Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 85.0%

Time: 13.9s

Alternatives: 9

Speedup: 85.0×

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: 33.4% 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: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.9 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \left(\ell + \ell\right)}{\sqrt{2} \cdot \left(t\_m \cdot x\right)}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 1.22 \cdot 10^{+19}:\\ \;\;\;\;\frac{t\_3}{\sqrt{2 \cdot \left(t\_m \cdot t\_m\right) + \frac{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \frac{\ell \cdot \ell}{x}\right) - \mathsf{fma}\left(t\_2, -2, \frac{t\_2}{-x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma 2.0 (* t_m t_m) (* l l))) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 5.9e-159)
      (/ t_3 (fma 0.5 (/ (* l (+ l l)) (* (sqrt 2.0) (* t_m x))) t_3))
      (if (<= t_m 1.22e+19)
        (/
         t_3
         (sqrt
          (+
           (* 2.0 (* t_m t_m))
           (/
            (-
             (fma 2.0 (/ (* t_m t_m) x) (/ (* l l) x))
             (fma t_2 -2.0 (/ t_2 (- x))))
            x))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma(2.0, (t_m * t_m), (l * l));
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 5.9e-159) {
		tmp = t_3 / fma(0.5, ((l * (l + l)) / (sqrt(2.0) * (t_m * x))), t_3);
	} else if (t_m <= 1.22e+19) {
		tmp = t_3 / sqrt(((2.0 * (t_m * t_m)) + ((fma(2.0, ((t_m * t_m) / x), ((l * l) / x)) - fma(t_2, -2.0, (t_2 / -x))) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(2.0, Float64(t_m * t_m), Float64(l * l))
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 5.9e-159)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(l * Float64(l + l)) / Float64(sqrt(2.0) * Float64(t_m * x))), t_3));
	elseif (t_m <= 1.22e+19)
		tmp = Float64(t_3 / sqrt(Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), Float64(Float64(l * l) / x)) - fma(t_2, -2.0, Float64(t_2 / Float64(-x)))) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.9e-159], N[(t$95$3 / N[(0.5 * N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.22e+19], N[(t$95$3 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * -2.0 + N[(t$95$2 / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.9e-159

   1. Initial program 32.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   5. Simplified 14.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 14.9

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(x \cdot \color{blue}{\sqrt{2}}\right)}, t \cdot \sqrt{2}\right)} \]

   8. Simplified 14.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}}, t \cdot \sqrt{2}\right)} \]

   9. Taylor expanded in l around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt1-in N/A

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

       7. metadata-eval N/A

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

       8. cancel-sign-sub-inv N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. cancel-sign-sub-inv N/A

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

       12. metadata-eval N/A

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

       13. *-lft-identity N/A

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

       14. lower-+.f64 N/A

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

       15. *-commutative N/A

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

       16. *-commutative N/A

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

       17. associate-*l* N/A

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

       18. lower-*.f64 N/A

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

       19. lower-sqrt.f64 N/A

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

       20. lower-*.f64 14.9

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

   11. Simplified 14.9%

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

   if 5.9e-159 < t < 1.22e19

   1. Initial program 48.4%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   5. Simplified 78.2%

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

   if 1.22e19 < t

   1. Initial program 35.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 93.8

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

   5. Simplified 93.8%

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 95.3

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

   7. Applied egg-rr 95.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-rgt-identity 95.3

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

   9. Applied egg-rr 95.3%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.9 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \left(\ell + \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+19}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right) - \mathsf{fma}\left(\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), -2, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{-x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \left(\ell + \ell\right)}{\sqrt{2} \cdot \left(t\_m \cdot x\right)}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 1.22 \cdot 10^{+19}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.8e-158)
      (/ t_2 (fma 0.5 (/ (* l (+ l l)) (* (sqrt 2.0) (* t_m x))) t_2))
      (if (<= t_m 1.22e+19)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            (fma 2.0 (/ (* t_m t_m) x) (fma 2.0 (* t_m t_m) (/ (* l l) x)))
            (/ (fma 2.0 (* t_m t_m) (* l l)) x)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.8e-158) {
		tmp = t_2 / fma(0.5, ((l * (l + l)) / (sqrt(2.0) * (t_m * x))), t_2);
	} else if (t_m <= 1.22e+19) {
		tmp = t_m * sqrt((2.0 / (fma(2.0, ((t_m * t_m) / x), fma(2.0, (t_m * t_m), ((l * l) / x))) + (fma(2.0, (t_m * t_m), (l * l)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.8e-158)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(l * Float64(l + l)) / Float64(sqrt(2.0) * Float64(t_m * x))), t_2));
	elseif (t_m <= 1.22e+19)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), fma(2.0, Float64(t_m * t_m), Float64(Float64(l * l) / x))) + Float64(fma(2.0, Float64(t_m * t_m), Float64(l * l)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.8e-158], N[(t$95$2 / N[(0.5 * N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.22e+19], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.79999999999999995e-158

   1. Initial program 32.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   5. Simplified 14.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 14.9

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(x \cdot \color{blue}{\sqrt{2}}\right)}, t \cdot \sqrt{2}\right)} \]

   8. Simplified 14.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}}, t \cdot \sqrt{2}\right)} \]

   9. Taylor expanded in l around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt1-in N/A

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

       7. metadata-eval N/A

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

       8. cancel-sign-sub-inv N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. cancel-sign-sub-inv N/A

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

       12. metadata-eval N/A

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

       13. *-lft-identity N/A

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

       14. lower-+.f64 N/A

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

       15. *-commutative N/A

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

       16. *-commutative N/A

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

       17. associate-*l* N/A

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

       18. lower-*.f64 N/A

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

       19. lower-sqrt.f64 N/A

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

       20. lower-*.f64 14.9

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

   11. Simplified 14.9%

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

   if 1.79999999999999995e-158 < t < 1.22e19

   1. Initial program 48.4%

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

      1. lift-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

   4. Applied egg-rr 48.7%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

   7. Simplified 77.8%

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

   if 1.22e19 < t

   1. Initial program 35.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 93.8

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

   5. Simplified 93.8%

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 95.3

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

   7. Applied egg-rr 95.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-rgt-identity 95.3

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

   9. Applied egg-rr 95.3%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \left(\ell + \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+19}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 3.8e+126)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (/ (* t_m (sqrt 2.0)) (* (* (sqrt 2.0) l) (sqrt (/ 1.0 x)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 3.8e+126) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = (t_m * sqrt(2.0)) / ((sqrt(2.0) * l) * sqrt((1.0 / x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 3.8d+126) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = (t_m * sqrt(2.0d0)) / ((sqrt(2.0d0) * l) * sqrt((1.0d0 / x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 3.8e+126) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = (t_m * Math.sqrt(2.0)) / ((Math.sqrt(2.0) * l) * Math.sqrt((1.0 / x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 3.8e+126:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = (t_m * math.sqrt(2.0)) / ((math.sqrt(2.0) * l) * math.sqrt((1.0 / x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 3.8e+126)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 3.8e+126)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = (t_m * sqrt(2.0)) / ((sqrt(2.0) * l) * sqrt((1.0 / x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 3.8e+126], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.80000000000000017e126

   1. Initial program 39.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 41.8

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

   5. Simplified 41.8%

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 42.4

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

   7. Applied egg-rr 42.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-rgt-identity 42.4

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

   9. Applied egg-rr 42.4%

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

   if 3.80000000000000017e126 < l

   1. Initial program 0.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 22.0

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

   4. Applied egg-rr 22.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 0.7

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

   7. Simplified 0.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 67.9

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

   10. Simplified 67.9%

       \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{0.5} \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 1.2e+131)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (* (sqrt 0.5) (* (/ (sqrt 2.0) l) (sqrt x)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.2e+131) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt(0.5) * ((sqrt(2.0) / l) * sqrt(x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 1.2d+131) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt(0.5d0) * ((sqrt(2.0d0) / l) * sqrt(x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.2e+131) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt(0.5) * ((Math.sqrt(2.0) / l) * Math.sqrt(x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 1.2e+131:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt(0.5) * ((math.sqrt(2.0) / l) * math.sqrt(x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 1.2e+131)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(0.5) * Float64(Float64(sqrt(2.0) / l) * sqrt(x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 1.2e+131)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt(0.5) * ((sqrt(2.0) / l) * sqrt(x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 1.2e+131], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.2e131

   1. Initial program 39.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 41.8

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

   5. Simplified 41.8%

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 42.4

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

   7. Applied egg-rr 42.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-rgt-identity 42.4

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

   9. Applied egg-rr 42.4%

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

   if 1.2e131 < l

   1. Initial program 0.6%

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

      1. lift-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

   4. Applied egg-rr 0.6%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 25.7

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

   7. Simplified 25.7%

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

   8. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 67.4

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

   10. Simplified 67.4%

       \[\leadsto t \cdot \color{blue}{\left(\sqrt{0.5} \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{x}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-213}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell + \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-213)
    (* t_m (sqrt (/ 2.0 (* l (/ (+ l l) x)))))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.2e-213) {
		tmp = t_m * sqrt((2.0 / (l * ((l + l) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.2d-213) then
        tmp = t_m * sqrt((2.0d0 / (l * ((l + l) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.2e-213) {
		tmp = t_m * Math.sqrt((2.0 / (l * ((l + l) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 1.2e-213:
		tmp = t_m * math.sqrt((2.0 / (l * ((l + l) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.2e-213)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(l * Float64(Float64(l + l) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 1.2e-213)
		tmp = t_m * sqrt((2.0 / (l * ((l + l) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.2e-213], N[(t$95$m * N[Sqrt[N[(2.0 / N[(l * N[(N[(l + l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.19999999999999998e-213

   1. Initial program 34.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 39.7

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

   4. Applied egg-rr 39.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 2.7

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

   7. Simplified 2.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-undiv N/A

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

   9. Applied egg-rr 1.8%

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

   10. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. lower-+.f64 21.2

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

   12. Simplified 21.2%

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

   if 1.19999999999999998e-213 < t

   1. Initial program 36.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 80.3

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

   5. Simplified 80.3%

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 81.5

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

   7. Applied egg-rr 81.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-rgt-identity 81.5

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

   9. Applied egg-rr 81.5%

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

Alternative 6: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-214}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2 \cdot x}{\ell \cdot \left(\ell + \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e-214)
    (* t_m (sqrt (/ (* 2.0 x) (* l (+ l l)))))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3e-214) {
		tmp = t_m * sqrt(((2.0 * x) / (l * (l + l))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3d-214) then
        tmp = t_m * sqrt(((2.0d0 * x) / (l * (l + l))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3e-214) {
		tmp = t_m * Math.sqrt(((2.0 * x) / (l * (l + l))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3e-214:
		tmp = t_m * math.sqrt(((2.0 * x) / (l * (l + l))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3e-214)
		tmp = Float64(t_m * sqrt(Float64(Float64(2.0 * x) / Float64(l * Float64(l + l)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3e-214)
		tmp = t_m * sqrt(((2.0 * x) / (l * (l + l))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3e-214], N[(t$95$m * N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] / N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.99999999999999994e-214

   1. Initial program 34.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 39.7

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

   4. Applied egg-rr 39.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 2.7

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

   7. Simplified 2.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-undiv N/A

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

   9. Applied egg-rr 1.8%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. cancel-sign-sub-inv N/A

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

       4. metadata-eval N/A

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

       5. distribute-rgt1-in N/A

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

       6. metadata-eval N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l/ N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. metadata-eval N/A

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

       15. distribute-rgt1-in N/A

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

       16. metadata-eval N/A

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

       17. cancel-sign-sub-inv N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 N/A

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

       20. cancel-sign-sub-inv N/A

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

       21. metadata-eval N/A

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

       22. *-lft-identity N/A

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

       23. lower-+.f64 18.5

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

   12. Simplified 18.5%

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

   if 2.99999999999999994e-214 < t

   1. Initial program 36.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 80.3

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

   5. Simplified 80.3%

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 81.5

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

   7. Applied egg-rr 81.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-rgt-identity 81.5

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

   9. Applied egg-rr 81.5%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-214}:\\ \;\;\;\;t \cdot \sqrt{\frac{2 \cdot x}{\ell \cdot \left(\ell + \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \sqrt{\frac{x + -1}{x + 1}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (sqrt (/ (+ x -1.0) (+ x 1.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * sqrt(((x + -1.0) / (x + 1.0)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * Math.sqrt(((x + -1.0) / (x + 1.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * math.sqrt(((x + -1.0) / (x + 1.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * sqrt(((x + -1.0) / (x + 1.0)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-sqrt.f64 39.6

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

5. Simplified 39.6%

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval 40.2

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

7. Applied egg-rr 40.2%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. *-rgt-identity 40.2

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

9. Applied egg-rr 40.2%

   \[\leadsto \color{blue}{\sqrt{\frac{x + -1}{x + 1}}} \]
10. Add Preprocessing

Alternative 8: 76.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1}{x}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((-1.0d0) / x))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * (1.0 + (-1.0 / x))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * (1.0 + (-1.0 / x));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-sqrt.f64 39.6

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

5. Simplified 39.6%

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval 40.2

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

7. Applied egg-rr 40.2%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
9. Step-by-step derivation

   1. sub-neg N/A

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

   2. lower-+.f64 N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 39.5

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

10. Simplified 39.5%

    \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]
11. Add Preprocessing

Alternative 9: 75.7% accurate, 85.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m) :precision binary64 (* t_s 1.0))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * 1.0;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * 1.0;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * 1.0

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * 1.0;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{2}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{2}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{1}{2}} \]

   4. lower-sqrt.f64 38.6

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{0.5}} \]

5. Simplified 38.6%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{1}{2}}} \]

   2. metadata-eval N/A

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

   3. metadata-eval 39.2

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

7. Applied egg-rr 39.2%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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)))))