Toniolo and Linder, Equation (7)

Percentage Accurate: 34.2% → 86.6%

Time: 17.1s

Alternatives: 9

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.2% 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: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+95}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+134}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.02e+95)
     (- t_1)
     (if (<= t 4.5e+134)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt (+ (* 2.0 (* t (+ t (/ t x)))) (* 2.0 (/ l (/ x l)))))))
       t_1))))

C

double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.02e+95) {
		tmp = -t_1;
	} else if (t <= 4.5e+134) {
		tmp = t * (sqrt(2.0) / sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l / (x / l))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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 <= (-1.02d+95)) then
        tmp = -t_1
    else if (t <= 4.5d+134) then
        tmp = t * (sqrt(2.0d0) / sqrt(((2.0d0 * (t * (t + (t / x)))) + (2.0d0 * (l / (x / l))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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 <= -1.02e+95) {
		tmp = -t_1;
	} else if (t <= 4.5e+134) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l / (x / l))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -1.02e+95:
		tmp = -t_1
	elif t <= 4.5e+134:
		tmp = t * (math.sqrt(2.0) / math.sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l / (x / l))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.02e+95)
		tmp = Float64(-t_1);
	elseif (t <= 4.5e+134)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(2.0 * Float64(l / Float64(x / l)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1.02e+95)
		tmp = -t_1;
	elseif (t <= 4.5e+134)
		tmp = t * (sqrt(2.0) / sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l / (x / l))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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, -1.02e+95], (-t$95$1), If[LessEqual[t, 4.5e+134], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.0200000000000001e95

   1. Initial program 21.6%

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

      1. associate-*r/ 21.6%

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

      2. fma-neg 21.6%

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

      3. sub-neg 21.6%

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

      4. metadata-eval 21.6%

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

      5. +-commutative 21.6%

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

      6. fma-def 21.6%

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

      7. distribute-rgt-neg-in 21.6%

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

   3. Simplified 21.6%

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

   5. Applied egg-rr 87.4%

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

   6. Taylor expanded in t around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   if -1.0200000000000001e95 < t < 4.4999999999999997e134

   1. Initial program 44.3%

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

      1. associate-*l/ 44.3%

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

   3. Simplified 44.3%

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

   4. Taylor expanded in x around inf 76.4%

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

      1. associate--l+ 76.4%

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

      2. unpow2 76.4%

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

      3. distribute-lft-out 76.4%

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

      4. unpow2 76.4%

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

      5. unpow2 76.4%

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

      6. associate-*r/ 76.4%

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

      7. mul-1-neg 76.4%

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

      8. +-commutative 76.4%

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

      9. unpow2 76.4%

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

      10. associate-*l* 76.4%

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

      11. unpow2 76.4%

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

      12. fma-udef 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in t around 0 76.4%

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

      1. associate-*r/ 76.4%

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

      2. mul-1-neg 76.4%

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

      3. unpow2 76.4%

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

      4. distribute-rgt-neg-in 76.4%

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

   9. Simplified 76.4%

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

       1. *-un-lft-identity 76.4%

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

       2. associate-*l/ 76.4%

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

       3. *-commutative 76.4%

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

       4. +-commutative 76.4%

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

       5. associate-/l* 76.4%

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

   11. Applied egg-rr 76.4%

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

       1. *-lft-identity 76.4%

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

       2. associate-+r- 76.4%

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

       3. sub-neg 76.4%

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

       4. +-commutative 76.4%

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

       5. distribute-rgt-neg-out 76.4%

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

       6. unpow2 76.4%

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

       7. distribute-frac-neg 76.4%

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

       8. unpow2 76.4%

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

       9. associate-*r/ 82.4%

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

       10. remove-double-neg 82.4%

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

       11. associate-+l+ 82.4%

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

   13. Simplified 82.3%

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

   if 4.4999999999999997e134 < t

   1. Initial program 7.6%

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

      1. associate-*r/ 7.6%

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

      2. fma-neg 7.6%

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

      3. sub-neg 7.6%

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

      4. metadata-eval 7.6%

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

      5. +-commutative 7.6%

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

      6. fma-def 7.6%

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

      7. distribute-rgt-neg-in 7.6%

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

   3. Simplified 7.6%

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

   5. Applied egg-rr 83.4%

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

   6. Taylor expanded in t around inf 100.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+95}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+134}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 76.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-262} \lor \neg \left(t \leq 1.75 \cdot 10^{-202}\right) \land t \leq 4.6 \cdot 10^{-119}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -5.8e-261)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (if (or (<= t 3e-262) (and (not (<= t 1.75e-202)) (<= t 4.6e-119)))
     (* t (/ (sqrt x) l))
     (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.8e-261) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if ((t <= 3e-262) || (!(t <= 1.75e-202) && (t <= 4.6e-119))) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.8d-261)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else if ((t <= 3d-262) .or. (.not. (t <= 1.75d-202)) .and. (t <= 4.6d-119)) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.8e-261) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else if ((t <= 3e-262) || (!(t <= 1.75e-202) && (t <= 4.6e-119))) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -5.8e-261:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	elif (t <= 3e-262) or (not (t <= 1.75e-202) and (t <= 4.6e-119)):
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -5.8e-261)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	elseif ((t <= 3e-262) || (!(t <= 1.75e-202) && (t <= 4.6e-119)))
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5.8e-261)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	elseif ((t <= 3e-262) || (~((t <= 1.75e-202)) && (t <= 4.6e-119)))
		tmp = t * (sqrt(x) / l);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -5.8e-261], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 3e-262], And[N[Not[LessEqual[t, 1.75e-202]], $MachinePrecision], LessEqual[t, 4.6e-119]]], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $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 < -5.7999999999999997e-261

   1. Initial program 33.0%

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

      1. associate-/l* 33.0%

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

      2. fma-neg 33.0%

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

      3. remove-double-neg 33.0%

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

      4. fma-neg 33.0%

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

      5. sub-neg 33.0%

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

      6. metadata-eval 33.0%

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

      7. remove-double-neg 33.0%

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

      8. fma-def 33.0%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in t around -inf 86.5%

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

      1. associate-*r* 86.5%

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

      2. neg-mul-1 86.5%

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

      3. +-commutative 86.5%

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

      4. sub-neg 86.5%

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

      5. metadata-eval 86.5%

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

      6. +-commutative 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in x around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. metadata-eval 86.0%

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

      3. unpow2 86.0%

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

   9. Simplified 86.0%

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

   if -5.7999999999999997e-261 < t < 3.00000000000000018e-262 or 1.75e-202 < t < 4.59999999999999987e-119

   1. Initial program 9.9%

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

      1. associate-*l/ 9.9%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in x around inf 59.9%

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

      1. associate--l+ 59.9%

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

      2. unpow2 59.9%

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

      3. distribute-lft-out 59.9%

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

      4. unpow2 59.9%

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

      5. unpow2 59.9%

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

      6. associate-*r/ 59.9%

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

      7. mul-1-neg 59.9%

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

      8. +-commutative 59.9%

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

      9. unpow2 59.9%

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

      10. associate-*l* 59.9%

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

      11. unpow2 59.9%

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

      12. fma-udef 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in t around 0 54.2%

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

      1. sub-neg 54.2%

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

      2. unpow2 54.2%

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

      3. associate-/l* 54.2%

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

      4. mul-1-neg 54.2%

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

      5. unpow2 54.2%

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

      6. remove-double-neg 54.2%

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

      7. associate-/l* 64.6%

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

   9. Simplified 64.6%

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

   10. Taylor expanded in l around 0 64.1%

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

       1. associate-*l/ 64.1%

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

       2. *-lft-identity 64.1%

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

   12. Simplified 64.1%

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

   if 3.00000000000000018e-262 < t < 1.75e-202 or 4.59999999999999987e-119 < t

   1. Initial program 33.2%

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

      1. associate-*r/ 33.2%

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

      2. fma-neg 33.2%

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

      3. sub-neg 33.2%

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

      4. metadata-eval 33.2%

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

      5. +-commutative 33.2%

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

      6. fma-def 33.2%

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

      7. distribute-rgt-neg-in 33.2%

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

   3. Simplified 33.2%

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

   5. Applied egg-rr 72.2%

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

   6. Taylor expanded in t around inf 90.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-262} \lor \neg \left(t \leq 1.75 \cdot 10^{-202}\right) \land t \leq 4.6 \cdot 10^{-119}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-261}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-262} \lor \neg \left(t \leq 5 \cdot 10^{-203}\right) \land t \leq 4.5 \cdot 10^{-119}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -2.55e-261)
     (- t_1)
     (if (or (<= t 3e-262) (and (not (<= t 5e-203)) (<= t 4.5e-119)))
       (* t (/ (sqrt x) l))
       t_1))))

C

double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -2.55e-261) {
		tmp = -t_1;
	} else if ((t <= 3e-262) || (!(t <= 5e-203) && (t <= 4.5e-119))) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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 <= (-2.55d-261)) then
        tmp = -t_1
    else if ((t <= 3d-262) .or. (.not. (t <= 5d-203)) .and. (t <= 4.5d-119)) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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 <= -2.55e-261) {
		tmp = -t_1;
	} else if ((t <= 3e-262) || (!(t <= 5e-203) && (t <= 4.5e-119))) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -2.55e-261:
		tmp = -t_1
	elif (t <= 3e-262) or (not (t <= 5e-203) and (t <= 4.5e-119)):
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -2.55e-261)
		tmp = Float64(-t_1);
	elseif ((t <= 3e-262) || (!(t <= 5e-203) && (t <= 4.5e-119)))
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -2.55e-261)
		tmp = -t_1;
	elseif ((t <= 3e-262) || (~((t <= 5e-203)) && (t <= 4.5e-119)))
		tmp = t * (sqrt(x) / l);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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, -2.55e-261], (-t$95$1), If[Or[LessEqual[t, 3e-262], And[N[Not[LessEqual[t, 5e-203]], $MachinePrecision], LessEqual[t, 4.5e-119]]], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.54999999999999978e-261

   1. Initial program 33.0%

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

      1. associate-*r/ 32.9%

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

      2. fma-neg 32.9%

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

      3. sub-neg 32.9%

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

      4. metadata-eval 32.9%

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

      5. +-commutative 32.9%

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

      6. fma-def 32.9%

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

      7. distribute-rgt-neg-in 32.9%

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

   3. Simplified 32.9%

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

   5. Applied egg-rr 70.2%

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

   6. Taylor expanded in t around -inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. sub-neg 86.5%

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

      3. metadata-eval 86.5%

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

   8. Simplified 86.5%

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

   if -2.54999999999999978e-261 < t < 3.00000000000000018e-262 or 5.0000000000000002e-203 < t < 4.5000000000000003e-119

   1. Initial program 9.9%

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

      1. associate-*l/ 9.9%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in x around inf 59.9%

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

      1. associate--l+ 59.9%

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

      2. unpow2 59.9%

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

      3. distribute-lft-out 59.9%

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

      4. unpow2 59.9%

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

      5. unpow2 59.9%

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

      6. associate-*r/ 59.9%

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

      7. mul-1-neg 59.9%

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

      8. +-commutative 59.9%

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

      9. unpow2 59.9%

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

      10. associate-*l* 59.9%

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

      11. unpow2 59.9%

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

      12. fma-udef 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in t around 0 54.2%

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

      1. sub-neg 54.2%

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

      2. unpow2 54.2%

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

      3. associate-/l* 54.2%

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

      4. mul-1-neg 54.2%

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

      5. unpow2 54.2%

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

      6. remove-double-neg 54.2%

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

      7. associate-/l* 64.6%

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

   9. Simplified 64.6%

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

   10. Taylor expanded in l around 0 64.1%

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

       1. associate-*l/ 64.1%

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

       2. *-lft-identity 64.1%

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

   12. Simplified 64.1%

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

   if 3.00000000000000018e-262 < t < 5.0000000000000002e-203 or 4.5000000000000003e-119 < t

   1. Initial program 33.2%

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

      1. associate-*r/ 33.2%

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

      2. fma-neg 33.2%

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

      3. sub-neg 33.2%

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

      4. metadata-eval 33.2%

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

      5. +-commutative 33.2%

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

      6. fma-def 33.2%

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

      7. distribute-rgt-neg-in 33.2%

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

   3. Simplified 33.2%

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

   5. Applied egg-rr 72.2%

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

   6. Taylor expanded in t around inf 90.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{-261}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-262} \lor \neg \left(t \leq 5 \cdot 10^{-203}\right) \land t \leq 4.5 \cdot 10^{-119}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 46.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -5.8e-261)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (* (sqrt x) (/ t l))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.8e-261) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = sqrt(x) * (t / l);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.8d-261)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else
        tmp = sqrt(x) * (t / l)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.8e-261) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = Math.sqrt(x) * (t / l);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -5.8e-261:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	else:
		tmp = math.sqrt(x) * (t / l)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -5.8e-261)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(sqrt(x) * Float64(t / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5.8e-261)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	else
		tmp = sqrt(x) * (t / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -5.8e-261], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.7999999999999997e-261

   1. Initial program 33.0%

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

      1. associate-/l* 33.0%

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

      2. fma-neg 33.0%

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

      3. remove-double-neg 33.0%

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

      4. fma-neg 33.0%

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

      5. sub-neg 33.0%

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

      6. metadata-eval 33.0%

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

      7. remove-double-neg 33.0%

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

      8. fma-def 33.0%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in t around -inf 86.5%

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

      1. associate-*r* 86.5%

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

      2. neg-mul-1 86.5%

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

      3. +-commutative 86.5%

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

      4. sub-neg 86.5%

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

      5. metadata-eval 86.5%

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

      6. +-commutative 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in x around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. metadata-eval 86.0%

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

      3. unpow2 86.0%

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

   9. Simplified 86.0%

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

   if -5.7999999999999997e-261 < t

   1. Initial program 30.2%

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

      1. associate-*l/ 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in x around inf 47.4%

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

      1. associate--l+ 47.4%

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

      2. unpow2 47.4%

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

      3. distribute-lft-out 47.4%

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

      4. unpow2 47.4%

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

      5. unpow2 47.4%

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

      6. associate-*r/ 47.4%

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

      7. mul-1-neg 47.4%

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

      8. +-commutative 47.4%

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

      9. unpow2 47.4%

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

      10. associate-*l* 47.4%

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

      11. unpow2 47.4%

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

      12. fma-udef 47.4%

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

   6. Simplified 47.4%

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

   7. Taylor expanded in t around 0 14.2%

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

      1. sub-neg 14.2%

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

      2. unpow2 14.2%

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

      3. associate-/l* 14.2%

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

      4. mul-1-neg 14.2%

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

      5. unpow2 14.2%

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

      6. remove-double-neg 14.2%

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

      7. associate-/l* 16.8%

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

   9. Simplified 16.8%

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

   10. Taylor expanded in l around 0 10.6%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \end{array} \]

Alternative 5: 47.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -5.8e-261)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (* t (/ (sqrt x) l))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.8e-261) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = t * (sqrt(x) / l);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.8d-261)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else
        tmp = t * (sqrt(x) / l)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.8e-261) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = t * (Math.sqrt(x) / l);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -5.8e-261:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	else:
		tmp = t * (math.sqrt(x) / l)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -5.8e-261)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(t * Float64(sqrt(x) / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5.8e-261)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	else
		tmp = t * (sqrt(x) / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -5.8e-261], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.7999999999999997e-261

   1. Initial program 33.0%

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

      1. associate-/l* 33.0%

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

      2. fma-neg 33.0%

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

      3. remove-double-neg 33.0%

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

      4. fma-neg 33.0%

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

      5. sub-neg 33.0%

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

      6. metadata-eval 33.0%

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

      7. remove-double-neg 33.0%

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

      8. fma-def 33.0%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in t around -inf 86.5%

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

      1. associate-*r* 86.5%

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

      2. neg-mul-1 86.5%

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

      3. +-commutative 86.5%

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

      4. sub-neg 86.5%

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

      5. metadata-eval 86.5%

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

      6. +-commutative 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in x around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. metadata-eval 86.0%

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

      3. unpow2 86.0%

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

   9. Simplified 86.0%

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

   if -5.7999999999999997e-261 < t

   1. Initial program 30.2%

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

      1. associate-*l/ 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in x around inf 47.4%

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

      1. associate--l+ 47.4%

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

      2. unpow2 47.4%

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

      3. distribute-lft-out 47.4%

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

      4. unpow2 47.4%

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

      5. unpow2 47.4%

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

      6. associate-*r/ 47.4%

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

      7. mul-1-neg 47.4%

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

      8. +-commutative 47.4%

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

      9. unpow2 47.4%

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

      10. associate-*l* 47.4%

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

      11. unpow2 47.4%

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

      12. fma-udef 47.4%

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

   6. Simplified 47.4%

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

   7. Taylor expanded in t around 0 14.2%

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

      1. sub-neg 14.2%

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

      2. unpow2 14.2%

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

      3. associate-/l* 14.2%

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

      4. mul-1-neg 14.2%

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

      5. unpow2 14.2%

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

      6. remove-double-neg 14.2%

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

      7. associate-/l* 16.8%

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

   9. Simplified 16.8%

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

   10. Taylor expanded in l around 0 13.4%

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

       1. associate-*l/ 13.4%

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

       2. *-lft-identity 13.4%

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

   12. Simplified 13.4%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 6: 40.8% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.15e-261)
   (+ (/ 1.0 x) (- -1.0 (/ 0.5 (* x x))))
   (/ -0.5 (* x x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.15e-261) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = -0.5 / (x * x);
	}
	return tmp;
}

Fortran

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 <= (-1.15d-261)) then
        tmp = (1.0d0 / x) + ((-1.0d0) - (0.5d0 / (x * x)))
    else
        tmp = (-0.5d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.15e-261) {
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	} else {
		tmp = -0.5 / (x * x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.15e-261:
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)))
	else:
		tmp = -0.5 / (x * x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.15e-261)
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(-0.5 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.15e-261)
		tmp = (1.0 / x) + (-1.0 - (0.5 / (x * x)));
	else
		tmp = -0.5 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.15e-261], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.15e-261

   1. Initial program 33.0%

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

      1. associate-/l* 33.0%

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

      2. fma-neg 33.0%

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

      3. remove-double-neg 33.0%

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

      4. fma-neg 33.0%

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

      5. sub-neg 33.0%

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

      6. metadata-eval 33.0%

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

      7. remove-double-neg 33.0%

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

      8. fma-def 33.0%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in t around -inf 86.5%

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

      1. associate-*r* 86.5%

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

      2. neg-mul-1 86.5%

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

      3. +-commutative 86.5%

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

      4. sub-neg 86.5%

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

      5. metadata-eval 86.5%

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

      6. +-commutative 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in x around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. metadata-eval 86.0%

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

      3. unpow2 86.0%

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

   9. Simplified 86.0%

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

   if -1.15e-261 < t

   1. Initial program 30.2%

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

      1. associate-/l* 30.1%

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

      2. fma-neg 30.2%

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

      3. remove-double-neg 30.2%

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

      4. fma-neg 30.1%

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

      5. sub-neg 30.1%

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

      6. metadata-eval 30.1%

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

      7. remove-double-neg 30.1%

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

      8. fma-def 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in t around -inf 1.8%

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

      1. associate-*r* 1.8%

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

      2. neg-mul-1 1.8%

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

      3. +-commutative 1.8%

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

      4. sub-neg 1.8%

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

      5. metadata-eval 1.8%

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

      6. +-commutative 1.8%

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

   6. Simplified 1.8%

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

   7. Taylor expanded in x around inf 1.8%

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

      1. associate-*r/ 1.8%

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

      2. metadata-eval 1.8%

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

      3. unpow2 1.8%

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

   9. Simplified 1.8%

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

   10. Taylor expanded in x around 0 5.3%

       \[\leadsto \color{blue}{\frac{-0.5}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 5.3%

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

   12. Simplified 5.3%

       \[\leadsto \color{blue}{\frac{-0.5}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x \cdot x}\\ \end{array} \]

Alternative 7: 40.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.15e-261) (+ (/ 1.0 x) -1.0) (/ -0.5 (* x x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.15e-261) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = -0.5 / (x * x);
	}
	return tmp;
}

Fortran

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 <= (-1.15d-261)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else
        tmp = (-0.5d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.15e-261) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = -0.5 / (x * x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.15e-261:
		tmp = (1.0 / x) + -1.0
	else:
		tmp = -0.5 / (x * x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.15e-261)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	else
		tmp = Float64(-0.5 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.15e-261)
		tmp = (1.0 / x) + -1.0;
	else
		tmp = -0.5 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.15e-261], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.15e-261

   1. Initial program 33.0%

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

      1. associate-/l* 33.0%

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

      2. fma-neg 33.0%

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

      3. remove-double-neg 33.0%

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

      4. fma-neg 33.0%

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

      5. sub-neg 33.0%

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

      6. metadata-eval 33.0%

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

      7. remove-double-neg 33.0%

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

      8. fma-def 33.0%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in t around -inf 86.5%

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

      1. associate-*r* 86.5%

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

      2. neg-mul-1 86.5%

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

      3. +-commutative 86.5%

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

      4. sub-neg 86.5%

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

      5. metadata-eval 86.5%

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

      6. +-commutative 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in x around inf 85.8%

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

   if -1.15e-261 < t

   1. Initial program 30.2%

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

      1. associate-/l* 30.1%

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

      2. fma-neg 30.2%

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

      3. remove-double-neg 30.2%

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

      4. fma-neg 30.1%

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

      5. sub-neg 30.1%

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

      6. metadata-eval 30.1%

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

      7. remove-double-neg 30.1%

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

      8. fma-def 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in t around -inf 1.8%

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

      1. associate-*r* 1.8%

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

      2. neg-mul-1 1.8%

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

      3. +-commutative 1.8%

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

      4. sub-neg 1.8%

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

      5. metadata-eval 1.8%

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

      6. +-commutative 1.8%

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

   6. Simplified 1.8%

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

   7. Taylor expanded in x around inf 1.8%

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

      1. associate-*r/ 1.8%

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

      2. metadata-eval 1.8%

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

      3. unpow2 1.8%

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

   9. Simplified 1.8%

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

   10. Taylor expanded in x around 0 5.3%

       \[\leadsto \color{blue}{\frac{-0.5}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 5.3%

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

   12. Simplified 5.3%

       \[\leadsto \color{blue}{\frac{-0.5}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-261}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x \cdot x}\\ \end{array} \]

Alternative 8: 39.4% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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 / x) + (-1.0d0)
end function

Java

public static double code(double x, double l, double t) {
	return (1.0 / x) + -1.0;
}

Python

def code(x, l, t):
	return (1.0 / x) + -1.0

Julia

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

MATLAB

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

Wolfram

code[x_, l_, t_] := N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

   1. associate-/l* 31.5%

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

   2. fma-neg 31.5%

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

   3. remove-double-neg 31.5%

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

   4. fma-neg 31.5%

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

   5. sub-neg 31.5%

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

   6. metadata-eval 31.5%

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

   7. remove-double-neg 31.5%

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

   8. fma-def 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in t around -inf 43.1%

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

   1. associate-*r* 43.1%

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

   2. neg-mul-1 43.1%

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

   3. +-commutative 43.1%

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

   4. sub-neg 43.1%

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

   5. metadata-eval 43.1%

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

   6. +-commutative 43.1%

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

6. Simplified 43.1%

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

7. Taylor expanded in x around inf 42.8%

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

8. Final simplification 42.8%

   \[\leadsto \frac{1}{x} + -1 \]

Alternative 9: 39.1% accurate, 225.0× speedup

Math

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

FPCore

(FPCore (x l t) :precision binary64 -1.0)

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, l_, t_] := -1.0

Derivation

1. Initial program 31.5%

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

   1. associate-/l* 31.5%

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

   2. fma-neg 31.5%

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

   3. remove-double-neg 31.5%

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

   4. fma-neg 31.5%

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

   5. sub-neg 31.5%

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

   6. metadata-eval 31.5%

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

   7. remove-double-neg 31.5%

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

   8. fma-def 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in t around -inf 43.1%

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

   1. associate-*r* 43.1%

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

   2. neg-mul-1 43.1%

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

   3. +-commutative 43.1%

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

   4. sub-neg 43.1%

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

   5. metadata-eval 43.1%

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

   6. +-commutative 43.1%

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

6. Simplified 43.1%

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

7. Taylor expanded in x around inf 42.5%

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

8. Final simplification 42.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023213 
(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)))))