-310 < t

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in x around inf 75.9%

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

4. Final simplification 70.5%

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

Alternative 14: 38.6% accurate, 75.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{t}{t} \end{array} \]

FPCore

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

C

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

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 = t / t
end function

Java

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

Python

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

Julia

l = abs(l)
function code(x, l, t)
	return Float64(t / t)
end

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := N[(t / t), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 33.4%

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

4. Taylor expanded in x around inf 40.0%

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

5. Final simplification 40.0%

   \[\leadsto \frac{t}{t} \]

Reproduce

herbie shell --seed 2023306 
(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)))))