Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 85.1%

Time: 12.4s

Alternatives: 14

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: 33.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{t\_2 \cdot 2}{t\_m}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, \frac{\mathsf{fma}\left(t\_2, -2, \frac{\frac{t\_2}{x} + \mathsf{fma}\left(2, t\_2, \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{\ell \cdot \ell}{x}\right)\right)}{-x}\right)}{-x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{-1 - x}{1 - x}} \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

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 (* t_m t_m) 2.0 (* l l))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 7.2e-165)
      (/ t_3 (fma (/ 0.5 (* x (sqrt 2.0))) (/ (* t_2 2.0) t_m) t_3))
      (if (<= t_m 2.55e+57)
        (/
         t_3
         (sqrt
          (fma
           (* 2.0 t_m)
           t_m
           (/
            (fma
             t_2
             -2.0
             (/
              (+
               (/ t_2 x)
               (fma 2.0 t_2 (fma (/ (* t_m t_m) x) 2.0 (/ (* l l) x))))
              (- x)))
            (- x)))))
        (/ t_3 (* (sqrt (/ (- -1.0 x) (- 1.0 x))) t_3)))))))

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((t_m * t_m), 2.0, (l * l));
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 7.2e-165) {
		tmp = t_3 / fma((0.5 / (x * sqrt(2.0))), ((t_2 * 2.0) / t_m), t_3);
	} else if (t_m <= 2.55e+57) {
		tmp = t_3 / sqrt(fma((2.0 * t_m), t_m, (fma(t_2, -2.0, (((t_2 / x) + fma(2.0, t_2, fma(((t_m * t_m) / x), 2.0, ((l * l) / x)))) / -x)) / -x)));
	} else {
		tmp = t_3 / (sqrt(((-1.0 - x) / (1.0 - x))) * t_3);
	}
	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(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 7.2e-165)
		tmp = Float64(t_3 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(t_2 * 2.0) / t_m), t_3));
	elseif (t_m <= 2.55e+57)
		tmp = Float64(t_3 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(fma(t_2, -2.0, Float64(Float64(Float64(t_2 / x) + fma(2.0, t_2, fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(l * l) / x)))) / Float64(-x))) / Float64(-x)))));
	else
		tmp = Float64(t_3 / Float64(sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))) * t_3));
	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[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-165], N[(t$95$3 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * 2.0), $MachinePrecision] / t$95$m), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+57], N[(t$95$3 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(t$95$2 * -2.0 + N[(N[(N[(t$95$2 / x), $MachinePrecision] + N[(2.0 * t$95$2 + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.19999999999999969e-165

   1. Initial program 23.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 15.2%

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

   if 7.19999999999999969e-165 < t < 2.55000000000000011e57

   1. Initial program 62.4%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 89.6%

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

   if 2.55000000000000011e57 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{t}, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2 \cdot t, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), -2, \frac{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\ell \cdot \ell}{x}\right)\right)}{-x}\right)}{-x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{-1 - x}{1 - x}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{-1 - x}{1 - x}\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right) \cdot 2}{t\_m}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\_m\right) \cdot t\_2, t\_m, \frac{\mathsf{fma}\left(\ell, \ell, \frac{\mathsf{fma}\left(\ell, \ell, \frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}\right) + \ell \cdot \ell}{x}\right) + \ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2} \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

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 (/ (- -1.0 x) (- 1.0 x))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 7.2e-165)
      (/
       t_3
       (fma
        (/ 0.5 (* x (sqrt 2.0)))
        (/ (* (fma (* t_m t_m) 2.0 (* l l)) 2.0) t_m)
        t_3))
      (if (<= t_m 2.55e+57)
        (/
         t_3
         (sqrt
          (fma
           (* (* 2.0 t_m) t_2)
           t_m
           (/
            (+
             (fma l l (/ (+ (fma l l (/ (fma l l (* l l)) x)) (* l l)) x))
             (* l l))
            x))))
        (/ t_3 (* (sqrt t_2) t_3)))))))

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 = (-1.0 - x) / (1.0 - x);
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 7.2e-165) {
		tmp = t_3 / fma((0.5 / (x * sqrt(2.0))), ((fma((t_m * t_m), 2.0, (l * l)) * 2.0) / t_m), t_3);
	} else if (t_m <= 2.55e+57) {
		tmp = t_3 / sqrt(fma(((2.0 * t_m) * t_2), t_m, ((fma(l, l, ((fma(l, l, (fma(l, l, (l * l)) / x)) + (l * l)) / x)) + (l * l)) / x)));
	} else {
		tmp = t_3 / (sqrt(t_2) * t_3);
	}
	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(Float64(-1.0 - x) / Float64(1.0 - x))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 7.2e-165)
		tmp = Float64(t_3 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) * 2.0) / t_m), t_3));
	elseif (t_m <= 2.55e+57)
		tmp = Float64(t_3 / sqrt(fma(Float64(Float64(2.0 * t_m) * t_2), t_m, Float64(Float64(fma(l, l, Float64(Float64(fma(l, l, Float64(fma(l, l, Float64(l * l)) / x)) + Float64(l * l)) / x)) + Float64(l * l)) / x))));
	else
		tmp = Float64(t_3 / Float64(sqrt(t_2) * t_3));
	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[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-165], N[(t$95$3 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / t$95$m), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+57], N[(t$95$3 / N[Sqrt[N[(N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$m + N[(N[(N[(l * l + N[(N[(N[(l * l + N[(N[(l * l + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.19999999999999969e-165

   1. Initial program 23.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 15.2%

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

   if 7.19999999999999969e-165 < t < 2.55000000000000011e57

   1. Initial program 62.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around -inf

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

   7. Applied rewrites 89.6%

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

   if 2.55000000000000011e57 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{t}, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\right) \cdot \frac{-1 - x}{1 - x}, t, \frac{\mathsf{fma}\left(\ell, \ell, \frac{\mathsf{fma}\left(\ell, \ell, \frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}\right) + \ell \cdot \ell}{x}\right) + \ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{-1 - x}{1 - x}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{-1 - x}{1 - x}\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right) \cdot 2}{t\_m}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\_m\right) \cdot t\_2, t\_m, \frac{\mathsf{fma}\left(\ell, \ell, \frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}\right) + \ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2} \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

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 (/ (- -1.0 x) (- 1.0 x))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 7.2e-165)
      (/
       t_3
       (fma
        (/ 0.5 (* x (sqrt 2.0)))
        (/ (* (fma (* t_m t_m) 2.0 (* l l)) 2.0) t_m)
        t_3))
      (if (<= t_m 2.55e+57)
        (/
         t_3
         (sqrt
          (fma
           (* (* 2.0 t_m) t_2)
           t_m
           (/ (+ (fma l l (/ (fma l l (* l l)) x)) (* l l)) x))))
        (/ t_3 (* (sqrt t_2) t_3)))))))

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 = (-1.0 - x) / (1.0 - x);
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 7.2e-165) {
		tmp = t_3 / fma((0.5 / (x * sqrt(2.0))), ((fma((t_m * t_m), 2.0, (l * l)) * 2.0) / t_m), t_3);
	} else if (t_m <= 2.55e+57) {
		tmp = t_3 / sqrt(fma(((2.0 * t_m) * t_2), t_m, ((fma(l, l, (fma(l, l, (l * l)) / x)) + (l * l)) / x)));
	} else {
		tmp = t_3 / (sqrt(t_2) * t_3);
	}
	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(Float64(-1.0 - x) / Float64(1.0 - x))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 7.2e-165)
		tmp = Float64(t_3 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) * 2.0) / t_m), t_3));
	elseif (t_m <= 2.55e+57)
		tmp = Float64(t_3 / sqrt(fma(Float64(Float64(2.0 * t_m) * t_2), t_m, Float64(Float64(fma(l, l, Float64(fma(l, l, Float64(l * l)) / x)) + Float64(l * l)) / x))));
	else
		tmp = Float64(t_3 / Float64(sqrt(t_2) * t_3));
	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[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-165], N[(t$95$3 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / t$95$m), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+57], N[(t$95$3 / N[Sqrt[N[(N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$m + N[(N[(N[(l * l + N[(N[(l * l + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.19999999999999969e-165

   1. Initial program 23.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 15.2%

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

   if 7.19999999999999969e-165 < t < 2.55000000000000011e57

   1. Initial program 62.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 89.3%

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

   if 2.55000000000000011e57 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{t}, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\right) \cdot \frac{-1 - x}{1 - x}, t, \frac{\mathsf{fma}\left(\ell, \ell, \frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}\right) + \ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{-1 - x}{1 - x}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{-1 - x}{1 - x}\\ t_3 := \left(2 \cdot t\_m\right) \cdot t\_2\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-286}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(t\_3, t\_m, \frac{\left(\ell \cdot \ell\right) \cdot 2}{x}\right)}}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(t\_3, t\_m, \frac{\frac{2}{x} + 2}{x} \cdot \left(\ell \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{\sqrt{t\_2} \cdot t\_4}\\ \end{array} \end{array} \end{array} \]

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 (/ (- -1.0 x) (- 1.0 x)))
        (t_3 (* (* 2.0 t_m) t_2))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.2e-286)
      (/ t_4 (sqrt (fma t_3 t_m (/ (* (* l l) 2.0) x))))
      (if (<= t_m 7.2e-165)
        (/ t_4 (* (sqrt (/ (fma 2.0 x 2.0) (- x 1.0))) t_m))
        (if (<= t_m 2.55e+57)
          (/ t_4 (sqrt (fma t_3 t_m (* (/ (+ (/ 2.0 x) 2.0) x) (* l l)))))
          (/ t_4 (* (sqrt t_2) t_4))))))))

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 = (-1.0 - x) / (1.0 - x);
	double t_3 = (2.0 * t_m) * t_2;
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.2e-286) {
		tmp = t_4 / sqrt(fma(t_3, t_m, (((l * l) * 2.0) / x)));
	} else if (t_m <= 7.2e-165) {
		tmp = t_4 / (sqrt((fma(2.0, x, 2.0) / (x - 1.0))) * t_m);
	} else if (t_m <= 2.55e+57) {
		tmp = t_4 / sqrt(fma(t_3, t_m, ((((2.0 / x) + 2.0) / x) * (l * l))));
	} else {
		tmp = t_4 / (sqrt(t_2) * t_4);
	}
	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(Float64(-1.0 - x) / Float64(1.0 - x))
	t_3 = Float64(Float64(2.0 * t_m) * t_2)
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.2e-286)
		tmp = Float64(t_4 / sqrt(fma(t_3, t_m, Float64(Float64(Float64(l * l) * 2.0) / x))));
	elseif (t_m <= 7.2e-165)
		tmp = Float64(t_4 / Float64(sqrt(Float64(fma(2.0, x, 2.0) / Float64(x - 1.0))) * t_m));
	elseif (t_m <= 2.55e+57)
		tmp = Float64(t_4 / sqrt(fma(t_3, t_m, Float64(Float64(Float64(Float64(2.0 / x) + 2.0) / x) * Float64(l * l)))));
	else
		tmp = Float64(t_4 / Float64(sqrt(t_2) * t_4));
	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[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.2e-286], N[(t$95$4 / N[Sqrt[N[(t$95$3 * t$95$m + N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e-165], N[(t$95$4 / N[(N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+57], N[(t$95$4 / N[Sqrt[N[(t$95$3 * t$95$m + N[(N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$4 / N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.19999999999999997e-286

   1. Initial program 26.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 31.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 48.0%

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

   if 1.19999999999999997e-286 < t < 7.19999999999999969e-165

   1. Initial program 6.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 55.0%

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

   9. Applied rewrites 54.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-/.f64 55.0

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

   11. Applied rewrites 55.0%

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

   if 7.19999999999999969e-165 < t < 2.55000000000000011e57

   1. Initial program 62.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 89.2%

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

   if 2.55000000000000011e57 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-286}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\right) \cdot \frac{-1 - x}{1 - x}, t, \frac{\left(\ell \cdot \ell\right) \cdot 2}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\right) \cdot \frac{-1 - x}{1 - x}, t, \frac{\frac{2}{x} + 2}{x} \cdot \left(\ell \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{-1 - x}{1 - x}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{-1 - x}{1 - x}\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right) \cdot 2}{t\_m}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\_m\right) \cdot t\_2, t\_m, \frac{\frac{2}{x} + 2}{x} \cdot \left(\ell \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2} \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

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 (/ (- -1.0 x) (- 1.0 x))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 7.2e-165)
      (/
       t_3
       (fma
        (/ 0.5 (* x (sqrt 2.0)))
        (/ (* (fma (* t_m t_m) 2.0 (* l l)) 2.0) t_m)
        t_3))
      (if (<= t_m 2.55e+57)
        (/
         t_3
         (sqrt
          (fma (* (* 2.0 t_m) t_2) t_m (* (/ (+ (/ 2.0 x) 2.0) x) (* l l)))))
        (/ t_3 (* (sqrt t_2) t_3)))))))

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 = (-1.0 - x) / (1.0 - x);
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 7.2e-165) {
		tmp = t_3 / fma((0.5 / (x * sqrt(2.0))), ((fma((t_m * t_m), 2.0, (l * l)) * 2.0) / t_m), t_3);
	} else if (t_m <= 2.55e+57) {
		tmp = t_3 / sqrt(fma(((2.0 * t_m) * t_2), t_m, ((((2.0 / x) + 2.0) / x) * (l * l))));
	} else {
		tmp = t_3 / (sqrt(t_2) * t_3);
	}
	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(Float64(-1.0 - x) / Float64(1.0 - x))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 7.2e-165)
		tmp = Float64(t_3 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) * 2.0) / t_m), t_3));
	elseif (t_m <= 2.55e+57)
		tmp = Float64(t_3 / sqrt(fma(Float64(Float64(2.0 * t_m) * t_2), t_m, Float64(Float64(Float64(Float64(2.0 / x) + 2.0) / x) * Float64(l * l)))));
	else
		tmp = Float64(t_3 / Float64(sqrt(t_2) * t_3));
	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[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-165], N[(t$95$3 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / t$95$m), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+57], N[(t$95$3 / N[Sqrt[N[(N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$m + N[(N[(N[(N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] / x), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.19999999999999969e-165

   1. Initial program 23.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 15.2%

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

   if 7.19999999999999969e-165 < t < 2.55000000000000011e57

   1. Initial program 62.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 89.3%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 89.2%

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

   if 2.55000000000000011e57 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{t}, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\right) \cdot \frac{-1 - x}{1 - x}, t, \frac{\frac{2}{x} + 2}{x} \cdot \left(\ell \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{-1 - x}{1 - x}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{-1 - x}{1 - x}\\ t_3 := \sqrt{2} \cdot t\_m\\ t_4 := \frac{t\_3}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\_m\right) \cdot t\_2, t\_m, \frac{\left(\ell \cdot \ell\right) \cdot 2}{x}\right)}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-286}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2} \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

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 (/ (- -1.0 x) (- 1.0 x)))
        (t_3 (* (sqrt 2.0) t_m))
        (t_4
         (/ t_3 (sqrt (fma (* (* 2.0 t_m) t_2) t_m (/ (* (* l l) 2.0) x))))))
   (*
    t_s
    (if (<= t_m 1.2e-286)
      t_4
      (if (<= t_m 7.2e-165)
        (/ t_3 (* (sqrt (/ (fma 2.0 x 2.0) (- x 1.0))) t_m))
        (if (<= t_m 2.55e+57) t_4 (/ t_3 (* (sqrt t_2) t_3))))))))

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 = (-1.0 - x) / (1.0 - x);
	double t_3 = sqrt(2.0) * t_m;
	double t_4 = t_3 / sqrt(fma(((2.0 * t_m) * t_2), t_m, (((l * l) * 2.0) / x)));
	double tmp;
	if (t_m <= 1.2e-286) {
		tmp = t_4;
	} else if (t_m <= 7.2e-165) {
		tmp = t_3 / (sqrt((fma(2.0, x, 2.0) / (x - 1.0))) * t_m);
	} else if (t_m <= 2.55e+57) {
		tmp = t_4;
	} else {
		tmp = t_3 / (sqrt(t_2) * t_3);
	}
	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(Float64(-1.0 - x) / Float64(1.0 - x))
	t_3 = Float64(sqrt(2.0) * t_m)
	t_4 = Float64(t_3 / sqrt(fma(Float64(Float64(2.0 * t_m) * t_2), t_m, Float64(Float64(Float64(l * l) * 2.0) / x))))
	tmp = 0.0
	if (t_m <= 1.2e-286)
		tmp = t_4;
	elseif (t_m <= 7.2e-165)
		tmp = Float64(t_3 / Float64(sqrt(Float64(fma(2.0, x, 2.0) / Float64(x - 1.0))) * t_m));
	elseif (t_m <= 2.55e+57)
		tmp = t_4;
	else
		tmp = Float64(t_3 / Float64(sqrt(t_2) * t_3));
	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[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[Sqrt[N[(N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$m + N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.2e-286], t$95$4, If[LessEqual[t$95$m, 7.2e-165], N[(t$95$3 / N[(N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+57], t$95$4, N[(t$95$3 / N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.19999999999999997e-286 or 7.19999999999999969e-165 < t < 2.55000000000000011e57

   1. Initial program 36.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 43.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 59.9%

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

   if 1.19999999999999997e-286 < t < 7.19999999999999969e-165

   1. Initial program 6.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 55.0%

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

   9. Applied rewrites 54.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lower-/.f64 55.0

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

   11. Applied rewrites 55.0%

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

   if 2.55000000000000011e57 < t

   1. Initial program 27.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-286}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\right) \cdot \frac{-1 - x}{1 - x}, t, \frac{\left(\ell \cdot \ell\right) \cdot 2}{x}\right)}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\left(2 \cdot t\right) \cdot \frac{-1 - x}{1 - x}, t, \frac{\left(\ell \cdot \ell\right) \cdot 2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{-1 - x}{1 - x}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.7 \cdot 10^{-287}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\left(\frac{x}{x - 1} - 1\right) - \frac{-1}{x - 1}} \cdot \ell} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{-1 - x}{1 - x}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 4.7e-287)
      (*
       (/ t_m (* (sqrt (- (- (/ x (- x 1.0)) 1.0) (/ -1.0 (- x 1.0)))) l))
       (sqrt 2.0))
      (/ t_2 (* (sqrt (/ (- -1.0 x) (- 1.0 x))) t_2))))))

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 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 4.7e-287) {
		tmp = (t_m / (sqrt((((x / (x - 1.0)) - 1.0) - (-1.0 / (x - 1.0)))) * l)) * sqrt(2.0);
	} else {
		tmp = t_2 / (sqrt(((-1.0 - x) / (1.0 - x))) * t_2);
	}
	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 = sqrt(2.0d0) * t_m
    if (t_m <= 4.7d-287) then
        tmp = (t_m / (sqrt((((x / (x - 1.0d0)) - 1.0d0) - ((-1.0d0) / (x - 1.0d0)))) * l)) * sqrt(2.0d0)
    else
        tmp = t_2 / (sqrt((((-1.0d0) - x) / (1.0d0 - x))) * t_2)
    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 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 4.7e-287) {
		tmp = (t_m / (Math.sqrt((((x / (x - 1.0)) - 1.0) - (-1.0 / (x - 1.0)))) * l)) * Math.sqrt(2.0);
	} else {
		tmp = t_2 / (Math.sqrt(((-1.0 - x) / (1.0 - x))) * t_2);
	}
	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 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 4.7e-287:
		tmp = (t_m / (math.sqrt((((x / (x - 1.0)) - 1.0) - (-1.0 / (x - 1.0)))) * l)) * math.sqrt(2.0)
	else:
		tmp = t_2 / (math.sqrt(((-1.0 - x) / (1.0 - x))) * t_2)
	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(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 4.7e-287)
		tmp = Float64(Float64(t_m / Float64(sqrt(Float64(Float64(Float64(x / Float64(x - 1.0)) - 1.0) - Float64(-1.0 / Float64(x - 1.0)))) * l)) * sqrt(2.0));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))) * t_2));
	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 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 4.7e-287)
		tmp = (t_m / (sqrt((((x / (x - 1.0)) - 1.0) - (-1.0 / (x - 1.0)))) * l)) * sqrt(2.0);
	else
		tmp = t_2 / (sqrt(((-1.0 - x) / (1.0 - x))) * t_2);
	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[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.7e-287], N[(N[(t$95$m / N[(N[Sqrt[N[(N[(N[(x / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] - N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.6999999999999999e-287

   1. Initial program 26.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 1.9

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

   5. Applied rewrites 1.9%

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

   7. Applied rewrites 1.9%

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

   9. Applied rewrites 1.9%

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

   10. Taylor expanded in l around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. associate--l+ N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower--.f64 9.5

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

   12. Applied rewrites 9.5%

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

   if 4.6999999999999999e-287 < t

   1. Initial program 37.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 82.8

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

   5. Applied rewrites 82.8%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-287}:\\ \;\;\;\;\frac{t}{\sqrt{\left(\frac{x}{x - 1} - 1\right) - \frac{-1}{x - 1}} \cdot \ell} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{-1 - x}{1 - x}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \frac{t\_2}{\sqrt{\frac{-1 - x}{1 - x}} \cdot t\_2} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)))
   (* t_s (/ t_2 (* (sqrt (/ (- -1.0 x) (- 1.0 x))) t_2)))))

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 = sqrt(2.0) * t_m;
	return t_s * (t_2 / (sqrt(((-1.0 - x) / (1.0 - x))) * t_2));
}

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
    t_2 = sqrt(2.0d0) * t_m
    code = t_s * (t_2 / (sqrt((((-1.0d0) - x) / (1.0d0 - x))) * t_2))
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 = Math.sqrt(2.0) * t_m;
	return t_s * (t_2 / (Math.sqrt(((-1.0 - x) / (1.0 - x))) * t_2));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = math.sqrt(2.0) * t_m
	return t_s * (t_2 / (math.sqrt(((-1.0 - x) / (1.0 - x))) * t_2))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	return Float64(t_s * Float64(t_2 / Float64(sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))) * t_2)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	t_2 = sqrt(2.0) * t_m;
	tmp = t_s * (t_2 / (sqrt(((-1.0 - x) / (1.0 - x))) * t_2));
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[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * N[(t$95$2 / N[(N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 43.6

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

5. Applied rewrites 43.6%

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

6. Final simplification 43.6%

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

Alternative 9: 76.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t\_m} \end{array} \]

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 2.0) t_m) (* (sqrt (/ (fma 2.0 x 2.0) (- x 1.0))) t_m))))

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(2.0) * t_m) / (sqrt((fma(2.0, x, 2.0) / (x - 1.0))) * t_m));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(fma(2.0, x, 2.0) / Float64(x - 1.0))) * t_m)))
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[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 43.6

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

5. Applied rewrites 43.6%

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

7. Applied rewrites 43.6%

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

9. Applied rewrites 43.5%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*l/ N/A

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

    4. *-commutative N/A

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

    5. lift-*.f64 N/A

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

    6. lower-/.f64 43.6

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

11. Applied rewrites 43.6%

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

Alternative 10: 76.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t\_m} \cdot t\_m\right) \end{array} \]

FPCore

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

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(2.0) / (sqrt((fma(2.0, x, 2.0) / (x - 1.0))) * t_m)) * t_m);
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(sqrt(2.0) / Float64(sqrt(Float64(fma(2.0, x, 2.0) / Float64(x - 1.0))) * t_m)) * t_m))
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[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

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 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 43.6

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

5. Applied rewrites 43.6%

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

7. Applied rewrites 43.6%

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

9. Applied rewrites 43.5%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*l/ N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 43.4

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

11. Applied rewrites 43.4%

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

12. Final simplification 43.4%

    \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x - 1}} \cdot t} \cdot t \]
13. Add Preprocessing

Alternative 11: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\sqrt{\frac{1 - x}{-1 - x}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \end{array} \]

FPCore

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

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(((1.0 - x) / (-1.0 - x))) * sqrt(0.5)) * sqrt(2.0));
}

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(((1.0d0 - x) / ((-1.0d0) - x))) * sqrt(0.5d0)) * sqrt(2.0d0))
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(((1.0 - x) / (-1.0 - x))) * Math.sqrt(0.5)) * Math.sqrt(2.0));
}

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(((1.0 - x) / (-1.0 - x))) * math.sqrt(0.5)) * math.sqrt(2.0))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x))) * sqrt(0.5)) * sqrt(2.0)))
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(((1.0 - x) / (-1.0 - x))) * sqrt(0.5)) * sqrt(2.0));
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[(N[(N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 43.6

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

5. Applied rewrites 43.6%

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

7. Applied rewrites 43.6%

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

9. Applied rewrites 43.5%

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

10. Taylor expanded in l around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-sqrt.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower--.f64 N/A

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

    6. lower-+.f64 43.0

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

12. Applied rewrites 43.0%

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

13. Final simplification 43.0%

    \[\leadsto \left(\sqrt{\frac{1 - x}{-1 - x}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2} \]
14. Add Preprocessing

Alternative 12: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 - x}{-1 - x}}\right) \end{array} \]

FPCore

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

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * ((sqrt(0.5) * sqrt(2.0)) * sqrt(((1.0 - x) / (-1.0 - x))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * ((sqrt(0.5d0) * sqrt(2.0d0)) * sqrt(((1.0d0 - x) / ((-1.0d0) - x))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * ((Math.sqrt(0.5) * Math.sqrt(2.0)) * Math.sqrt(((1.0 - x) / (-1.0 - x))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * ((math.sqrt(0.5) * math.sqrt(2.0)) * math.sqrt(((1.0 - x) / (-1.0 - x))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(sqrt(0.5) * sqrt(2.0)) * sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * ((sqrt(0.5) * sqrt(2.0)) * sqrt(((1.0 - x) / (-1.0 - x))));
end

Wolfram

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

Derivation

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 l around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 43.0

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

5. Applied rewrites 43.0%

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

6. Final simplification 43.0%

   \[\leadsto \left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 - x}{-1 - x}} \]
7. Add Preprocessing

Alternative 13: 76.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{4}{x} + 2} \cdot t\_m} \end{array} \]

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 2.0) t_m) (* (sqrt (+ (/ 4.0 x) 2.0)) t_m))))

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(2.0) * t_m) / (sqrt(((4.0 / x) + 2.0)) * t_m));
}

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(2.0d0) * t_m) / (sqrt(((4.0d0 / x) + 2.0d0)) * t_m))
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(2.0) * t_m) / (Math.sqrt(((4.0 / x) + 2.0)) * t_m));
}

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(2.0) * t_m) / (math.sqrt(((4.0 / x) + 2.0)) * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(Float64(4.0 / x) + 2.0)) * t_m)))
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(2.0) * t_m) / (sqrt(((4.0 / x) + 2.0)) * t_m));
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[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(N[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 43.6

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

5. Applied rewrites 43.6%

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

7. Applied rewrites 43.6%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 + 4 \cdot \frac{1}{x}} \cdot t} \]
9. Step-by-step derivation

10. Applied rewrites 42.9%

    \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{x} + 2} \cdot t} \]
11. Add Preprocessing

Alternative 14: 75.7% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 42.0

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

5. Applied rewrites 42.0%

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

7. Applied rewrites 42.6%

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

Reproduce

herbie shell --seed 2024288 
(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)))))