Toniolo and Linder, Equation (7)

Percentage Accurate: 33.8% → 80.9%

Time: 20.9s

Alternatives: 9

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot {t}^{2}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-163}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(t_1 + \frac{{\ell}^{2}}{x}\right)\right) + \frac{t_1 + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{-254}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-211}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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))))
   (if (<= t -3.4e+87)
     (+ (/ 1.0 x) -1.0)
     (if (<= t -3e-163)
       (/
        t
        (/
         (sqrt
          (+
           (+ (* 2.0 (/ (pow t 2.0) x)) (+ t_1 (/ (pow l 2.0) x)))
           (/ (+ t_1 (pow l 2.0)) x)))
         (sqrt 2.0)))
       (if (<= t -6.6e-254)
         -1.0
         (if (<= t 4.1e-211)
           (* (/ t l) (sqrt x))
           (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 tmp;
	if (t <= -3.4e+87) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -3e-163) {
		tmp = t / (sqrt((((2.0 * (pow(t, 2.0) / x)) + (t_1 + (pow(l, 2.0) / x))) + ((t_1 + pow(l, 2.0)) / x))) / sqrt(2.0));
	} else if (t <= -6.6e-254) {
		tmp = -1.0;
	} else if (t <= 4.1e-211) {
		tmp = (t / l) * sqrt(x);
	} 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 = 2.0d0 * (t ** 2.0d0)
    if (t <= (-3.4d+87)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= (-3d-163)) then
        tmp = t / (sqrt((((2.0d0 * ((t ** 2.0d0) / x)) + (t_1 + ((l ** 2.0d0) / x))) + ((t_1 + (l ** 2.0d0)) / x))) / sqrt(2.0d0))
    else if (t <= (-6.6d-254)) then
        tmp = -1.0d0
    else if (t <= 4.1d-211) then
        tmp = (t / l) * sqrt(x)
    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 tmp;
	if (t <= -3.4e+87) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -3e-163) {
		tmp = t / (Math.sqrt((((2.0 * (Math.pow(t, 2.0) / x)) + (t_1 + (Math.pow(l, 2.0) / x))) + ((t_1 + Math.pow(l, 2.0)) / x))) / Math.sqrt(2.0));
	} else if (t <= -6.6e-254) {
		tmp = -1.0;
	} else if (t <= 4.1e-211) {
		tmp = (t / l) * Math.sqrt(x);
	} 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)
	tmp = 0
	if t <= -3.4e+87:
		tmp = (1.0 / x) + -1.0
	elif t <= -3e-163:
		tmp = t / (math.sqrt((((2.0 * (math.pow(t, 2.0) / x)) + (t_1 + (math.pow(l, 2.0) / x))) + ((t_1 + math.pow(l, 2.0)) / x))) / math.sqrt(2.0))
	elif t <= -6.6e-254:
		tmp = -1.0
	elif t <= 4.1e-211:
		tmp = (t / l) * math.sqrt(x)
	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))
	tmp = 0.0
	if (t <= -3.4e+87)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= -3e-163)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(t_1 + Float64((l ^ 2.0) / x))) + Float64(Float64(t_1 + (l ^ 2.0)) / x))) / sqrt(2.0)));
	elseif (t <= -6.6e-254)
		tmp = -1.0;
	elseif (t <= 4.1e-211)
		tmp = Float64(Float64(t / l) * sqrt(x));
	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);
	tmp = 0.0;
	if (t <= -3.4e+87)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= -3e-163)
		tmp = t / (sqrt((((2.0 * ((t ^ 2.0) / x)) + (t_1 + ((l ^ 2.0) / x))) + ((t_1 + (l ^ 2.0)) / x))) / sqrt(2.0));
	elseif (t <= -6.6e-254)
		tmp = -1.0;
	elseif (t <= 4.1e-211)
		tmp = (t / l) * sqrt(x);
	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]}, If[LessEqual[t, -3.4e+87], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, -3e-163], N[(t / N[(N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6.6e-254], -1.0, If[LessEqual[t, 4.1e-211], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $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 < -3.4000000000000002e87

   1. Initial program 33.1%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in t around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. +-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -3.4000000000000002e87 < t < -3.0000000000000002e-163

   1. Initial program 47.4%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in x around inf 81.6%

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

   if -3.0000000000000002e-163 < t < -6.60000000000000033e-254

   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 t around -inf 69.3%

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

      1. associate-*r* 69.3%

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

      2. neg-mul-1 69.3%

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

      3. +-commutative 69.3%

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

      4. sub-neg 69.3%

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

      5. metadata-eval 69.3%

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

      6. +-commutative 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around inf 69.3%

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

   if -6.60000000000000033e-254 < t < 4.1000000000000002e-211

   1. Initial program 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in x around inf 61.0%

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

   5. Taylor expanded in t around 0 60.8%

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

      1. cancel-sign-sub-inv 60.8%

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

      2. metadata-eval 60.8%

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

      3. distribute-rgt1-in 60.8%

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

      4. metadata-eval 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in t around 0 50.2%

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

   if 4.1000000000000002e-211 < t

   1. Initial program 38.9%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. sub-neg 85.9%

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

      3. metadata-eval 85.9%

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

      4. +-commutative 85.9%

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

   6. Simplified 85.9%

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

   7. Taylor expanded in t around 0 85.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-163}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{-254}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-211}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 2: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-163}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x} + \frac{1}{x + -1}} \cdot \frac{\ell}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-247}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2.95e-92)
   (+ (/ 1.0 x) -1.0)
   (if (<= t -1.26e-163)
     (/ t (* (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0)))) (/ l (sqrt 2.0))))
     (if (<= t -2.25e-247)
       -1.0
       (if (<= t 6.6e-212)
         (* (/ t l) (sqrt x))
         (sqrt (/ (+ x -1.0) (+ 1.0 x))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.95e-92) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -1.26e-163) {
		tmp = t / (sqrt(((1.0 / x) + (1.0 / (x + -1.0)))) * (l / sqrt(2.0)));
	} else if (t <= -2.25e-247) {
		tmp = -1.0;
	} else if (t <= 6.6e-212) {
		tmp = (t / l) * sqrt(x);
	} 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 <= (-2.95d-92)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= (-1.26d-163)) then
        tmp = t / (sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0))))) * (l / sqrt(2.0d0)))
    else if (t <= (-2.25d-247)) then
        tmp = -1.0d0
    else if (t <= 6.6d-212) then
        tmp = (t / l) * sqrt(x)
    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 <= -2.95e-92) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -1.26e-163) {
		tmp = t / (Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0)))) * (l / Math.sqrt(2.0)));
	} else if (t <= -2.25e-247) {
		tmp = -1.0;
	} else if (t <= 6.6e-212) {
		tmp = (t / l) * Math.sqrt(x);
	} 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 <= -2.95e-92:
		tmp = (1.0 / x) + -1.0
	elif t <= -1.26e-163:
		tmp = t / (math.sqrt(((1.0 / x) + (1.0 / (x + -1.0)))) * (l / math.sqrt(2.0)))
	elif t <= -2.25e-247:
		tmp = -1.0
	elif t <= 6.6e-212:
		tmp = (t / l) * math.sqrt(x)
	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 <= -2.95e-92)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= -1.26e-163)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0)))) * Float64(l / sqrt(2.0))));
	elseif (t <= -2.25e-247)
		tmp = -1.0;
	elseif (t <= 6.6e-212)
		tmp = Float64(Float64(t / l) * sqrt(x));
	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 <= -2.95e-92)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= -1.26e-163)
		tmp = t / (sqrt(((1.0 / x) + (1.0 / (x + -1.0)))) * (l / sqrt(2.0)));
	elseif (t <= -2.25e-247)
		tmp = -1.0;
	elseif (t <= 6.6e-212)
		tmp = (t / l) * sqrt(x);
	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, -2.95e-92], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, -1.26e-163], N[(t / N[(N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.25e-247], -1.0, If[LessEqual[t, 6.6e-212], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $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 < -2.95e-92

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in t around -inf 91.2%

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

      1. associate-*r* 91.2%

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

      2. neg-mul-1 91.2%

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

      3. +-commutative 91.2%

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

      4. sub-neg 91.2%

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

      5. metadata-eval 91.2%

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

      6. +-commutative 91.2%

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

   6. Simplified 91.2%

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

   7. Taylor expanded in x around inf 91.2%

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

   if -2.95e-92 < t < -1.26000000000000002e-163

   1. Initial program 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in l around inf 1.7%

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

      1. *-commutative 1.7%

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

      2. associate--l+ 16.0%

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

      3. sub-neg 16.0%

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

      4. metadata-eval 16.0%

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

      5. +-commutative 16.0%

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

      6. sub-neg 16.0%

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

      7. metadata-eval 16.0%

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

      8. +-commutative 16.0%

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

   6. Simplified 16.0%

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

   7. Taylor expanded in x around inf 46.4%

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

   if -1.26000000000000002e-163 < t < -2.2500000000000001e-247

   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 t around -inf 69.3%

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

      1. associate-*r* 69.3%

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

      2. neg-mul-1 69.3%

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

      3. +-commutative 69.3%

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

      4. sub-neg 69.3%

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

      5. metadata-eval 69.3%

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

      6. +-commutative 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around inf 69.3%

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

   if -2.2500000000000001e-247 < t < 6.6000000000000004e-212

   1. Initial program 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in x around inf 61.0%

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

   5. Taylor expanded in t around 0 60.8%

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

      1. cancel-sign-sub-inv 60.8%

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

      2. metadata-eval 60.8%

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

      3. distribute-rgt1-in 60.8%

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

      4. metadata-eval 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in t around 0 50.2%

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

   if 6.6000000000000004e-212 < t

   1. Initial program 38.9%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. sub-neg 85.9%

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

      3. metadata-eval 85.9%

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

      4. +-commutative 85.9%

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

   6. Simplified 85.9%

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

   7. Taylor expanded in t around 0 85.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-163}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x} + \frac{1}{x + -1}} \cdot \frac{\ell}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-247}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 3: 76.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{x \cdot 0.5} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-245}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.6e-81)
   (+ (/ 1.0 x) -1.0)
   (if (<= t -3.2e-163)
     (* (sqrt (* x 0.5)) (/ t (/ l (sqrt 2.0))))
     (if (<= t -4.4e-245)
       -1.0
       (if (<= t 7.2e-212)
         (* (/ t l) (sqrt x))
         (sqrt (/ (+ x -1.0) (+ 1.0 x))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.6e-81) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -3.2e-163) {
		tmp = sqrt((x * 0.5)) * (t / (l / sqrt(2.0)));
	} else if (t <= -4.4e-245) {
		tmp = -1.0;
	} else if (t <= 7.2e-212) {
		tmp = (t / l) * sqrt(x);
	} 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 <= (-1.6d-81)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= (-3.2d-163)) then
        tmp = sqrt((x * 0.5d0)) * (t / (l / sqrt(2.0d0)))
    else if (t <= (-4.4d-245)) then
        tmp = -1.0d0
    else if (t <= 7.2d-212) then
        tmp = (t / l) * sqrt(x)
    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 <= -1.6e-81) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -3.2e-163) {
		tmp = Math.sqrt((x * 0.5)) * (t / (l / Math.sqrt(2.0)));
	} else if (t <= -4.4e-245) {
		tmp = -1.0;
	} else if (t <= 7.2e-212) {
		tmp = (t / l) * Math.sqrt(x);
	} 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 <= -1.6e-81:
		tmp = (1.0 / x) + -1.0
	elif t <= -3.2e-163:
		tmp = math.sqrt((x * 0.5)) * (t / (l / math.sqrt(2.0)))
	elif t <= -4.4e-245:
		tmp = -1.0
	elif t <= 7.2e-212:
		tmp = (t / l) * math.sqrt(x)
	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 <= -1.6e-81)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= -3.2e-163)
		tmp = Float64(sqrt(Float64(x * 0.5)) * Float64(t / Float64(l / sqrt(2.0))));
	elseif (t <= -4.4e-245)
		tmp = -1.0;
	elseif (t <= 7.2e-212)
		tmp = Float64(Float64(t / l) * sqrt(x));
	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 <= -1.6e-81)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= -3.2e-163)
		tmp = sqrt((x * 0.5)) * (t / (l / sqrt(2.0)));
	elseif (t <= -4.4e-245)
		tmp = -1.0;
	elseif (t <= 7.2e-212)
		tmp = (t / l) * sqrt(x);
	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, -1.6e-81], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, -3.2e-163], N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.4e-245], -1.0, If[LessEqual[t, 7.2e-212], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $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 < -1.6e-81

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in t around -inf 91.2%

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

      1. associate-*r* 91.2%

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

      2. neg-mul-1 91.2%

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

      3. +-commutative 91.2%

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

      4. sub-neg 91.2%

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

      5. metadata-eval 91.2%

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

      6. +-commutative 91.2%

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

   6. Simplified 91.2%

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

   7. Taylor expanded in x around inf 91.2%

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

   if -1.6e-81 < t < -3.19999999999999988e-163

   1. Initial program 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 16.3%

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

      3. sub-neg 16.3%

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

      4. metadata-eval 16.3%

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

      5. +-commutative 16.3%

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

      6. sub-neg 16.3%

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

      7. metadata-eval 16.3%

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

      8. +-commutative 16.3%

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

      9. associate-/l* 16.3%

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

   6. Simplified 16.3%

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

   7. Taylor expanded in x around inf 43.5%

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

   8. Taylor expanded in x around inf 43.5%

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

      1. *-commutative 43.5%

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

   10. Simplified 43.5%

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

   if -3.19999999999999988e-163 < t < -4.39999999999999986e-245

   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 t around -inf 69.3%

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

      1. associate-*r* 69.3%

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

      2. neg-mul-1 69.3%

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

      3. +-commutative 69.3%

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

      4. sub-neg 69.3%

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

      5. metadata-eval 69.3%

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

      6. +-commutative 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around inf 69.3%

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

   if -4.39999999999999986e-245 < t < 7.2000000000000002e-212

   1. Initial program 1.7%

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

   3. Simplified 1.7%

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

   4. Taylor expanded in x around inf 61.0%

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

   5. Taylor expanded in t around 0 60.8%

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

      1. cancel-sign-sub-inv 60.8%

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

      2. metadata-eval 60.8%

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

      3. distribute-rgt1-in 60.8%

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

      4. metadata-eval 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in t around 0 50.2%

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

   if 7.2000000000000002e-212 < t

   1. Initial program 38.9%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. sub-neg 85.9%

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

      3. metadata-eval 85.9%

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

      4. +-commutative 85.9%

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

   6. Simplified 85.9%

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

   7. Taylor expanded in t around 0 85.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{x \cdot 0.5} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-245}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 4: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-250}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (/ t l) (sqrt x))))
   (if (<= t -4.3e-92)
     (+ (/ 1.0 x) -1.0)
     (if (<= t -1.65e-163)
       t_1
       (if (<= t -2.7e-250)
         -1.0
         (if (<= t 5e-212) 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 / l) * sqrt(x);
	double tmp;
	if (t <= -4.3e-92) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -1.65e-163) {
		tmp = t_1;
	} else if (t <= -2.7e-250) {
		tmp = -1.0;
	} else if (t <= 5e-212) {
		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 / l) * sqrt(x)
    if (t <= (-4.3d-92)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= (-1.65d-163)) then
        tmp = t_1
    else if (t <= (-2.7d-250)) then
        tmp = -1.0d0
    else if (t <= 5d-212) 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 / l) * Math.sqrt(x);
	double tmp;
	if (t <= -4.3e-92) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -1.65e-163) {
		tmp = t_1;
	} else if (t <= -2.7e-250) {
		tmp = -1.0;
	} else if (t <= 5e-212) {
		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 / l) * math.sqrt(x)
	tmp = 0
	if t <= -4.3e-92:
		tmp = (1.0 / x) + -1.0
	elif t <= -1.65e-163:
		tmp = t_1
	elif t <= -2.7e-250:
		tmp = -1.0
	elif t <= 5e-212:
		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 / l) * sqrt(x))
	tmp = 0.0
	if (t <= -4.3e-92)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= -1.65e-163)
		tmp = t_1;
	elseif (t <= -2.7e-250)
		tmp = -1.0;
	elseif (t <= 5e-212)
		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 / l) * sqrt(x);
	tmp = 0.0;
	if (t <= -4.3e-92)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= -1.65e-163)
		tmp = t_1;
	elseif (t <= -2.7e-250)
		tmp = -1.0;
	elseif (t <= 5e-212)
		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 / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.3e-92], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, -1.65e-163], t$95$1, If[LessEqual[t, -2.7e-250], -1.0, If[LessEqual[t, 5e-212], 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 < -4.30000000000000014e-92

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in t around -inf 91.2%

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

      1. associate-*r* 91.2%

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

      2. neg-mul-1 91.2%

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

      3. +-commutative 91.2%

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

      4. sub-neg 91.2%

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

      5. metadata-eval 91.2%

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

      6. +-commutative 91.2%

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

   6. Simplified 91.2%

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

   7. Taylor expanded in x around inf 91.2%

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

   if -4.30000000000000014e-92 < t < -1.65e-163 or -2.70000000000000002e-250 < t < 5.00000000000000043e-212

   1. Initial program 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in x around inf 66.5%

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

   5. Taylor expanded in t around 0 58.9%

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

      1. cancel-sign-sub-inv 58.9%

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

      2. metadata-eval 58.9%

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

      3. distribute-rgt1-in 58.9%

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

      4. metadata-eval 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in t around 0 47.8%

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

   if -1.65e-163 < t < -2.70000000000000002e-250

   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 t around -inf 69.3%

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

      1. associate-*r* 69.3%

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

      2. neg-mul-1 69.3%

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

      3. +-commutative 69.3%

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

      4. sub-neg 69.3%

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

      5. metadata-eval 69.3%

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

      6. +-commutative 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around inf 69.3%

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

   if 5.00000000000000043e-212 < t

   1. Initial program 38.9%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. sub-neg 85.9%

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

      3. metadata-eval 85.9%

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

      4. +-commutative 85.9%

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

   6. Simplified 85.9%

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

   7. Taylor expanded in t around 0 85.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-163}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-250}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 5: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{if}\;t \leq -1.38 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -1.06 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-245}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (/ t l) (sqrt x))))
   (if (<= t -1.38e-92)
     (+ (/ 1.0 x) -1.0)
     (if (<= t -1.06e-163)
       t_1
       (if (<= t -8.8e-245)
         -1.0
         (if (<= t 3.2e-211) t_1 (+ 1.0 (/ -1.0 x))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (t / l) * sqrt(x);
	double tmp;
	if (t <= -1.38e-92) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -1.06e-163) {
		tmp = t_1;
	} else if (t <= -8.8e-245) {
		tmp = -1.0;
	} else if (t <= 3.2e-211) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / l) * sqrt(x)
    if (t <= (-1.38d-92)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= (-1.06d-163)) then
        tmp = t_1
    else if (t <= (-8.8d-245)) then
        tmp = -1.0d0
    else if (t <= 3.2d-211) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = (t / l) * Math.sqrt(x);
	double tmp;
	if (t <= -1.38e-92) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -1.06e-163) {
		tmp = t_1;
	} else if (t <= -8.8e-245) {
		tmp = -1.0;
	} else if (t <= 3.2e-211) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (t / l) * math.sqrt(x)
	tmp = 0
	if t <= -1.38e-92:
		tmp = (1.0 / x) + -1.0
	elif t <= -1.06e-163:
		tmp = t_1
	elif t <= -8.8e-245:
		tmp = -1.0
	elif t <= 3.2e-211:
		tmp = t_1
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(t / l) * sqrt(x))
	tmp = 0.0
	if (t <= -1.38e-92)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= -1.06e-163)
		tmp = t_1;
	elseif (t <= -8.8e-245)
		tmp = -1.0;
	elseif (t <= 3.2e-211)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (t / l) * sqrt(x);
	tmp = 0.0;
	if (t <= -1.38e-92)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= -1.06e-163)
		tmp = t_1;
	elseif (t <= -8.8e-245)
		tmp = -1.0;
	elseif (t <= 3.2e-211)
		tmp = t_1;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.38e-92], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, -1.06e-163], t$95$1, If[LessEqual[t, -8.8e-245], -1.0, If[LessEqual[t, 3.2e-211], t$95$1, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.38000000000000006e-92

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in t around -inf 91.2%

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

      1. associate-*r* 91.2%

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

      2. neg-mul-1 91.2%

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

      3. +-commutative 91.2%

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

      4. sub-neg 91.2%

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

      5. metadata-eval 91.2%

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

      6. +-commutative 91.2%

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

   6. Simplified 91.2%

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

   7. Taylor expanded in x around inf 91.2%

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

   if -1.38000000000000006e-92 < t < -1.06000000000000006e-163 or -8.79999999999999971e-245 < t < 3.19999999999999985e-211

   1. Initial program 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in x around inf 66.5%

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

   5. Taylor expanded in t around 0 58.9%

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

      1. cancel-sign-sub-inv 58.9%

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

      2. metadata-eval 58.9%

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

      3. distribute-rgt1-in 58.9%

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

      4. metadata-eval 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in t around 0 47.8%

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

   if -1.06000000000000006e-163 < t < -8.79999999999999971e-245

   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 t around -inf 69.3%

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

      1. associate-*r* 69.3%

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

      2. neg-mul-1 69.3%

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

      3. +-commutative 69.3%

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

      4. sub-neg 69.3%

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

      5. metadata-eval 69.3%

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

      6. +-commutative 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in x around inf 69.3%

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

   if 3.19999999999999985e-211 < t

   1. Initial program 38.9%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. sub-neg 85.9%

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

      3. metadata-eval 85.9%

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

      4. +-commutative 85.9%

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

   6. Simplified 85.9%

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

   7. Taylor expanded in x around inf 84.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.38 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -1.06 \cdot 10^{-163}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-245}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-211}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 6: 75.9% accurate, 31.9× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) -1.0 (+ 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;
	} 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
    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;
	} 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
	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 = -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;
	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], -1.0, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in t around -inf 79.7%

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

      1. associate-*r* 79.7%

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

      2. neg-mul-1 79.7%

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

      3. +-commutative 79.7%

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

      4. sub-neg 79.7%

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

      5. metadata-eval 79.7%

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

      6. +-commutative 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in x around inf 79.2%

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

   if -4.999999999999985e-310 < t

   1. Initial program 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. sub-neg 77.3%

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

      3. metadata-eval 77.3%

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

      4. +-commutative 77.3%

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

   6. Simplified 77.3%

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

   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 77.7%

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

Alternative 7: 76.3% 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 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in t around -inf 79.7%

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

      1. associate-*r* 79.7%

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

      2. neg-mul-1 79.7%

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

      3. +-commutative 79.7%

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

      4. sub-neg 79.7%

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

      5. metadata-eval 79.7%

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

      6. +-commutative 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in x around inf 79.7%

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

   if -4.999999999999985e-310 < t

   1. Initial program 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. sub-neg 77.3%

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

      3. metadata-eval 77.3%

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

      4. +-commutative 77.3%

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

   6. Simplified 77.3%

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

   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 78.0%

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

Alternative 8: 75.6% accurate, 73.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in t around -inf 79.7%

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

      1. associate-*r* 79.7%

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

      2. neg-mul-1 79.7%

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

      3. +-commutative 79.7%

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

      4. sub-neg 79.7%

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

      5. metadata-eval 79.7%

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

      6. +-commutative 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in x around inf 79.2%

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

   if -4.999999999999985e-310 < t

   1. Initial program 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around -inf 1.8%

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

      1. associate-*r* 1.8%

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

      2. neg-mul-1 1.8%

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

      3. +-commutative 1.8%

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

      4. sub-neg 1.8%

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

      5. metadata-eval 1.8%

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

      6. +-commutative 1.8%

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

   6. Simplified 1.8%

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

      1. div-inv 1.8%

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

      2. +-commutative 1.8%

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

   8. Applied egg-rr 1.8%

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

   9. Taylor expanded in x around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 75.1%

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

       4. associate-*r* 75.1%

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

       5. metadata-eval 75.1%

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

       6. *-lft-identity 75.1%

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

   11. Simplified 75.1%

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

   12. Taylor expanded in t around 0 75.3%

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

4. Final simplification 77.3%

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

Alternative 9: 38.3% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 t around -inf 41.4%

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

   1. associate-*r* 41.4%

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

   2. neg-mul-1 41.4%

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

   3. +-commutative 41.4%

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

   4. sub-neg 41.4%

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

   5. metadata-eval 41.4%

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

   6. +-commutative 41.4%

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

6. Simplified 41.4%

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

7. Taylor expanded in x around inf 41.1%

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

8. Final simplification 41.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023318 
(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)))))