Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 85.3%

Time: 16.2s

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: 33.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-157}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_3 \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+80}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \frac{\ell \cdot \ell}{x}\right) + \left(\frac{t\_2}{x} - t\_2 \cdot -2\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 2.45e-157)
      (/ t_3 (fma 0.5 (/ (* 2.0 (* l l)) (* t_3 x)) t_3))
      (if (<= t_m 6e+80)
        (/
         t_3
         (sqrt
          (fma
           2.0
           (* t_m t_m)
           (/
            (+
             (fma 2.0 (/ (* t_m t_m) x) (/ (* l l) x))
             (- (/ t_2 x) (* t_2 -2.0)))
            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 <= 2.45e-157) {
		tmp = t_3 / fma(0.5, ((2.0 * (l * l)) / (t_3 * x)), t_3);
	} else if (t_m <= 6e+80) {
		tmp = t_3 / sqrt(fma(2.0, (t_m * t_m), ((fma(2.0, ((t_m * t_m) / x), ((l * l) / x)) + ((t_2 / x) - (t_2 * -2.0))) / 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 <= 2.45e-157)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * Float64(l * l)) / Float64(t_3 * x)), t_3));
	elseif (t_m <= 6e+80)
		tmp = Float64(t_3 / sqrt(fma(2.0, Float64(t_m * t_m), Float64(Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), Float64(Float64(l * l) / x)) + Float64(Float64(t_2 / x) - Float64(t_2 * -2.0))) / 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, 2.45e-157], N[(t$95$3 / N[(0.5 * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * x), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+80], N[(t$95$3 / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m), $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[(N[(t$95$2 / x), $MachinePrecision] - N[(t$95$2 * -2.0), $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 < 2.4499999999999999e-157

   1. Initial program 28.1%

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

   3. Taylor expanded in 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. Applied rewrites 17.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 18.1%

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

   if 2.4499999999999999e-157 < t < 5.99999999999999974e80

   1. Initial program 56.1%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

   5. Applied rewrites 86.1%

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

   if 5.99999999999999974e80 < t

   1. Initial program 24.8%

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

   3. Taylor expanded in 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 95.7

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

   5. Applied rewrites 95.7%

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 98.5

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

   10. Applied rewrites 98.5%

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

   12. Applied rewrites 100.0%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-157}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+80}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} - \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot -2\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_2 \cdot x}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+80}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, t\_m \cdot \left(t\_m + \frac{t\_m}{x}\right), \frac{1}{x} \cdot \mathsf{fma}\left(\ell, \ell, \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 7.5e-155)
      (/ t_2 (fma 0.5 (/ (* 2.0 (* l l)) (* t_2 x)) t_2))
      (if (<= t_m 6e+80)
        (*
         t_m
         (sqrt
          (/
           2.0
           (fma
            2.0
            (* t_m (+ t_m (/ t_m x)))
            (* (/ 1.0 x) (fma l l (fma 2.0 (* t_m t_m) (* 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 t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 7.5e-155) {
		tmp = t_2 / fma(0.5, ((2.0 * (l * l)) / (t_2 * x)), t_2);
	} else if (t_m <= 6e+80) {
		tmp = t_m * sqrt((2.0 / fma(2.0, (t_m * (t_m + (t_m / x))), ((1.0 / x) * fma(l, l, fma(2.0, (t_m * t_m), (l * l)))))));
	} 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 <= 7.5e-155)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(2.0 * Float64(l * l)) / Float64(t_2 * x)), t_2));
	elseif (t_m <= 6e+80)
		tmp = Float64(t_m * sqrt(Float64(2.0 / fma(2.0, Float64(t_m * Float64(t_m + Float64(t_m / x))), Float64(Float64(1.0 / x) * fma(l, l, fma(2.0, Float64(t_m * t_m), Float64(l * l))))))));
	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, 7.5e-155], N[(t$95$2 / N[(0.5 * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+80], N[(t$95$m * N[Sqrt[N[(2.0 / N[(2.0 * N[(t$95$m * N[(t$95$m + N[(t$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(l * l + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 7.5000000000000006e-155

   1. Initial program 28.1%

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

   3. Taylor expanded in 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. Applied rewrites 17.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 18.1%

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

   if 7.5000000000000006e-155 < t < 5.99999999999999974e80

   1. Initial program 56.1%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Applied rewrites 86.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 86.2%

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

   if 5.99999999999999974e80 < t

   1. Initial program 24.8%

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

   3. Taylor expanded in 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 95.7

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

   5. Applied rewrites 95.7%

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 98.5

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

   10. Applied rewrites 98.5%

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

   12. Applied rewrites 100.0%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+80}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \frac{1}{x} \cdot \mathsf{fma}\left(\ell, \ell, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-157}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_2 \cdot x}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+19}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(2, x \cdot \left(t\_m \cdot t\_m\right), \mathsf{fma}\left(2, \ell \cdot \ell, \left(t\_m \cdot t\_m\right) \cdot 4\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 2.45e-157)
      (/ t_2 (fma 0.5 (/ (* 2.0 (* l l)) (* t_2 x)) t_2))
      (if (<= t_m 1.25e+19)
        (/
         t_2
         (sqrt
          (/
           (fma 2.0 (* x (* t_m t_m)) (fma 2.0 (* l l) (* (* t_m t_m) 4.0)))
           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 <= 2.45e-157) {
		tmp = t_2 / fma(0.5, ((2.0 * (l * l)) / (t_2 * x)), t_2);
	} else if (t_m <= 1.25e+19) {
		tmp = t_2 / sqrt((fma(2.0, (x * (t_m * t_m)), fma(2.0, (l * l), ((t_m * t_m) * 4.0))) / 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 <= 2.45e-157)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(2.0 * Float64(l * l)) / Float64(t_2 * x)), t_2));
	elseif (t_m <= 1.25e+19)
		tmp = Float64(t_2 / sqrt(Float64(fma(2.0, Float64(x * Float64(t_m * t_m)), fma(2.0, Float64(l * l), Float64(Float64(t_m * t_m) * 4.0))) / 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, 2.45e-157], N[(t$95$2 / N[(0.5 * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+19], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * N[(x * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * l), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 2.4499999999999999e-157

   1. Initial program 28.1%

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

   3. Taylor expanded in 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. Applied rewrites 17.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 18.1%

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

   if 2.4499999999999999e-157 < t < 1.25e19

   1. Initial program 54.4%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(2, \left(t \cdot t\right) \cdot x, \mathsf{fma}\left(2, \ell \cdot \ell, \left(t \cdot t\right) \cdot 4\right)\right)}{\color{blue}{x}}}} \]

   if 1.25e19 < t

   1. Initial program 32.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 91.2

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

   10. Applied rewrites 91.2%

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

   12. Applied rewrites 92.6%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-157}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+19}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(2, x \cdot \left(t \cdot t\right), \mathsf{fma}\left(2, \ell \cdot \ell, \left(t \cdot t\right) \cdot 4\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+225}:\\ \;\;\;\;\frac{t\_2}{t\_2 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \end{array} \end{array} \end{array} \]

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 (<= l 5.2e+225)
      (/ t_2 (* t_2 (sqrt (/ (+ x 1.0) (+ x -1.0)))))
      (* (sqrt 2.0) (/ t_m (* l (sqrt (/ (+ 2.0 (/ 2.0 x)) 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 t_2 = t_m * sqrt(2.0);
	double tmp;
	if (l <= 5.2e+225) {
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	} else {
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((2.0 + (2.0 / x)) / 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) :: t_2
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    if (l <= 5.2d+225) then
        tmp = t_2 / (t_2 * sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    else
        tmp = sqrt(2.0d0) * (t_m / (l * sqrt(((2.0d0 + (2.0d0 / x)) / 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 t_2 = t_m * Math.sqrt(2.0);
	double tmp;
	if (l <= 5.2e+225) {
		tmp = t_2 / (t_2 * Math.sqrt(((x + 1.0) / (x + -1.0))));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l * Math.sqrt(((2.0 + (2.0 / x)) / 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):
	t_2 = t_m * math.sqrt(2.0)
	tmp = 0
	if l <= 5.2e+225:
		tmp = t_2 / (t_2 * math.sqrt(((x + 1.0) / (x + -1.0))))
	else:
		tmp = math.sqrt(2.0) * (t_m / (l * math.sqrt(((2.0 + (2.0 / x)) / x))))
	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 (l <= 5.2e+225)
		tmp = Float64(t_2 / Float64(t_2 * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / 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)
	t_2 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (l <= 5.2e+225)
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	else
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((2.0 + (2.0 / x)) / 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_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 5.2e+225], N[(t$95$2 / N[(t$95$2 * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.20000000000000009e225

   1. Initial program 35.3%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 42.8

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

   5. Applied rewrites 42.8%

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

   if 5.20000000000000009e225 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 16.5

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 23.2

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 65.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 65.4%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+225}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \end{array} \end{array} \]

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 5.2e+225)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* (sqrt 2.0) (/ t_m (* l (sqrt (/ (+ 2.0 (/ 2.0 x)) 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 <= 5.2e+225) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((2.0 + (2.0 / x)) / 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 <= 5.2d+225) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = sqrt(2.0d0) * (t_m / (l * sqrt(((2.0d0 + (2.0d0 / x)) / 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 <= 5.2e+225) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l * Math.sqrt(((2.0 + (2.0 / x)) / 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 <= 5.2e+225:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = math.sqrt(2.0) * (t_m / (l * math.sqrt(((2.0 + (2.0 / x)) / 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 <= 5.2e+225)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / 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 <= 5.2e+225)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((2.0 + (2.0 / x)) / 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, 5.2e+225], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.20000000000000009e225

   1. Initial program 35.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 41.1

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

   5. Applied rewrites 41.1%

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 42.1

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

   10. Applied rewrites 42.1%

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

   12. Applied rewrites 42.8%

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

   if 5.20000000000000009e225 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 16.5

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 23.2

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 65.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 65.4%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \end{array} \end{array} \]

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 5.2e+225)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (/ (sqrt 2.0) (* l (sqrt (/ (+ 2.0 (/ 2.0 x)) 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 <= 5.2e+225) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt(2.0) / (l * sqrt(((2.0 + (2.0 / x)) / 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 <= 5.2d+225) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt(2.0d0) / (l * sqrt(((2.0d0 + (2.0d0 / x)) / 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 <= 5.2e+225) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 + (2.0 / x)) / 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 <= 5.2e+225:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt(2.0) / (l * math.sqrt(((2.0 + (2.0 / x)) / 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 <= 5.2e+225)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / 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 <= 5.2e+225)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt(2.0) / (l * sqrt(((2.0 + (2.0 / x)) / 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, 5.2e+225], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.20000000000000009e225

   1. Initial program 35.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 41.1

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

   5. Applied rewrites 41.1%

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 42.1

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

   10. Applied rewrites 42.1%

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

   12. Applied rewrites 42.8%

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

   if 5.20000000000000009e225 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 16.5

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 23.2

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 65.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 65.3

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

   13. Applied rewrites 65.3%

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

Alternative 7: 78.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \end{array} \end{array} \]

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 5.2e+225)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (/ (* t_m (sqrt 2.0)) (* l (sqrt (/ 2.0 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 <= 5.2e+225) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / 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 <= 5.2d+225) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = (t_m * sqrt(2.0d0)) / (l * sqrt((2.0d0 / 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 <= 5.2e+225) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = (t_m * Math.sqrt(2.0)) / (l * Math.sqrt((2.0 / 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 <= 5.2e+225:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = (t_m * math.sqrt(2.0)) / (l * math.sqrt((2.0 / 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 <= 5.2e+225)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l * sqrt(Float64(2.0 / 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 <= 5.2e+225)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / 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, 5.2e+225], 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[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.20000000000000009e225

   1. Initial program 35.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 41.1

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

   5. Applied rewrites 41.1%

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 42.1

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

   10. Applied rewrites 42.1%

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

   12. Applied rewrites 42.8%

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

   if 5.20000000000000009e225 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 16.5

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 23.2

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.9%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% 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 33.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 39.8

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

5. Applied rewrites 39.8%

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

7. Applied rewrites 40.4%

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

8. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 40.7

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

10. Applied rewrites 40.7%

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

12. Applied rewrites 41.3%

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

Alternative 9: 75.6% 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 33.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 39.8

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

5. Applied rewrites 39.8%

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

7. Applied rewrites 40.4%

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

Reproduce

herbie shell --seed 2024219 
(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)))))