Toniolo and Linder, Equation (7)

Percentage Accurate: 32.7% → 80.9%

Time: 14.7s

Alternatives: 8

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: 32.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-224}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2.15e-224)
   (/ -1.0 (sqrt (/ (+ x 1.0) (+ -1.0 x))))
   (if (<= t 2.7e-167) (/ t (* l (sqrt (/ 1.0 x)))) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.15e-224) {
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= 2.7e-167) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.15d-224)) then
        tmp = (-1.0d0) / sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    else if (t <= 2.7d-167) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.15e-224) {
		tmp = -1.0 / Math.sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= 2.7e-167) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2.15e-224:
		tmp = -1.0 / math.sqrt(((x + 1.0) / (-1.0 + x)))
	elif t <= 2.7e-167:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2.15e-224)
		tmp = Float64(-1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))));
	elseif (t <= 2.7e-167)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2.15e-224)
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	elseif (t <= 2.7e-167)
		tmp = t / (l * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2.15e-224], N[(-1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e-167], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.15e-224

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around -inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt-neg-in 82.5%

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

      4. +-commutative 82.5%

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

      5. sub-neg 82.5%

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

      6. metadata-eval 82.5%

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

      7. +-commutative 82.5%

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

   6. Simplified 82.5%

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

      1. frac-2neg 82.5%

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

      2. neg-sub0 82.5%

         \[\leadsto \frac{\color{blue}{0 - t}}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      3. div-sub 82.5%

         \[\leadsto \color{blue}{\frac{0}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} - \frac{t}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)}} \]

      4. distribute-rgt-neg-out 82.5%

         \[\leadsto \frac{0}{-\color{blue}{\left(-\sqrt{\frac{x + 1}{-1 + x}} \cdot t\right)}} - \frac{t}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      5. remove-double-neg 82.5%

         \[\leadsto \frac{0}{\color{blue}{\sqrt{\frac{x + 1}{-1 + x}} \cdot t}} - \frac{t}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      6. *-commutative 82.5%

         \[\leadsto \frac{0}{\color{blue}{t \cdot \sqrt{\frac{x + 1}{-1 + x}}}} - \frac{t}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      7. +-commutative 82.5%

         \[\leadsto \frac{0}{t \cdot \sqrt{\frac{x + 1}{\color{blue}{x + -1}}}} - \frac{t}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto \frac{0}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} - \frac{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      9. sqrt-unprod 3.2%

         \[\leadsto \frac{0}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} - \frac{\color{blue}{\sqrt{t \cdot t}}}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      10. sqr-neg 3.2%

          \[\leadsto \frac{0}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} - \frac{\sqrt{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      11. sqrt-unprod 1.7%

          \[\leadsto \frac{0}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} - \frac{\color{blue}{\sqrt{-t} \cdot \sqrt{-t}}}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      12. add-sqr-sqrt 1.7%

          \[\leadsto \frac{0}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} - \frac{\color{blue}{-t}}{-\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)} \]

      13. frac-2neg 1.7%

          \[\leadsto \frac{0}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} - \color{blue}{\frac{t}{\sqrt{\frac{x + 1}{-1 + x}} \cdot \left(-t\right)}} \]

      14. *-commutative 1.7%

          \[\leadsto \frac{0}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} - \frac{t}{\color{blue}{\left(-t\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   8. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\frac{0}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} - \frac{t}{t \cdot \sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. div0 82.5%

         \[\leadsto \color{blue}{0} - \frac{t}{t \cdot \sqrt{\frac{x + 1}{x + -1}}} \]

      2. neg-sub0 82.5%

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

      3. associate-/r* 82.5%

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

      4. *-inverses 82.5%

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

      5. distribute-neg-frac 82.5%

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

      6. metadata-eval 82.5%

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

      7. +-commutative 82.5%

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

   10. Simplified 82.5%

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

   if -2.15e-224 < t < 2.7000000000000001e-167

   1. Initial program 4.1%

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

   3. Simplified 4.1%

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

   4. Taylor expanded in x around inf 47.5%

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

   5. Taylor expanded in t around 0 47.1%

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

      1. cancel-sign-sub-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. distribute-rgt1-in 47.1%

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

      4. metadata-eval 47.1%

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

   7. Simplified 47.1%

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

   8. Taylor expanded in l around 0 52.9%

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

   if 2.7000000000000001e-167 < t

   1. Initial program 37.1%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 88.5%

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

      1. +-commutative 88.5%

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

      2. sub-neg 88.5%

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

      3. metadata-eval 88.5%

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

      4. +-commutative 88.5%

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

   6. Simplified 88.5%

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

   7. Taylor expanded in x around inf 88.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-224}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2: 80.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{-222}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2.55e-222)
   (+ -1.0 (/ 1.0 x))
   (if (<= t 3.1e-167) (/ t (* l (sqrt (/ 1.0 x)))) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.55e-222) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 3.1e-167) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.55d-222)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 3.1d-167) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.55e-222) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 3.1e-167) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2.55e-222:
		tmp = -1.0 + (1.0 / x)
	elif t <= 3.1e-167:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2.55e-222)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 3.1e-167)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2.55e-222)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 3.1e-167)
		tmp = t / (l * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2.55e-222], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.1e-167], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.5500000000000001e-222

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around -inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt-neg-in 82.5%

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

      4. +-commutative 82.5%

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

      5. sub-neg 82.5%

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

      6. metadata-eval 82.5%

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

      7. +-commutative 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in x around inf 82.2%

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

   if -2.5500000000000001e-222 < t < 3.1e-167

   1. Initial program 4.1%

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

   3. Simplified 4.1%

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

   4. Taylor expanded in x around inf 47.5%

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

   5. Taylor expanded in t around 0 47.1%

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

      1. cancel-sign-sub-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. distribute-rgt1-in 47.1%

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

      4. metadata-eval 47.1%

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

   7. Simplified 47.1%

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

   8. Taylor expanded in l around 0 52.9%

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

   if 3.1e-167 < t

   1. Initial program 37.1%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 88.5%

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

      1. +-commutative 88.5%

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

      2. sub-neg 88.5%

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

      3. metadata-eval 88.5%

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

      4. +-commutative 88.5%

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

   6. Simplified 88.5%

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

   7. Taylor expanded in x around inf 88.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{-222}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 80.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-224}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -8e-224)
   (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
   (if (<= t 4.4e-167) (/ t (* l (sqrt (/ 1.0 x)))) (+ 1.0 (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -8e-224) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= 4.4e-167) {
		tmp = t / (l * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8d-224)) then
        tmp = -sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else if (t <= 4.4d-167) then
        tmp = t / (l * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -8e-224) {
		tmp = -Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= 4.4e-167) {
		tmp = t / (l * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -8e-224:
		tmp = -math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif t <= 4.4e-167:
		tmp = t / (l * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -8e-224)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= 4.4e-167)
		tmp = Float64(t / Float64(l * sqrt(Float64(1.0 / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -8e-224)
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	elseif (t <= 4.4e-167)
		tmp = t / (l * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -8e-224], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, 4.4e-167], N[(t / N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.0000000000000002e-224

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around -inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt-neg-in 82.5%

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

      4. +-commutative 82.5%

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

      5. sub-neg 82.5%

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

      6. metadata-eval 82.5%

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

      7. +-commutative 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in t around 0 82.5%

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

      1. mul-1-neg 82.5%

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

      2. sub-neg 82.5%

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

      3. metadata-eval 82.5%

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

      4. +-commutative 82.5%

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

      5. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   if -8.0000000000000002e-224 < t < 4.3999999999999999e-167

   1. Initial program 4.1%

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

   3. Simplified 4.1%

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

   4. Taylor expanded in x around inf 47.5%

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

   5. Taylor expanded in t around 0 47.1%

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

      1. cancel-sign-sub-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. distribute-rgt1-in 47.1%

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

      4. metadata-eval 47.1%

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

   7. Simplified 47.1%

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

   8. Taylor expanded in l around 0 52.9%

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

   if 4.3999999999999999e-167 < t

   1. Initial program 37.1%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 88.5%

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

      1. +-commutative 88.5%

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

      2. sub-neg 88.5%

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

      3. metadata-eval 88.5%

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

      4. +-commutative 88.5%

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

   6. Simplified 88.5%

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

   7. Taylor expanded in x around inf 88.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-224}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4: 79.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-246}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2.4e-246)
   (+ -1.0 (/ 1.0 x))
   (if (<= t 2.6e-167) (* (/ 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 <= -2.4e-246) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 2.6e-167) {
		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 <= (-2.4d-246)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 2.6d-167) 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 <= -2.4e-246) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 2.6e-167) {
		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 <= -2.4e-246:
		tmp = -1.0 + (1.0 / x)
	elif t <= 2.6e-167:
		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 <= -2.4e-246)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 2.6e-167)
		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 <= -2.4e-246)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 2.6e-167)
		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, -2.4e-246], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e-167], 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 < -2.3999999999999998e-246

   1. Initial program 34.4%

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

   3. Simplified 34.4%

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

   4. Taylor expanded in t around -inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. *-commutative 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. +-commutative 78.9%

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

      5. sub-neg 78.9%

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

      6. metadata-eval 78.9%

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

      7. +-commutative 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in x around inf 78.5%

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

   if -2.3999999999999998e-246 < t < 2.5999999999999999e-167

   1. Initial program 4.6%

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

   3. Simplified 4.6%

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

   4. Taylor expanded in x around inf 46.5%

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

   5. Taylor expanded in t around 0 46.1%

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

      1. cancel-sign-sub-inv 46.1%

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

      2. metadata-eval 46.1%

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

      3. distribute-rgt1-in 46.1%

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

      4. metadata-eval 46.1%

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

   7. Simplified 46.1%

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

   8. Taylor expanded in t around 0 55.0%

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

   if 2.5999999999999999e-167 < t

   1. Initial program 37.1%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around inf 88.5%

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

      1. +-commutative 88.5%

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

      2. sub-neg 88.5%

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

      3. metadata-eval 88.5%

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

      4. +-commutative 88.5%

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

   6. Simplified 88.5%

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

   7. Taylor expanded in x around inf 88.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-246}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 5: 76.3% accurate, 31.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 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 32.8%

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

   4. Taylor expanded in t around -inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. +-commutative 75.9%

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

      5. sub-neg 75.9%

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

      6. metadata-eval 75.9%

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

      7. +-commutative 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in x around inf 75.0%

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

   if -4.999999999999985e-310 < t

   1. Initial program 29.7%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in t around inf 75.6%

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

      1. +-commutative 75.6%

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

      2. sub-neg 75.6%

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

      3. metadata-eval 75.6%

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

      4. +-commutative 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in x around inf 75.6%

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

4. Final simplification 75.3%

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

Alternative 6: 76.6% 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 32.8%

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

   4. Taylor expanded in t around -inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. +-commutative 75.9%

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

      5. sub-neg 75.9%

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

      6. metadata-eval 75.9%

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

      7. +-commutative 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in x around inf 75.6%

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

   if -4.999999999999985e-310 < t

   1. Initial program 29.7%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in t around inf 75.6%

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

      1. +-commutative 75.6%

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

      2. sub-neg 75.6%

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

      3. metadata-eval 75.6%

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

      4. +-commutative 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in x around inf 75.6%

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

4. Final simplification 75.6%

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

Alternative 7: 76.0% 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 32.8%

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

   4. Taylor expanded in t around -inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. +-commutative 75.9%

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

      5. sub-neg 75.9%

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

      6. metadata-eval 75.9%

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

      7. +-commutative 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in x around inf 75.0%

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

   if -4.999999999999985e-310 < t

   1. Initial program 29.7%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in t around inf 75.6%

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

      1. +-commutative 75.6%

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

      2. sub-neg 75.6%

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

      3. metadata-eval 75.6%

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

      4. +-commutative 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in x around inf 75.3%

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

4. Final simplification 75.2%

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

Alternative 8: 39.0% 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 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in t around -inf 42.0%

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

   1. mul-1-neg 42.0%

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

   2. *-commutative 42.0%

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

   3. distribute-rgt-neg-in 42.0%

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

   4. +-commutative 42.0%

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

   5. sub-neg 42.0%

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

   6. metadata-eval 42.0%

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

   7. +-commutative 42.0%

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

6. Simplified 42.0%

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

7. Taylor expanded in x around inf 41.5%

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

8. Final simplification 41.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023309 
(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)))))