Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 81.1%

Time: 13.3s

Alternatives: 8

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: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.06 \cdot 10^{+24}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\mathsf{fma}\left(-\ell, \ell, \left(-\ell\right) \cdot \ell\right)}{-x}\right)\right)}^{-0.5} \cdot \sqrt{2}\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x - -1}{x - 1}} \cdot \sqrt{2}} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-220)
    1.0
    (if (<= t_m 1.06e+24)
      (*
       (*
        (pow (fma (* t_m t_m) 2.0 (/ (fma (- l) l (* (- l) l)) (- x))) -0.5)
        (sqrt 2.0))
       t_m)
      (* (/ 1.0 (* (sqrt (/ (- x -1.0) (- x 1.0))) (sqrt 2.0))) (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) {
	double tmp;
	if (t_m <= 3.3e-220) {
		tmp = 1.0;
	} else if (t_m <= 1.06e+24) {
		tmp = (pow(fma((t_m * t_m), 2.0, (fma(-l, l, (-l * l)) / -x)), -0.5) * sqrt(2.0)) * t_m;
	} else {
		tmp = (1.0 / (sqrt(((x - -1.0) / (x - 1.0))) * sqrt(2.0))) * sqrt(2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.3e-220)
		tmp = 1.0;
	elseif (t_m <= 1.06e+24)
		tmp = Float64(Float64((fma(Float64(t_m * t_m), 2.0, Float64(fma(Float64(-l), l, Float64(Float64(-l) * l)) / Float64(-x))) ^ -0.5) * sqrt(2.0)) * t_m);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * sqrt(2.0))) * sqrt(2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e-220], 1.0, If[LessEqual[t$95$m, 1.06e+24], N[(N[(N[Power[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[((-l) * l + N[((-l) * l), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.29999999999999999e-220

   1. Initial program 27.0%

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

   3. Taylor expanded in 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 9.7

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

   5. Applied rewrites 9.7%

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

   7. Applied rewrites 9.8%

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

   if 3.29999999999999999e-220 < t < 1.06e24

   1. Initial program 47.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing
   3. 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 52.4%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 73.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 73.6%

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

   12. Applied rewrites 73.8%

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

   if 1.06e24 < t

   1. Initial program 35.1%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.1%

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. metadata-eval N/A

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

       7. sub-neg N/A

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

       8. lower--.f64 N/A

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

       9. lower--.f64 N/A

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

       10. lower-sqrt.f64 97.5

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

   12. Applied rewrites 97.5%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+24}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(t \cdot t, 2, \frac{\mathsf{fma}\left(-\ell, \ell, \left(-\ell\right) \cdot \ell\right)}{-x}\right)\right)}^{-0.5} \cdot \sqrt{2}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x - -1}{x - 1}} \cdot \sqrt{2}} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.06 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\mathsf{fma}\left(-\ell, \ell, \left(-\ell\right) \cdot \ell\right)}{-x}\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x - -1}{x - 1}} \cdot \sqrt{2}} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-220)
    1.0
    (if (<= t_m 1.06e+24)
      (*
       (/
        (sqrt 2.0)
        (sqrt (fma (* t_m t_m) 2.0 (/ (fma (- l) l (* (- l) l)) (- x)))))
       t_m)
      (* (/ 1.0 (* (sqrt (/ (- x -1.0) (- x 1.0))) (sqrt 2.0))) (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) {
	double tmp;
	if (t_m <= 3.3e-220) {
		tmp = 1.0;
	} else if (t_m <= 1.06e+24) {
		tmp = (sqrt(2.0) / sqrt(fma((t_m * t_m), 2.0, (fma(-l, l, (-l * l)) / -x)))) * t_m;
	} else {
		tmp = (1.0 / (sqrt(((x - -1.0) / (x - 1.0))) * sqrt(2.0))) * sqrt(2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.3e-220)
		tmp = 1.0;
	elseif (t_m <= 1.06e+24)
		tmp = Float64(Float64(sqrt(2.0) / sqrt(fma(Float64(t_m * t_m), 2.0, Float64(fma(Float64(-l), l, Float64(Float64(-l) * l)) / Float64(-x))))) * t_m);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * sqrt(2.0))) * sqrt(2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e-220], 1.0, If[LessEqual[t$95$m, 1.06e+24], N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[((-l) * l + N[((-l) * l), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.29999999999999999e-220

   1. Initial program 27.0%

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

   3. Taylor expanded in 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 9.7

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

   5. Applied rewrites 9.7%

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

   7. Applied rewrites 9.8%

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

   if 3.29999999999999999e-220 < t < 1.06e24

   1. Initial program 47.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing
   3. 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 52.4%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 73.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 73.6%

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

   12. Applied rewrites 73.8%

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

   if 1.06e24 < t

   1. Initial program 35.1%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.1%

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. metadata-eval N/A

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

       7. sub-neg N/A

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

       8. lower--.f64 N/A

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

       9. lower--.f64 N/A

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

       10. lower-sqrt.f64 97.5

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

   12. Applied rewrites 97.5%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, \frac{\mathsf{fma}\left(-\ell, \ell, \left(-\ell\right) \cdot \ell\right)}{-x}\right)}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x - -1}{x - 1}} \cdot \sqrt{2}} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-220}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.06 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\ell \cdot \ell}{x} \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x - -1}{x - 1}} \cdot \sqrt{2}} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-220)
    1.0
    (if (<= t_m 1.06e+24)
      (/ (* (sqrt 2.0) t_m) (sqrt (fma (* t_m t_m) 2.0 (* (/ (* l l) x) 2.0))))
      (* (/ 1.0 (* (sqrt (/ (- x -1.0) (- x 1.0))) (sqrt 2.0))) (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) {
	double tmp;
	if (t_m <= 3.3e-220) {
		tmp = 1.0;
	} else if (t_m <= 1.06e+24) {
		tmp = (sqrt(2.0) * t_m) / sqrt(fma((t_m * t_m), 2.0, (((l * l) / x) * 2.0)));
	} else {
		tmp = (1.0 / (sqrt(((x - -1.0) / (x - 1.0))) * sqrt(2.0))) * sqrt(2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.3e-220)
		tmp = 1.0;
	elseif (t_m <= 1.06e+24)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(fma(Float64(t_m * t_m), 2.0, Float64(Float64(Float64(l * l) / x) * 2.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * sqrt(2.0))) * sqrt(2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e-220], 1.0, If[LessEqual[t$95$m, 1.06e+24], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.29999999999999999e-220

   1. Initial program 27.0%

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

   3. Taylor expanded in 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 9.7

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

   5. Applied rewrites 9.7%

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

   7. Applied rewrites 9.8%

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

   if 3.29999999999999999e-220 < t < 1.06e24

   1. Initial program 47.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing
   3. 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 52.4%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 73.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 73.6%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 73.6%

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

   if 1.06e24 < t

   1. Initial program 35.1%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.1%

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. metadata-eval N/A

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

       7. sub-neg N/A

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

       8. lower--.f64 N/A

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

       9. lower--.f64 N/A

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

       10. lower-sqrt.f64 97.5

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

   12. Applied rewrites 97.5%

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

Alternative 4: 77.7% accurate, 1.2× speedup

Math

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

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 (* (/ 1.0 (* (sqrt (/ (- x -1.0) (- x 1.0))) (sqrt 2.0))) (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 * ((1.0 / (sqrt(((x - -1.0) / (x - 1.0))) * sqrt(2.0))) * 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 * ((1.0d0 / (sqrt(((x - (-1.0d0)) / (x - 1.0d0))) * sqrt(2.0d0))) * 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 * ((1.0 / (Math.sqrt(((x - -1.0) / (x - 1.0))) * Math.sqrt(2.0))) * 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 * ((1.0 / (math.sqrt(((x - -1.0) / (x - 1.0))) * math.sqrt(2.0))) * 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(1.0 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * sqrt(2.0))) * 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 * ((1.0 / (sqrt(((x - -1.0) / (x - 1.0))) * sqrt(2.0))) * 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[(1.0 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.9%

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

3. Taylor expanded in t around inf

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
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 42.9

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

5. Applied rewrites 42.9%

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

7. Applied rewrites 42.8%

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

9. Applied rewrites 42.8%

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

10. Taylor expanded in t around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-sqrt.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. +-commutative N/A

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

    6. metadata-eval N/A

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

    7. sub-neg N/A

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

    8. lower--.f64 N/A

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

    9. lower--.f64 N/A

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

    10. lower-sqrt.f64 42.9

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

12. Applied rewrites 42.9%

    \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{x - -1}{x - 1}} \cdot \sqrt{2}}} \cdot \sqrt{2} \]
13. Add Preprocessing

Alternative 5: 77.6% accurate, 1.3× speedup

Math

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

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
  (* (/ t_m (* (sqrt (* (/ (- -1.0 x) (- 1.0 x)) 2.0)) t_m)) (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 * ((t_m / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * 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 * ((t_m / (sqrt(((((-1.0d0) - x) / (1.0d0 - x)) * 2.0d0)) * t_m)) * 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 * ((t_m / (Math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * 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 * ((t_m / (math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * 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(t_m / Float64(sqrt(Float64(Float64(Float64(-1.0 - x) / Float64(1.0 - x)) * 2.0)) * t_m)) * 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 * ((t_m / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * 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[(t$95$m / N[(N[Sqrt[N[(N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.9%

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

3. Taylor expanded in t around inf

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
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 42.9

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

5. Applied rewrites 42.9%

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

7. Applied rewrites 42.8%

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

8. Final simplification 42.8%

   \[\leadsto \frac{t}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t} \cdot \sqrt{2} \]
9. Add Preprocessing

Alternative 6: 77.4% accurate, 1.3× speedup

Math

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

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 (* (/ (- -1.0 x) (- 1.0 x)) 2.0)) 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((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * 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) / (sqrt(((((-1.0d0) - x) / (1.0d0 - x)) * 2.0d0)) * t_m)) * 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) / (Math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * t_m)) * 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) / (math.sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * 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(Float64(Float64(-1.0 - x) / Float64(1.0 - x)) * 2.0)) * t_m)) * 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) / (sqrt((((-1.0 - x) / (1.0 - x)) * 2.0)) * 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[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.9%

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

3. Taylor expanded in t around inf

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
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 42.9

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

5. Applied rewrites 42.9%

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

7. Applied rewrites 42.7%

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

8. Final simplification 42.7%

   \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{-1 - x}{1 - x} \cdot 2} \cdot t} \cdot t \]
9. Add Preprocessing

Alternative 7: 77.3% accurate, 1.4× speedup

Math

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

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 32.9%

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

3. Taylor expanded in t around inf

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
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 42.9

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

5. Applied rewrites 42.9%

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

7. Applied rewrites 42.8%

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

9. Applied rewrites 42.8%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-/.f64 N/A

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

    4. un-div-inv N/A

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

    5. lift-/.f64 N/A

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

11. Applied rewrites 42.7%

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

Alternative 8: 76.5% 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 32.9%

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

3. Taylor expanded in 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 41.7

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

5. Applied rewrites 41.7%

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

7. Applied rewrites 42.4%

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

Reproduce

herbie shell --seed 2024249 
(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)))))