Toniolo and Linder, Equation (7)

Percentage Accurate: 33.0% → 81.2%

Time: 16.7s

Alternatives: 10

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: 33.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := 2 \cdot {t}^{2}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-161}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(t_2 + \frac{{\ell}^{2}}{x}\right)\right) + \frac{t_2 + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))) (t_2 (* 2.0 (pow t 2.0))))
   (if (<= t -5.2e-161)
     (- t_1)
     (if (<= t 1.05e-304)
       (/
        t
        (*
         (sqrt (+ (/ 1.0 (+ x -1.0)) (+ (/ 1.0 x) (/ 1.0 (pow x 2.0)))))
         (/ l (sqrt 2.0))))
       (if (<= t 5e-160)
         1.0
         (if (<= t 6.4e+18)
           (/
            t
            (/
             (sqrt
              (+
               (+ (* 2.0 (/ (pow t 2.0) x)) (+ t_2 (/ (pow l 2.0) x)))
               (/ (+ t_2 (pow l 2.0)) x)))
             (sqrt 2.0)))
           t_1))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = 2.0 * pow(t, 2.0);
	double tmp;
	if (t <= -5.2e-161) {
		tmp = -t_1;
	} else if (t <= 1.05e-304) {
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / pow(x, 2.0))))) * (l / sqrt(2.0)));
	} else if (t <= 5e-160) {
		tmp = 1.0;
	} else if (t <= 6.4e+18) {
		tmp = t / (sqrt((((2.0 * (pow(t, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))) + ((t_2 + pow(l, 2.0)) / x))) / sqrt(2.0));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    t_2 = 2.0d0 * (t ** 2.0d0)
    if (t <= (-5.2d-161)) then
        tmp = -t_1
    else if (t <= 1.05d-304) then
        tmp = t / (sqrt(((1.0d0 / (x + (-1.0d0))) + ((1.0d0 / x) + (1.0d0 / (x ** 2.0d0))))) * (l / sqrt(2.0d0)))
    else if (t <= 5d-160) then
        tmp = 1.0d0
    else if (t <= 6.4d+18) then
        tmp = t / (sqrt((((2.0d0 * ((t ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))) + ((t_2 + (l ** 2.0d0)) / x))) / sqrt(2.0d0))
    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) / (x + 1.0)));
	double t_2 = 2.0 * Math.pow(t, 2.0);
	double tmp;
	if (t <= -5.2e-161) {
		tmp = -t_1;
	} else if (t <= 1.05e-304) {
		tmp = t / (Math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / Math.pow(x, 2.0))))) * (l / Math.sqrt(2.0)));
	} else if (t <= 5e-160) {
		tmp = 1.0;
	} else if (t <= 6.4e+18) {
		tmp = t / (Math.sqrt((((2.0 * (Math.pow(t, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))) + ((t_2 + Math.pow(l, 2.0)) / x))) / Math.sqrt(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	t_2 = 2.0 * math.pow(t, 2.0)
	tmp = 0
	if t <= -5.2e-161:
		tmp = -t_1
	elif t <= 1.05e-304:
		tmp = t / (math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / math.pow(x, 2.0))))) * (l / math.sqrt(2.0)))
	elif t <= 5e-160:
		tmp = 1.0
	elif t <= 6.4e+18:
		tmp = t / (math.sqrt((((2.0 * (math.pow(t, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))) + ((t_2 + math.pow(l, 2.0)) / x))) / math.sqrt(2.0))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(2.0 * (t ^ 2.0))
	tmp = 0.0
	if (t <= -5.2e-161)
		tmp = Float64(-t_1);
	elseif (t <= 1.05e-304)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 / x) + Float64(1.0 / (x ^ 2.0))))) * Float64(l / sqrt(2.0))));
	elseif (t <= 5e-160)
		tmp = 1.0;
	elseif (t <= 6.4e+18)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))) + Float64(Float64(t_2 + (l ^ 2.0)) / x))) / sqrt(2.0)));
	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) / (x + 1.0)));
	t_2 = 2.0 * (t ^ 2.0);
	tmp = 0.0;
	if (t <= -5.2e-161)
		tmp = -t_1;
	elseif (t <= 1.05e-304)
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / (x ^ 2.0))))) * (l / sqrt(2.0)));
	elseif (t <= 5e-160)
		tmp = 1.0;
	elseif (t <= 6.4e+18)
		tmp = t / (sqrt((((2.0 * ((t ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))) + ((t_2 + (l ^ 2.0)) / x))) / sqrt(2.0));
	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[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.2e-161], (-t$95$1), If[LessEqual[t, 1.05e-304], N[(t / N[(N[Sqrt[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e-160], 1.0, If[LessEqual[t, 6.4e+18], N[(t / N[(N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.19999999999999991e-161

   1. Initial program 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around -inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. +-commutative 86.2%

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

      5. sub-neg 86.2%

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

      6. metadata-eval 86.2%

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

      7. +-commutative 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in t around 0 86.2%

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

      1. mul-1-neg 86.2%

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

      2. sub-neg 86.2%

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

      3. metadata-eval 86.2%

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

      4. +-commutative 86.2%

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

   9. Simplified 86.2%

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

   if -5.19999999999999991e-161 < t < 1.05000000000000004e-304

   1. Initial program 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in l around inf 4.0%

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

      1. *-commutative 4.0%

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

      2. associate--l+ 30.9%

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

      3. sub-neg 30.9%

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

      4. metadata-eval 30.9%

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

      5. +-commutative 30.9%

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

      6. sub-neg 30.9%

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

      7. metadata-eval 30.9%

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

      8. +-commutative 30.9%

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

   6. Simplified 30.9%

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

   7. Taylor expanded in x around inf 58.4%

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

   if 1.05000000000000004e-304 < t < 4.99999999999999994e-160

   1. Initial program 2.8%

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

   3. Simplified 2.8%

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

   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. mul-1-neg 1.8%

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

      2. *-commutative 1.8%

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

      3. distribute-rgt-neg-in 1.8%

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

      4. +-commutative 1.8%

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

      5. sub-neg 1.8%

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

      6. metadata-eval 1.8%

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

      7. +-commutative 1.8%

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

   6. Simplified 1.8%

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

   7. Taylor expanded in t around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

      4. +-commutative 1.8%

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

   9. Simplified 1.8%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 70.1%

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

   12. Simplified 70.1%

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

   if 4.99999999999999994e-160 < t < 6.4e18

   1. Initial program 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in x around inf 84.5%

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

   if 6.4e18 < t

   1. Initial program 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in t around inf 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 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-161}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{-153}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.65e-153)
     (- t_1)
     (if (<= t 1.05e-304)
       (/
        t
        (*
         (sqrt (+ (/ 1.0 (+ x -1.0)) (+ (/ 1.0 x) (/ 1.0 (pow x 2.0)))))
         (/ l (sqrt 2.0))))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.65e-153) {
		tmp = -t_1;
	} else if (t <= 1.05e-304) {
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / pow(x, 2.0))))) * (l / sqrt(2.0)));
	} 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)) / (x + 1.0d0)))
    if (t <= (-1.65d-153)) then
        tmp = -t_1
    else if (t <= 1.05d-304) then
        tmp = t / (sqrt(((1.0d0 / (x + (-1.0d0))) + ((1.0d0 / x) + (1.0d0 / (x ** 2.0d0))))) * (l / sqrt(2.0d0)))
    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) / (x + 1.0)));
	double tmp;
	if (t <= -1.65e-153) {
		tmp = -t_1;
	} else if (t <= 1.05e-304) {
		tmp = t / (Math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / Math.pow(x, 2.0))))) * (l / Math.sqrt(2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -1.65e-153:
		tmp = -t_1
	elif t <= 1.05e-304:
		tmp = t / (math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / math.pow(x, 2.0))))) * (l / math.sqrt(2.0)))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.65e-153)
		tmp = Float64(-t_1);
	elseif (t <= 1.05e-304)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 / x) + Float64(1.0 / (x ^ 2.0))))) * Float64(l / sqrt(2.0))));
	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) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1.65e-153)
		tmp = -t_1;
	elseif (t <= 1.05e-304)
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / (x ^ 2.0))))) * (l / sqrt(2.0)));
	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[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.65e-153], (-t$95$1), If[LessEqual[t, 1.05e-304], N[(t / N[(N[Sqrt[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.64999999999999994e-153

   1. Initial program 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around -inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. +-commutative 86.2%

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

      5. sub-neg 86.2%

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

      6. metadata-eval 86.2%

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

      7. +-commutative 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in t around 0 86.2%

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

      1. mul-1-neg 86.2%

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

      2. sub-neg 86.2%

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

      3. metadata-eval 86.2%

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

      4. +-commutative 86.2%

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

   9. Simplified 86.2%

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

   if -1.64999999999999994e-153 < t < 1.05000000000000004e-304

   1. Initial program 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in l around inf 4.0%

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

      1. *-commutative 4.0%

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

      2. associate--l+ 30.9%

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

      3. sub-neg 30.9%

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

      4. metadata-eval 30.9%

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

      5. +-commutative 30.9%

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

      6. sub-neg 30.9%

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

      7. metadata-eval 30.9%

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

      8. +-commutative 30.9%

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

   6. Simplified 30.9%

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

   7. Taylor expanded in x around inf 58.4%

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

   if 1.05000000000000004e-304 < t

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around inf 79.5%

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

      1. +-commutative 79.5%

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

      2. sub-neg 79.5%

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

      3. metadata-eval 79.5%

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

      4. +-commutative 79.5%

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

   6. Simplified 79.5%

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

   7. Taylor expanded in t around 0 79.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-153}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-161}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-306}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.7e-161)
   (+ -1.0 (/ 1.0 x))
   (if (<= t 1.26e-306) (* (/ t l) (sqrt x)) (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.7e-161) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 1.26e-306) {
		tmp = (t / l) * sqrt(x);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 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 <= (-1.7d-161)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 1.26d-306) then
        tmp = (t / l) * sqrt(x)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 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 <= -1.7e-161) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 1.26e-306) {
		tmp = (t / l) * Math.sqrt(x);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.7e-161:
		tmp = -1.0 + (1.0 / x)
	elif t <= 1.26e-306:
		tmp = (t / l) * math.sqrt(x)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.7e-161)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 1.26e-306)
		tmp = Float64(Float64(t / l) * sqrt(x));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.7e-161)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 1.26e-306)
		tmp = (t / l) * sqrt(x);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.7e-161], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.26e-306], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.69999999999999991e-161

   1. Initial program 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around -inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. +-commutative 86.2%

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

      5. sub-neg 86.2%

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

      6. metadata-eval 86.2%

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

      7. +-commutative 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in x around inf 84.5%

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

   if -1.69999999999999991e-161 < t < 1.2600000000000001e-306

   1. Initial program 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in l around inf 3.6%

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

      1. *-commutative 3.6%

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

      2. associate--l+ 30.4%

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

      3. sub-neg 30.4%

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

      4. metadata-eval 30.4%

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

      5. +-commutative 30.4%

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

      6. sub-neg 30.4%

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

      7. metadata-eval 30.4%

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

      8. +-commutative 30.4%

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

      9. associate-/l* 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in x around inf 53.1%

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

      1. sqrt-div 53.2%

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

      2. metadata-eval 53.2%

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

      3. frac-times 57.0%

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

      4. *-un-lft-identity 57.0%

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

      5. add-sqr-sqrt 57.0%

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

      6. add-sqr-sqrt 57.0%

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

      7. hypot-def 57.1%

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

      8. inv-pow 57.1%

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

      9. sqrt-pow1 57.1%

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

      10. +-commutative 57.1%

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

      11. metadata-eval 57.1%

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

      12. inv-pow 57.1%

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

      13. sqrt-pow1 57.1%

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

      14. metadata-eval 57.1%

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

   9. Applied egg-rr 57.1%

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

   10. Taylor expanded in x around inf 53.3%

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

   if 1.2600000000000001e-306 < t

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around inf 79.5%

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

      1. +-commutative 79.5%

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

      2. sub-neg 79.5%

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

      3. metadata-eval 79.5%

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

      4. +-commutative 79.5%

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

   6. Simplified 79.5%

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

   7. Taylor expanded in t around 0 79.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-161}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-306}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 78.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.55 \cdot 10^{-158}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-306}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -3.55e-158)
   (+ -1.0 (/ 1.0 x))
   (if (<= t 1.26e-306)
     (/ t (* l (sqrt (/ 1.0 x))))
     (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.55e-158) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 1.26e-306) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 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 <= (-3.55d-158)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 1.26d-306) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 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 <= -3.55e-158) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 1.26e-306) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -3.55e-158:
		tmp = -1.0 + (1.0 / x)
	elif t <= 1.26e-306:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -3.55e-158)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 1.26e-306)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.55e-158)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 1.26e-306)
		tmp = t / (l * sqrt((1.0 / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -3.55e-158], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.26e-306], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.55000000000000002e-158

   1. Initial program 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around -inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. +-commutative 86.2%

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

      5. sub-neg 86.2%

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

      6. metadata-eval 86.2%

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

      7. +-commutative 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in x around inf 84.5%

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

   if -3.55000000000000002e-158 < t < 1.2600000000000001e-306

   1. Initial program 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in l around inf 3.6%

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

      1. *-commutative 3.6%

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

      2. associate--l+ 30.4%

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

      3. sub-neg 30.4%

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

      4. metadata-eval 30.4%

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

      5. +-commutative 30.4%

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

      6. sub-neg 30.4%

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

      7. metadata-eval 30.4%

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

      8. +-commutative 30.4%

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

      9. associate-/l* 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in x around inf 53.1%

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

      1. sqrt-div 53.2%

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

      2. metadata-eval 53.2%

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

      3. frac-times 57.0%

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

      4. *-un-lft-identity 57.0%

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

      5. add-sqr-sqrt 57.0%

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

      6. add-sqr-sqrt 57.0%

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

      7. hypot-def 57.1%

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

      8. inv-pow 57.1%

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

      9. sqrt-pow1 57.1%

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

      10. +-commutative 57.1%

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

      11. metadata-eval 57.1%

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

      12. inv-pow 57.1%

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

      13. sqrt-pow1 57.1%

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

      14. metadata-eval 57.1%

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

   9. Applied egg-rr 57.1%

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

   10. Taylor expanded in x around inf 57.2%

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

   if 1.2600000000000001e-306 < t

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around inf 79.5%

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

      1. +-commutative 79.5%

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

      2. sub-neg 79.5%

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

      3. metadata-eval 79.5%

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

      4. +-commutative 79.5%

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

   6. Simplified 79.5%

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

   7. Taylor expanded in t around 0 79.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.55 \cdot 10^{-158}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-306}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 5: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{-152}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -8e-152)
     (- t_1)
     (if (<= t 1.05e-304) (/ t (* l (sqrt (/ 1.0 x)))) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -8e-152) {
		tmp = -t_1;
	} else if (t <= 1.05e-304) {
		tmp = t / (l * sqrt((1.0 / x)));
	} 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)) / (x + 1.0d0)))
    if (t <= (-8d-152)) then
        tmp = -t_1
    else if (t <= 1.05d-304) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    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) / (x + 1.0)));
	double tmp;
	if (t <= -8e-152) {
		tmp = -t_1;
	} else if (t <= 1.05e-304) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -8e-152:
		tmp = -t_1
	elif t <= 1.05e-304:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -8e-152)
		tmp = Float64(-t_1);
	elseif (t <= 1.05e-304)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	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) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -8e-152)
		tmp = -t_1;
	elseif (t <= 1.05e-304)
		tmp = t / (l * sqrt((1.0 / x)));
	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[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -8e-152], (-t$95$1), If[LessEqual[t, 1.05e-304], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.00000000000000053e-152

   1. Initial program 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around -inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. +-commutative 86.2%

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

      5. sub-neg 86.2%

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

      6. metadata-eval 86.2%

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

      7. +-commutative 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in t around 0 86.2%

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

      1. mul-1-neg 86.2%

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

      2. sub-neg 86.2%

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

      3. metadata-eval 86.2%

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

      4. +-commutative 86.2%

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

   9. Simplified 86.2%

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

   if -8.00000000000000053e-152 < t < 1.05000000000000004e-304

   1. Initial program 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in l around inf 3.6%

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

      1. *-commutative 3.6%

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

      2. associate--l+ 30.4%

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

      3. sub-neg 30.4%

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

      4. metadata-eval 30.4%

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

      5. +-commutative 30.4%

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

      6. sub-neg 30.4%

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

      7. metadata-eval 30.4%

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

      8. +-commutative 30.4%

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

      9. associate-/l* 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in x around inf 53.1%

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

      1. sqrt-div 53.2%

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

      2. metadata-eval 53.2%

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

      3. frac-times 57.0%

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

      4. *-un-lft-identity 57.0%

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

      5. add-sqr-sqrt 57.0%

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

      6. add-sqr-sqrt 57.0%

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

      7. hypot-def 57.1%

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

      8. inv-pow 57.1%

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

      9. sqrt-pow1 57.1%

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

      10. +-commutative 57.1%

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

      11. metadata-eval 57.1%

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

      12. inv-pow 57.1%

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

      13. sqrt-pow1 57.1%

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

      14. metadata-eval 57.1%

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

   9. Applied egg-rr 57.1%

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

   10. Taylor expanded in x around inf 57.2%

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

   if 1.05000000000000004e-304 < t

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around inf 79.5%

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

      1. +-commutative 79.5%

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

      2. sub-neg 79.5%

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

      3. metadata-eval 79.5%

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

      4. +-commutative 79.5%

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

   6. Simplified 79.5%

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

   7. Taylor expanded in t around 0 79.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-152}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 6: 77.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-160}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -4.3e-160)
   (+ -1.0 (/ 1.0 x))
   (if (<= t 8e-305) (* (/ 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 <= -4.3e-160) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 8e-305) {
		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 <= (-4.3d-160)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 8d-305) 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 <= -4.3e-160) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 8e-305) {
		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 <= -4.3e-160:
		tmp = -1.0 + (1.0 / x)
	elif t <= 8e-305:
		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 <= -4.3e-160)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 8e-305)
		tmp = Float64(Float64(t / 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 <= -4.3e-160)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 8e-305)
		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, -4.3e-160], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-305], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.30000000000000014e-160

   1. Initial program 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around -inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. +-commutative 86.2%

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

      5. sub-neg 86.2%

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

      6. metadata-eval 86.2%

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

      7. +-commutative 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in x around inf 84.5%

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

   if -4.30000000000000014e-160 < t < 7.99999999999999997e-305

   1. Initial program 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in l around inf 3.6%

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

      1. *-commutative 3.6%

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

      2. associate--l+ 30.4%

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

      3. sub-neg 30.4%

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

      4. metadata-eval 30.4%

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

      5. +-commutative 30.4%

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

      6. sub-neg 30.4%

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

      7. metadata-eval 30.4%

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

      8. +-commutative 30.4%

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

      9. associate-/l* 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in x around inf 53.1%

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

      1. sqrt-div 53.2%

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

      2. metadata-eval 53.2%

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

      3. frac-times 57.0%

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

      4. *-un-lft-identity 57.0%

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

      5. add-sqr-sqrt 57.0%

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

      6. add-sqr-sqrt 57.0%

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

      7. hypot-def 57.1%

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

      8. inv-pow 57.1%

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

      9. sqrt-pow1 57.1%

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

      10. +-commutative 57.1%

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

      11. metadata-eval 57.1%

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

      12. inv-pow 57.1%

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

      13. sqrt-pow1 57.1%

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

      14. metadata-eval 57.1%

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

   9. Applied egg-rr 57.1%

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

   10. Taylor expanded in x around inf 53.3%

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

   if 7.99999999999999997e-305 < t

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. distribute-rgt-neg-in 1.7%

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

      4. +-commutative 1.7%

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

      5. sub-neg 1.7%

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

      6. metadata-eval 1.7%

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

      7. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 78.9%

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

   12. Simplified 78.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-160}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-305}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 7: 76.5% accurate, 31.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 32.8%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in t around -inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. *-commutative 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

      4. +-commutative 75.5%

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

      5. sub-neg 75.5%

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

      6. metadata-eval 75.5%

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

      7. +-commutative 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in x around inf 74.1%

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

   if -4.999999999999985e-310 < t

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. distribute-rgt-neg-in 1.7%

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

      4. +-commutative 1.7%

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

      5. sub-neg 1.7%

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

      6. metadata-eval 1.7%

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

      7. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 78.3%

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

   12. Simplified 78.3%

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

4. Final simplification 76.1%

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

Alternative 8: 76.8% accurate, 31.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 32.8%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in t around -inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. *-commutative 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

      4. +-commutative 75.5%

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

      5. sub-neg 75.5%

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

      6. metadata-eval 75.5%

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

      7. +-commutative 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in x around inf 74.1%

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

   if -4.999999999999985e-310 < t

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. distribute-rgt-neg-in 1.7%

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

      4. +-commutative 1.7%

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

      5. sub-neg 1.7%

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

      6. metadata-eval 1.7%

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

      7. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 78.9%

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

   12. Simplified 78.9%

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

4. Final simplification 76.3%

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

Alternative 9: 76.2% 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 32.8%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in t around -inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. *-commutative 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

      4. +-commutative 75.5%

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

      5. sub-neg 75.5%

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

      6. metadata-eval 75.5%

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

      7. +-commutative 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in x around inf 73.3%

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

   if -4.999999999999985e-310 < t

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in t around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. distribute-rgt-neg-in 1.7%

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

      4. +-commutative 1.7%

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

      5. sub-neg 1.7%

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

      6. metadata-eval 1.7%

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

      7. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in t around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

   9. Simplified 1.7%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. unpow2 0.0%

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

       2. rem-square-sqrt 78.3%

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

   12. Simplified 78.3%

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

4. Final simplification 75.6%

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

Alternative 10: 38.4% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

3. Simplified 31.9%

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

4. Taylor expanded in t around -inf 40.9%

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

   1. mul-1-neg 40.9%

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

   2. *-commutative 40.9%

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

   3. distribute-rgt-neg-in 40.9%

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

   4. +-commutative 40.9%

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

   5. sub-neg 40.9%

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

   6. metadata-eval 40.9%

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

   7. +-commutative 40.9%

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

6. Simplified 40.9%

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

7. Taylor expanded in x around inf 39.7%

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

8. Final simplification 39.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023308 
(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)))))