Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 82.6%

Time: 24.8s

Alternatives: 10

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{-74}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -4.6e-74)
     (- t_1)
     (if (<= t 2.45e-5)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt
          (+
           (/ (* l l) x)
           (+
            (* 2.0 (+ (* t t) (/ (* t t) x)))
            (/ (fma (* t 2.0) t (* l l)) x))))))
       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 <= -4.6e-74) {
		tmp = -t_1;
	} else if (t <= 2.45e-5) {
		tmp = t * (sqrt(2.0) / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma((t * 2.0), t, (l * l)) / x)))));
	} 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 <= -4.6e-74)
		tmp = Float64(-t_1);
	elseif (t <= 2.45e-5)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(Float64(t * 2.0), t, Float64(l * l)) / x))))));
	else
		tmp = t_1;
	end
	return 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, -4.6e-74], (-t$95$1), If[LessEqual[t, 2.45e-5], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t * 2.0), $MachinePrecision] * t + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.59999999999999961e-74

   1. Initial program 37.2%

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

      1. associate-*r/ 37.2%

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

      2. fma-neg 37.2%

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

      3. sub-neg 37.2%

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

      4. metadata-eval 37.2%

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

      5. +-commutative 37.2%

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

      6. fma-def 37.2%

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

      7. distribute-rgt-neg-in 37.2%

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

   3. Simplified 37.2%

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

      1. associate-*r/ 37.2%

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

      2. metadata-eval 37.2%

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

      3. sub-neg 37.2%

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

      4. fma-udef 37.2%

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

      5. +-commutative 37.2%

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

      6. distribute-rgt-neg-out 37.2%

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

      7. fma-neg 37.2%

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

      8. *-commutative 37.2%

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

      9. fma-neg 37.2%

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

   5. Applied egg-rr 82.4%

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

   6. Taylor expanded in t around -inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. sub-neg 90.2%

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

      3. metadata-eval 90.2%

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

   8. Simplified 90.2%

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

   if -4.59999999999999961e-74 < t < 2.45e-5

   1. Initial program 20.6%

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

      1. associate-*l/ 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in x around inf 68.9%

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

      1. associate--l+ 68.9%

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

      2. unpow2 68.9%

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

      3. distribute-lft-out 68.9%

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

      4. unpow2 68.9%

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

      5. unpow2 68.9%

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

      6. associate-*r/ 68.9%

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

      7. mul-1-neg 68.9%

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

      8. +-commutative 68.9%

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

      9. unpow2 68.9%

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

      10. associate-*l* 68.9%

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

      11. unpow2 68.9%

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

      12. fma-udef 68.9%

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

   6. Simplified 68.9%

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

   if 2.45e-5 < t

   1. Initial program 37.8%

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

      1. associate-*r/ 37.7%

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

      2. fma-neg 37.7%

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

      3. sub-neg 37.7%

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

      4. metadata-eval 37.7%

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

      5. +-commutative 37.7%

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

      6. fma-def 37.7%

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

      7. distribute-rgt-neg-in 37.7%

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

   3. Simplified 37.7%

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

      1. associate-*r/ 37.8%

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

      2. metadata-eval 37.8%

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

      3. sub-neg 37.8%

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

      4. fma-udef 37.8%

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

      5. +-commutative 37.8%

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

      6. distribute-rgt-neg-out 37.8%

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

      7. fma-neg 37.8%

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

      8. *-commutative 37.8%

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

      9. fma-neg 37.8%

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

   5. Applied egg-rr 83.1%

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

   6. Taylor expanded in t around inf 93.3%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-74}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-103}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{\mathsf{hypot}\left(\ell, \ell\right)}{\sqrt{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.5e-103)
     (- t_1)
     (if (<= t 6.6e-183) (/ (sqrt 2.0) (/ (/ (hypot l l) (sqrt x)) t)) 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 <= -1.5e-103) {
		tmp = -t_1;
	} else if (t <= 6.6e-183) {
		tmp = sqrt(2.0) / ((hypot(l, l) / sqrt(x)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 <= -1.5e-103) {
		tmp = -t_1;
	} else if (t <= 6.6e-183) {
		tmp = Math.sqrt(2.0) / ((Math.hypot(l, l) / Math.sqrt(x)) / t);
	} 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 <= -1.5e-103:
		tmp = -t_1
	elif t <= 6.6e-183:
		tmp = math.sqrt(2.0) / ((math.hypot(l, l) / math.sqrt(x)) / t)
	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 <= -1.5e-103)
		tmp = Float64(-t_1);
	elseif (t <= 6.6e-183)
		tmp = Float64(sqrt(2.0) / Float64(Float64(hypot(l, l) / sqrt(x)) / t));
	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 <= -1.5e-103)
		tmp = -t_1;
	elseif (t <= 6.6e-183)
		tmp = sqrt(2.0) / ((hypot(l, l) / sqrt(x)) / t);
	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, -1.5e-103], (-t$95$1), If[LessEqual[t, 6.6e-183], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Sqrt[l ^ 2 + l ^ 2], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5e-103

   1. Initial program 38.0%

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

      1. associate-*r/ 38.0%

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

      2. fma-neg 38.0%

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

      3. sub-neg 38.0%

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

      4. metadata-eval 38.0%

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

      5. +-commutative 38.0%

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

      6. fma-def 38.0%

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

      7. distribute-rgt-neg-in 38.0%

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

   3. Simplified 38.0%

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

      1. associate-*r/ 38.0%

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

      2. metadata-eval 38.0%

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

      3. sub-neg 38.0%

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

      4. fma-udef 38.0%

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

      5. +-commutative 38.0%

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

      6. distribute-rgt-neg-out 38.0%

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

      7. fma-neg 38.0%

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

      8. *-commutative 38.0%

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

      9. fma-neg 38.0%

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

   5. Applied egg-rr 82.1%

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

   6. Taylor expanded in t around -inf 89.5%

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

      1. mul-1-neg 89.5%

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

      2. sub-neg 89.5%

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

      3. metadata-eval 89.5%

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

   8. Simplified 89.5%

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

   if -1.5e-103 < t < 6.5999999999999999e-183

   1. Initial program 11.3%

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

      1. associate-/l* 11.4%

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

      2. fma-neg 11.4%

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

      3. remove-double-neg 11.4%

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

      4. fma-neg 11.4%

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

      5. sub-neg 11.4%

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

      6. metadata-eval 11.4%

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

      7. remove-double-neg 11.4%

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

      8. fma-def 11.4%

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

   3. Simplified 11.4%

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

   4. Taylor expanded in t around 0 5.1%

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

      1. associate-*l/ 5.1%

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

      2. *-lft-identity 5.1%

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

      3. *-commutative 5.1%

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

      4. unpow2 5.1%

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

      5. +-commutative 5.1%

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

      6. sub-neg 5.1%

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

      7. metadata-eval 5.1%

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

      8. +-commutative 5.1%

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

      9. unpow2 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in x around inf 49.5%

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

      1. unpow2 49.5%

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

      2. neg-mul-1 49.5%

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

      3. unpow2 49.5%

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

   9. Simplified 49.5%

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

       1. sqrt-div 53.7%

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

       2. fma-neg 53.7%

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

       3. add-sqr-sqrt 0.3%

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

       4. sqrt-unprod 3.5%

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

       5. sqr-neg 3.5%

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

       6. sqrt-unprod 3.0%

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

       7. add-sqr-sqrt 3.0%

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

       8. add-sqr-sqrt 0.3%

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

       9. sqrt-unprod 37.8%

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

       10. sqr-neg 37.8%

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

       11. sqrt-unprod 53.7%

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

       12. add-sqr-sqrt 53.7%

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

   11. Applied egg-rr 53.7%

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

       1. fma-udef 53.7%

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

       2. hypot-def 57.3%

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

   13. Simplified 57.3%

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

   if 6.5999999999999999e-183 < t

   1. Initial program 37.4%

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

      1. associate-*r/ 37.2%

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

      2. fma-neg 37.2%

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

      3. sub-neg 37.2%

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

      4. metadata-eval 37.2%

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

      5. +-commutative 37.2%

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

      6. fma-def 37.2%

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

      7. distribute-rgt-neg-in 37.2%

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

   3. Simplified 37.2%

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

      1. associate-*r/ 37.4%

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

      2. metadata-eval 37.4%

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

      3. sub-neg 37.4%

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

      4. fma-udef 37.4%

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

      5. +-commutative 37.4%

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

      6. distribute-rgt-neg-out 37.4%

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

      7. fma-neg 37.4%

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

      8. *-commutative 37.4%

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

      9. fma-neg 37.4%

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

   5. Applied egg-rr 72.8%

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

   6. Taylor expanded in t around inf 86.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-103}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{\mathsf{hypot}\left(\ell, \ell\right)}{\sqrt{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.95 \cdot 10^{-226}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell + \ell \cdot \ell}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -2.95e-226)
     (- t_1)
     (if (<= t 1.85e-268)
       (/ (sqrt 2.0) (/ (sqrt (/ (+ (* l l) (* l l)) x)) t))
       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 <= -2.95e-226) {
		tmp = -t_1;
	} else if (t <= 1.85e-268) {
		tmp = sqrt(2.0) / (sqrt((((l * l) + (l * l)) / x)) / t);
	} 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 <= (-2.95d-226)) then
        tmp = -t_1
    else if (t <= 1.85d-268) then
        tmp = sqrt(2.0d0) / (sqrt((((l * l) + (l * l)) / x)) / t)
    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 <= -2.95e-226) {
		tmp = -t_1;
	} else if (t <= 1.85e-268) {
		tmp = Math.sqrt(2.0) / (Math.sqrt((((l * l) + (l * l)) / x)) / t);
	} 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 <= -2.95e-226:
		tmp = -t_1
	elif t <= 1.85e-268:
		tmp = math.sqrt(2.0) / (math.sqrt((((l * l) + (l * l)) / x)) / t)
	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 <= -2.95e-226)
		tmp = Float64(-t_1);
	elseif (t <= 1.85e-268)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(Float64(l * l) + Float64(l * l)) / x)) / t));
	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 <= -2.95e-226)
		tmp = -t_1;
	elseif (t <= 1.85e-268)
		tmp = sqrt(2.0) / (sqrt((((l * l) + (l * l)) / x)) / t);
	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, -2.95e-226], (-t$95$1), If[LessEqual[t, 1.85e-268], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.95e-226

   1. Initial program 34.6%

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

      1. associate-*r/ 34.6%

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

      2. fma-neg 34.6%

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

      3. sub-neg 34.6%

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

      4. metadata-eval 34.6%

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

      5. +-commutative 34.6%

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

      6. fma-def 34.6%

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

      7. distribute-rgt-neg-in 34.6%

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

   3. Simplified 34.6%

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

      1. associate-*r/ 34.6%

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

      2. metadata-eval 34.6%

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

      3. sub-neg 34.6%

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

      4. fma-udef 34.6%

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

      5. +-commutative 34.6%

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

      6. distribute-rgt-neg-out 34.6%

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

      7. fma-neg 34.6%

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

      8. *-commutative 34.6%

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

      9. fma-neg 34.6%

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

   5. Applied egg-rr 74.1%

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

   6. Taylor expanded in t around -inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. sub-neg 81.2%

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

      3. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

   if -2.95e-226 < t < 1.85000000000000009e-268

   1. Initial program 3.5%

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

      1. associate-/l* 3.5%

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

      2. fma-neg 3.5%

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

      3. remove-double-neg 3.5%

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

      4. fma-neg 3.5%

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

      5. sub-neg 3.5%

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

      6. metadata-eval 3.5%

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

      7. remove-double-neg 3.5%

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

      8. fma-def 3.5%

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

   3. Simplified 3.5%

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

   4. Taylor expanded in t around 0 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. *-commutative 4.3%

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

      4. unpow2 4.3%

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

      5. +-commutative 4.3%

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

      6. sub-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. unpow2 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in x around inf 73.3%

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

      1. unpow2 73.3%

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

      2. neg-mul-1 73.3%

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

      3. unpow2 73.3%

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

   9. Simplified 73.3%

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

   if 1.85000000000000009e-268 < t

   1. Initial program 32.6%

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

      1. associate-*r/ 32.5%

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

      2. fma-neg 32.5%

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

      3. sub-neg 32.5%

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

      4. metadata-eval 32.5%

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

      5. +-commutative 32.5%

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

      6. fma-def 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

   3. Simplified 32.5%

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

      1. associate-*r/ 32.6%

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

      2. metadata-eval 32.6%

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

      3. sub-neg 32.6%

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

      4. fma-udef 32.6%

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

      5. +-commutative 32.6%

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

      6. distribute-rgt-neg-out 32.6%

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

      7. fma-neg 32.6%

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

      8. *-commutative 32.6%

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

      9. fma-neg 32.6%

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

   5. Applied egg-rr 66.4%

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

   6. Taylor expanded in t around inf 80.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.95 \cdot 10^{-226}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell + \ell \cdot \ell}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-226}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \sqrt{x \cdot \frac{2}{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -4.5e-226)
     (- t_1)
     (if (<= t 2.3e-268) (* t (sqrt (* x (/ 2.0 (fma l l (* l 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 <= -4.5e-226) {
		tmp = -t_1;
	} else if (t <= 2.3e-268) {
		tmp = t * sqrt((x * (2.0 / fma(l, l, (l * 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 <= -4.5e-226)
		tmp = Float64(-t_1);
	elseif (t <= 2.3e-268)
		tmp = Float64(t * sqrt(Float64(x * Float64(2.0 / fma(l, l, Float64(l * l))))));
	else
		tmp = t_1;
	end
	return 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, -4.5e-226], (-t$95$1), If[LessEqual[t, 2.3e-268], N[(t * N[Sqrt[N[(x * N[(2.0 / N[(l * l + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.50000000000000011e-226

   1. Initial program 34.6%

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

      1. associate-*r/ 34.6%

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

      2. fma-neg 34.6%

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

      3. sub-neg 34.6%

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

      4. metadata-eval 34.6%

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

      5. +-commutative 34.6%

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

      6. fma-def 34.6%

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

      7. distribute-rgt-neg-in 34.6%

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

   3. Simplified 34.6%

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

      1. associate-*r/ 34.6%

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

      2. metadata-eval 34.6%

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

      3. sub-neg 34.6%

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

      4. fma-udef 34.6%

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

      5. +-commutative 34.6%

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

      6. distribute-rgt-neg-out 34.6%

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

      7. fma-neg 34.6%

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

      8. *-commutative 34.6%

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

      9. fma-neg 34.6%

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

   5. Applied egg-rr 74.1%

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

   6. Taylor expanded in t around -inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. sub-neg 81.2%

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

      3. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

   if -4.50000000000000011e-226 < t < 2.3000000000000001e-268

   1. Initial program 3.5%

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

      1. associate-/l* 3.5%

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

      2. fma-neg 3.5%

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

      3. remove-double-neg 3.5%

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

      4. fma-neg 3.5%

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

      5. sub-neg 3.5%

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

      6. metadata-eval 3.5%

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

      7. remove-double-neg 3.5%

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

      8. fma-def 3.5%

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

   3. Simplified 3.5%

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

   4. Taylor expanded in t around 0 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. *-commutative 4.3%

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

      4. unpow2 4.3%

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

      5. +-commutative 4.3%

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

      6. sub-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. unpow2 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in x around inf 73.3%

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

      1. unpow2 73.3%

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

      2. neg-mul-1 73.3%

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

      3. unpow2 73.3%

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

   9. Simplified 73.3%

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

       1. expm1-log1p-u 73.1%

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

       2. expm1-udef 39.1%

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

   11. Applied egg-rr 38.9%

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

       1. expm1-def 69.4%

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

       2. expm1-log1p 69.6%

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

       3. *-commutative 69.6%

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

       4. associate-/r/ 69.6%

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

   13. Simplified 69.6%

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

   if 2.3000000000000001e-268 < t

   1. Initial program 32.6%

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

      1. associate-*r/ 32.5%

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

      2. fma-neg 32.5%

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

      3. sub-neg 32.5%

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

      4. metadata-eval 32.5%

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

      5. +-commutative 32.5%

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

      6. fma-def 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

   3. Simplified 32.5%

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

      1. associate-*r/ 32.6%

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

      2. metadata-eval 32.6%

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

      3. sub-neg 32.6%

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

      4. fma-udef 32.6%

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

      5. +-commutative 32.6%

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

      6. distribute-rgt-neg-out 32.6%

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

      7. fma-neg 32.6%

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

      8. *-commutative 32.6%

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

      9. fma-neg 32.6%

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

   5. Applied egg-rr 66.4%

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

   6. Taylor expanded in t around inf 80.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-226}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \sqrt{x \cdot \frac{2}{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 5: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-227}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -5e-227)
     (- t_1)
     (if (<= t 4.4e-270)
       (/ (sqrt 2.0) (/ (sqrt (* 2.0 (/ l (/ x l)))) t))
       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 <= -5e-227) {
		tmp = -t_1;
	} else if (t <= 4.4e-270) {
		tmp = sqrt(2.0) / (sqrt((2.0 * (l / (x / l)))) / t);
	} 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 <= (-5d-227)) then
        tmp = -t_1
    else if (t <= 4.4d-270) then
        tmp = sqrt(2.0d0) / (sqrt((2.0d0 * (l / (x / l)))) / t)
    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 <= -5e-227) {
		tmp = -t_1;
	} else if (t <= 4.4e-270) {
		tmp = Math.sqrt(2.0) / (Math.sqrt((2.0 * (l / (x / l)))) / t);
	} 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 <= -5e-227:
		tmp = -t_1
	elif t <= 4.4e-270:
		tmp = math.sqrt(2.0) / (math.sqrt((2.0 * (l / (x / l)))) / t)
	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 <= -5e-227)
		tmp = Float64(-t_1);
	elseif (t <= 4.4e-270)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(2.0 * Float64(l / Float64(x / l)))) / t));
	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 <= -5e-227)
		tmp = -t_1;
	elseif (t <= 4.4e-270)
		tmp = sqrt(2.0) / (sqrt((2.0 * (l / (x / l)))) / t);
	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, -5e-227], (-t$95$1), If[LessEqual[t, 4.4e-270], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(2.0 * N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.99999999999999961e-227

   1. Initial program 34.6%

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

      1. associate-*r/ 34.6%

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

      2. fma-neg 34.6%

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

      3. sub-neg 34.6%

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

      4. metadata-eval 34.6%

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

      5. +-commutative 34.6%

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

      6. fma-def 34.6%

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

      7. distribute-rgt-neg-in 34.6%

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

   3. Simplified 34.6%

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

      1. associate-*r/ 34.6%

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

      2. metadata-eval 34.6%

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

      3. sub-neg 34.6%

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

      4. fma-udef 34.6%

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

      5. +-commutative 34.6%

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

      6. distribute-rgt-neg-out 34.6%

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

      7. fma-neg 34.6%

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

      8. *-commutative 34.6%

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

      9. fma-neg 34.6%

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

   5. Applied egg-rr 74.1%

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

   6. Taylor expanded in t around -inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. sub-neg 81.2%

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

      3. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

   if -4.99999999999999961e-227 < t < 4.3999999999999997e-270

   1. Initial program 3.5%

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

      1. associate-/l* 3.5%

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

      2. fma-neg 3.5%

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

      3. remove-double-neg 3.5%

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

      4. fma-neg 3.5%

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

      5. sub-neg 3.5%

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

      6. metadata-eval 3.5%

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

      7. remove-double-neg 3.5%

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

      8. fma-def 3.5%

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

   3. Simplified 3.5%

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

   4. Taylor expanded in t around 0 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. *-commutative 4.3%

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

      4. unpow2 4.3%

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

      5. +-commutative 4.3%

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

      6. sub-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. unpow2 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in x around inf 73.3%

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

      1. unpow2 73.3%

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

      2. neg-mul-1 73.3%

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

      3. unpow2 73.3%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in l around 0 73.1%

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

       1. unpow2 73.1%

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

       2. associate-/l* 73.1%

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

   12. Simplified 73.1%

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

   if 4.3999999999999997e-270 < t

   1. Initial program 32.6%

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

      1. associate-*r/ 32.5%

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

      2. fma-neg 32.5%

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

      3. sub-neg 32.5%

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

      4. metadata-eval 32.5%

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

      5. +-commutative 32.5%

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

      6. fma-def 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

   3. Simplified 32.5%

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

      1. associate-*r/ 32.6%

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

      2. metadata-eval 32.6%

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

      3. sub-neg 32.6%

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

      4. fma-udef 32.6%

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

      5. +-commutative 32.6%

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

      6. distribute-rgt-neg-out 32.6%

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

      7. fma-neg 32.6%

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

      8. *-commutative 32.6%

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

      9. fma-neg 32.6%

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

   5. Applied egg-rr 66.4%

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

   6. Taylor expanded in t around inf 80.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-227}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 6: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-226}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.45e-226)
   -1.0
   (if (<= t 6.8e-270)
     (* (sqrt x) (/ (- t) l))
     (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.45e-226) {
		tmp = -1.0;
	} else if (t <= 6.8e-270) {
		tmp = sqrt(x) * (-t / 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 <= (-1.45d-226)) then
        tmp = -1.0d0
    else if (t <= 6.8d-270) then
        tmp = sqrt(x) * (-t / 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 <= -1.45e-226) {
		tmp = -1.0;
	} else if (t <= 6.8e-270) {
		tmp = Math.sqrt(x) * (-t / l);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.45e-226:
		tmp = -1.0
	elif t <= 6.8e-270:
		tmp = math.sqrt(x) * (-t / l)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.45e-226)
		tmp = -1.0;
	elseif (t <= 6.8e-270)
		tmp = Float64(sqrt(x) * Float64(Float64(-t) / 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 <= -1.45e-226)
		tmp = -1.0;
	elseif (t <= 6.8e-270)
		tmp = sqrt(x) * (-t / l);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.45e-226], -1.0, If[LessEqual[t, 6.8e-270], N[(N[Sqrt[x], $MachinePrecision] * N[((-t) / 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 < -1.45000000000000001e-226

   1. Initial program 34.6%

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

      1. associate-*r/ 34.6%

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

      2. fma-neg 34.6%

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

      3. sub-neg 34.6%

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

      4. metadata-eval 34.6%

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

      5. +-commutative 34.6%

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

      6. fma-def 34.6%

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

      7. distribute-rgt-neg-in 34.6%

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

   3. Simplified 34.6%

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

      1. fma-udef 34.6%

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

      2. fma-udef 34.6%

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

      3. +-commutative 34.6%

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

      4. *-commutative 34.6%

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

      5. clear-num 34.6%

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

      6. metadata-eval 34.6%

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

      7. sub-neg 34.6%

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

      8. un-div-inv 34.6%

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

      9. +-commutative 34.6%

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

      10. associate-*r* 34.6%

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

      11. fma-def 34.6%

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

      12. sub-neg 34.6%

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

      13. metadata-eval 34.6%

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

      14. add-sqr-sqrt 15.6%

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

      15. sqrt-unprod 36.5%

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

   5. Applied egg-rr 37.4%

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

   6. Taylor expanded in x around inf 35.7%

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

      1. distribute-lft-out 35.7%

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

      2. unpow2 35.7%

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

      3. unpow2 35.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in t around -inf 79.3%

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

   if -1.45000000000000001e-226 < t < 6.8000000000000001e-270

   1. Initial program 3.5%

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

      1. associate-/l* 3.5%

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

      2. fma-neg 3.5%

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

      3. remove-double-neg 3.5%

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

      4. fma-neg 3.5%

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

      5. sub-neg 3.5%

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

      6. metadata-eval 3.5%

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

      7. remove-double-neg 3.5%

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

      8. fma-def 3.5%

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

   3. Simplified 3.5%

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

   4. Taylor expanded in t around 0 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. *-commutative 4.3%

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

      4. unpow2 4.3%

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

      5. +-commutative 4.3%

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

      6. sub-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. unpow2 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in x around inf 73.3%

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

      1. unpow2 73.3%

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

      2. neg-mul-1 73.3%

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

      3. unpow2 73.3%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in l around -inf 63.6%

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

       1. mul-1-neg 63.6%

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

       2. *-commutative 63.6%

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

       3. distribute-rgt-neg-in 63.6%

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

   12. Simplified 63.6%

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

   if 6.8000000000000001e-270 < t

   1. Initial program 32.6%

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

      1. associate-*r/ 32.5%

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

      2. fma-neg 32.5%

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

      3. sub-neg 32.5%

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

      4. metadata-eval 32.5%

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

      5. +-commutative 32.5%

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

      6. fma-def 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

   3. Simplified 32.5%

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

      1. associate-*r/ 32.6%

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

      2. metadata-eval 32.6%

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

      3. sub-neg 32.6%

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

      4. fma-udef 32.6%

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

      5. +-commutative 32.6%

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

      6. distribute-rgt-neg-out 32.6%

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

      7. fma-neg 32.6%

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

      8. *-commutative 32.6%

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

      9. fma-neg 32.6%

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

   5. Applied egg-rr 66.4%

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

   6. Taylor expanded in t around inf 80.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-226}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7: 78.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{-227}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -4.1e-227)
     (- t_1)
     (if (<= t 2.25e-268) (* (sqrt x) (/ (- t) 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 <= -4.1e-227) {
		tmp = -t_1;
	} else if (t <= 2.25e-268) {
		tmp = sqrt(x) * (-t / 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 <= (-4.1d-227)) then
        tmp = -t_1
    else if (t <= 2.25d-268) then
        tmp = sqrt(x) * (-t / 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 <= -4.1e-227) {
		tmp = -t_1;
	} else if (t <= 2.25e-268) {
		tmp = Math.sqrt(x) * (-t / 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 <= -4.1e-227:
		tmp = -t_1
	elif t <= 2.25e-268:
		tmp = math.sqrt(x) * (-t / 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 <= -4.1e-227)
		tmp = Float64(-t_1);
	elseif (t <= 2.25e-268)
		tmp = Float64(sqrt(x) * Float64(Float64(-t) / 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 <= -4.1e-227)
		tmp = -t_1;
	elseif (t <= 2.25e-268)
		tmp = sqrt(x) * (-t / 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, -4.1e-227], (-t$95$1), If[LessEqual[t, 2.25e-268], N[(N[Sqrt[x], $MachinePrecision] * N[((-t) / l), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.10000000000000009e-227

   1. Initial program 34.6%

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

      1. associate-*r/ 34.6%

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

      2. fma-neg 34.6%

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

      3. sub-neg 34.6%

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

      4. metadata-eval 34.6%

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

      5. +-commutative 34.6%

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

      6. fma-def 34.6%

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

      7. distribute-rgt-neg-in 34.6%

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

   3. Simplified 34.6%

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

      1. associate-*r/ 34.6%

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

      2. metadata-eval 34.6%

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

      3. sub-neg 34.6%

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

      4. fma-udef 34.6%

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

      5. +-commutative 34.6%

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

      6. distribute-rgt-neg-out 34.6%

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

      7. fma-neg 34.6%

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

      8. *-commutative 34.6%

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

      9. fma-neg 34.6%

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

   5. Applied egg-rr 74.1%

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

   6. Taylor expanded in t around -inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. sub-neg 81.2%

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

      3. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

   if -4.10000000000000009e-227 < t < 2.2500000000000001e-268

   1. Initial program 3.5%

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

      1. associate-/l* 3.5%

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

      2. fma-neg 3.5%

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

      3. remove-double-neg 3.5%

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

      4. fma-neg 3.5%

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

      5. sub-neg 3.5%

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

      6. metadata-eval 3.5%

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

      7. remove-double-neg 3.5%

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

      8. fma-def 3.5%

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

   3. Simplified 3.5%

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

   4. Taylor expanded in t around 0 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. *-commutative 4.3%

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

      4. unpow2 4.3%

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

      5. +-commutative 4.3%

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

      6. sub-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. unpow2 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in x around inf 73.3%

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

      1. unpow2 73.3%

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

      2. neg-mul-1 73.3%

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

      3. unpow2 73.3%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in l around -inf 63.6%

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

       1. mul-1-neg 63.6%

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

       2. *-commutative 63.6%

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

       3. distribute-rgt-neg-in 63.6%

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

   12. Simplified 63.6%

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

   if 2.2500000000000001e-268 < t

   1. Initial program 32.6%

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

      1. associate-*r/ 32.5%

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

      2. fma-neg 32.5%

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

      3. sub-neg 32.5%

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

      4. metadata-eval 32.5%

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

      5. +-commutative 32.5%

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

      6. fma-def 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

   3. Simplified 32.5%

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

      1. associate-*r/ 32.6%

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

      2. metadata-eval 32.6%

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

      3. sub-neg 32.6%

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

      4. fma-udef 32.6%

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

      5. +-commutative 32.6%

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

      6. distribute-rgt-neg-out 32.6%

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

      7. fma-neg 32.6%

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

      8. *-commutative 32.6%

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

      9. fma-neg 32.6%

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

   5. Applied egg-rr 66.4%

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

   6. Taylor expanded in t around inf 80.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-227}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 8: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-227}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -2.7e-227) -1.0 (if (<= t 8.8e-269) (* (sqrt x) (/ (- t) l)) 1.0)))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.7e-227) {
		tmp = -1.0;
	} else if (t <= 8.8e-269) {
		tmp = sqrt(x) * (-t / l);
	} else {
		tmp = 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 <= (-2.7d-227)) then
        tmp = -1.0d0
    else if (t <= 8.8d-269) then
        tmp = sqrt(x) * (-t / l)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.7e-227) {
		tmp = -1.0;
	} else if (t <= 8.8e-269) {
		tmp = Math.sqrt(x) * (-t / l);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2.7e-227:
		tmp = -1.0
	elif t <= 8.8e-269:
		tmp = math.sqrt(x) * (-t / l)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2.7e-227)
		tmp = -1.0;
	elseif (t <= 8.8e-269)
		tmp = Float64(sqrt(x) * Float64(Float64(-t) / l));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2.7e-227)
		tmp = -1.0;
	elseif (t <= 8.8e-269)
		tmp = sqrt(x) * (-t / l);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2.7e-227], -1.0, If[LessEqual[t, 8.8e-269], N[(N[Sqrt[x], $MachinePrecision] * N[((-t) / l), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if t < -2.7e-227

   1. Initial program 34.6%

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

      1. associate-*r/ 34.6%

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

      2. fma-neg 34.6%

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

      3. sub-neg 34.6%

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

      4. metadata-eval 34.6%

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

      5. +-commutative 34.6%

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

      6. fma-def 34.6%

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

      7. distribute-rgt-neg-in 34.6%

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

   3. Simplified 34.6%

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

      1. fma-udef 34.6%

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

      2. fma-udef 34.6%

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

      3. +-commutative 34.6%

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

      4. *-commutative 34.6%

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

      5. clear-num 34.6%

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

      6. metadata-eval 34.6%

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

      7. sub-neg 34.6%

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

      8. un-div-inv 34.6%

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

      9. +-commutative 34.6%

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

      10. associate-*r* 34.6%

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

      11. fma-def 34.6%

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

      12. sub-neg 34.6%

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

      13. metadata-eval 34.6%

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

      14. add-sqr-sqrt 15.6%

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

      15. sqrt-unprod 36.5%

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

   5. Applied egg-rr 37.4%

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

   6. Taylor expanded in x around inf 35.7%

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

      1. distribute-lft-out 35.7%

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

      2. unpow2 35.7%

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

      3. unpow2 35.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in t around -inf 79.3%

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

   if -2.7e-227 < t < 8.79999999999999936e-269

   1. Initial program 3.5%

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

      1. associate-/l* 3.5%

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

      2. fma-neg 3.5%

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

      3. remove-double-neg 3.5%

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

      4. fma-neg 3.5%

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

      5. sub-neg 3.5%

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

      6. metadata-eval 3.5%

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

      7. remove-double-neg 3.5%

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

      8. fma-def 3.5%

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

   3. Simplified 3.5%

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

   4. Taylor expanded in t around 0 4.3%

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

      1. associate-*l/ 4.3%

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

      2. *-lft-identity 4.3%

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

      3. *-commutative 4.3%

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

      4. unpow2 4.3%

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

      5. +-commutative 4.3%

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

      6. sub-neg 4.3%

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

      7. metadata-eval 4.3%

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

      8. +-commutative 4.3%

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

      9. unpow2 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in x around inf 73.3%

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

      1. unpow2 73.3%

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

      2. neg-mul-1 73.3%

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

      3. unpow2 73.3%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in l around -inf 63.6%

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

       1. mul-1-neg 63.6%

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

       2. *-commutative 63.6%

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

       3. distribute-rgt-neg-in 63.6%

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

   12. Simplified 63.6%

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

   if 8.79999999999999936e-269 < t

   1. Initial program 32.6%

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

      1. associate-*r/ 32.5%

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

      2. fma-neg 32.5%

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

      3. sub-neg 32.5%

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

      4. metadata-eval 32.5%

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

      5. +-commutative 32.5%

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

      6. fma-def 32.5%

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

      7. distribute-rgt-neg-in 32.5%

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

   3. Simplified 32.5%

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

      1. fma-udef 32.5%

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

      2. fma-udef 32.5%

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

      3. +-commutative 32.5%

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

      4. *-commutative 32.5%

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

      5. clear-num 32.5%

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

      6. metadata-eval 32.5%

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

      7. sub-neg 32.5%

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

      8. un-div-inv 32.5%

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

      9. +-commutative 32.5%

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

      10. associate-*r* 32.5%

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

      11. fma-def 32.5%

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

      12. sub-neg 32.5%

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

      13. metadata-eval 32.5%

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

      14. add-sqr-sqrt 15.4%

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

      15. sqrt-unprod 31.6%

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

   5. Applied egg-rr 33.7%

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

   6. Taylor expanded in x around inf 33.0%

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

      1. distribute-lft-out 33.0%

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

      2. unpow2 33.0%

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

      3. unpow2 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in t around inf 79.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-227}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 76.0% accurate, 73.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-302}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (if (<= t -1.02e-302) -1.0 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.02e-302) {
		tmp = -1.0;
	} else {
		tmp = 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 <= (-1.02d-302)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.02e-302) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.02e-302:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.02e-302)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.02e-302)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.02e-302], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.02e-302

   1. Initial program 32.1%

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

      1. associate-*r/ 32.0%

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

      2. fma-neg 32.0%

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

      3. sub-neg 32.0%

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

      4. metadata-eval 32.0%

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

      5. +-commutative 32.0%

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

      6. fma-def 32.0%

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

      7. distribute-rgt-neg-in 32.0%

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

   3. Simplified 32.0%

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

      1. fma-udef 32.0%

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

      2. fma-udef 32.0%

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

      3. +-commutative 32.0%

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

      4. *-commutative 32.0%

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

      5. clear-num 32.0%

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

      6. metadata-eval 32.0%

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

      7. sub-neg 32.0%

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

      8. un-div-inv 32.0%

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

      9. +-commutative 32.0%

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

      10. associate-*r* 32.0%

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

      11. fma-def 32.0%

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

      12. sub-neg 32.0%

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

      13. metadata-eval 32.0%

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

      14. add-sqr-sqrt 14.3%

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

      15. sqrt-unprod 36.2%

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

   5. Applied egg-rr 37.0%

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

   6. Taylor expanded in x around inf 35.5%

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

      1. distribute-lft-out 35.5%

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

      2. unpow2 35.5%

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

      3. unpow2 35.5%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in t around -inf 74.9%

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

   if -1.02e-302 < t

   1. Initial program 31.8%

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

      1. associate-*r/ 31.7%

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

      2. fma-neg 31.7%

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

      3. sub-neg 31.7%

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

      4. metadata-eval 31.7%

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

      5. +-commutative 31.7%

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

      6. fma-def 31.7%

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

      7. distribute-rgt-neg-in 31.7%

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

   3. Simplified 31.7%

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

      1. fma-udef 31.7%

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

      2. fma-udef 31.7%

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

      3. +-commutative 31.7%

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

      4. *-commutative 31.7%

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

      5. clear-num 31.7%

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

      6. metadata-eval 31.7%

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

      7. sub-neg 31.7%

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

      8. un-div-inv 31.7%

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

      9. +-commutative 31.7%

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

      10. associate-*r* 31.7%

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

      11. fma-def 31.7%

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

      12. sub-neg 31.7%

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

      13. metadata-eval 31.7%

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

      14. add-sqr-sqrt 15.0%

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

      15. sqrt-unprod 32.6%

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

   5. Applied egg-rr 34.6%

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

   6. Taylor expanded in x around inf 34.0%

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

      1. distribute-lft-out 34.0%

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

      2. unpow2 34.0%

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

      3. unpow2 34.0%

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

   8. Simplified 34.0%

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

   9. Taylor expanded in t around inf 77.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-302}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 38.4% 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 31.9%

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

   1. associate-*r/ 31.9%

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

   2. fma-neg 31.9%

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

   3. sub-neg 31.9%

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

   4. metadata-eval 31.9%

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

   5. +-commutative 31.9%

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

   6. fma-def 31.9%

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

   7. distribute-rgt-neg-in 31.9%

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

3. Simplified 31.9%

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

   1. fma-udef 31.9%

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

   2. fma-udef 31.9%

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

   3. +-commutative 31.9%

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

   4. *-commutative 31.9%

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

   5. clear-num 31.9%

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

   6. metadata-eval 31.9%

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

   7. sub-neg 31.9%

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

   8. un-div-inv 31.9%

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

   9. +-commutative 31.9%

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

   10. associate-*r* 31.9%

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

   11. fma-def 31.9%

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

   12. sub-neg 31.9%

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

   13. metadata-eval 31.9%

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

   14. add-sqr-sqrt 14.6%

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

   15. sqrt-unprod 34.6%

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

5. Applied egg-rr 36.0%

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

6. Taylor expanded in x around inf 34.8%

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

   1. distribute-lft-out 34.8%

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

   2. unpow2 34.8%

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

   3. unpow2 34.8%

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

8. Simplified 34.8%

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

9. Taylor expanded in t around -inf 42.6%

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

10. Final simplification 42.6%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023172 
(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)))))