Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 84.0%

Time: 11.4s

Alternatives: 12

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: 33.3% 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: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{x}\\ t_3 := \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)}{x}\\ t_4 := \frac{\ell \cdot \ell}{x}\\ t_5 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-238}:\\ \;\;\;\;\frac{t\_5}{\sqrt{\mathsf{fma}\left(2, t\_2 + t\_m \cdot t\_m, t\_3 + t\_4\right)}}\\ \mathbf{elif}\;t\_m \leq 2.4 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_5}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t\_m} \cdot 2, t\_5\right)}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(t\_2, 2, \mathsf{fma}\left(t\_m \cdot t\_m, 2, t\_4\right)\right) + t\_3}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_5}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (/ (* t_m t_m) x))
        (t_3 (/ (fma (* t_m t_m) 2.0 (* l l)) x))
        (t_4 (/ (* l l) x))
        (t_5 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 9e-238)
      (/ t_5 (sqrt (fma 2.0 (+ t_2 (* t_m t_m)) (+ t_3 t_4))))
      (if (<= t_m 2.4e-161)
        (/ t_5 (fma (/ 0.5 (* x (sqrt 2.0))) (* (/ (* l l) t_m) 2.0) t_5))
        (if (<= t_m 5e+66)
          (*
           (sqrt (/ 2.0 (+ (fma t_2 2.0 (fma (* t_m t_m) 2.0 t_4)) t_3)))
           t_m)
          (/ t_5 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_5))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = (t_m * t_m) / x;
	double t_3 = fma((t_m * t_m), 2.0, (l * l)) / x;
	double t_4 = (l * l) / x;
	double t_5 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 9e-238) {
		tmp = t_5 / sqrt(fma(2.0, (t_2 + (t_m * t_m)), (t_3 + t_4)));
	} else if (t_m <= 2.4e-161) {
		tmp = t_5 / fma((0.5 / (x * sqrt(2.0))), (((l * l) / t_m) * 2.0), t_5);
	} else if (t_m <= 5e+66) {
		tmp = sqrt((2.0 / (fma(t_2, 2.0, fma((t_m * t_m), 2.0, t_4)) + t_3))) * t_m;
	} else {
		tmp = t_5 / (sqrt(((x - -1.0) / (x - 1.0))) * t_5);
	}
	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(t_m * t_m) / x)
	t_3 = Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) / x)
	t_4 = Float64(Float64(l * l) / x)
	t_5 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 9e-238)
		tmp = Float64(t_5 / sqrt(fma(2.0, Float64(t_2 + Float64(t_m * t_m)), Float64(t_3 + t_4))));
	elseif (t_m <= 2.4e-161)
		tmp = Float64(t_5 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(Float64(l * l) / t_m) * 2.0), t_5));
	elseif (t_m <= 5e+66)
		tmp = Float64(sqrt(Float64(2.0 / Float64(fma(t_2, 2.0, fma(Float64(t_m * t_m), 2.0, t_4)) + t_3))) * t_m);
	else
		tmp = Float64(t_5 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_5));
	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] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-238], N[(t$95$5 / N[Sqrt[N[(2.0 * N[(t$95$2 + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e-161], N[(t$95$5 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+66], N[(N[Sqrt[N[(2.0 / N[(N[(t$95$2 * 2.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$5 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 8.99999999999999992e-238

   1. Initial program 29.1%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

   5. Applied rewrites 56.4%

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

   if 8.99999999999999992e-238 < t < 2.39999999999999999e-161

   1. Initial program 2.6%

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

   3. 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 78.0%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 78.0%

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

   if 2.39999999999999999e-161 < t < 4.99999999999999991e66

   1. Initial program 47.2%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 62.9

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

   5. Applied rewrites 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

   10. Applied rewrites 89.5%

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

   if 4.99999999999999991e66 < t

   1. Initial program 19.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 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 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-238}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x} + t \cdot t, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \mathsf{fma}\left(t \cdot t, 2, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)}{x}}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 2.4e-161)
      (/ t_2 (fma (/ 0.5 (* x (sqrt 2.0))) (* (/ (* l l) t_m) 2.0) t_2))
      (if (<= t_m 5e+66)
        (*
         (sqrt
          (/
           2.0
           (+
            (fma (/ (* t_m t_m) x) 2.0 (fma (* t_m t_m) 2.0 (/ (* l l) x)))
            (/ (fma (* t_m t_m) 2.0 (* l l)) x))))
         t_m)
        (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) 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 <= 2.4e-161) {
		tmp = t_2 / fma((0.5 / (x * sqrt(2.0))), (((l * l) / t_m) * 2.0), t_2);
	} else if (t_m <= 5e+66) {
		tmp = sqrt((2.0 / (fma(((t_m * t_m) / x), 2.0, fma((t_m * t_m), 2.0, ((l * l) / x))) + (fma((t_m * t_m), 2.0, (l * l)) / x)))) * t_m;
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * 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 <= 2.4e-161)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(Float64(l * l) / t_m) * 2.0), t_2));
	elseif (t_m <= 5e+66)
		tmp = Float64(sqrt(Float64(2.0 / Float64(fma(Float64(Float64(t_m * t_m) / x), 2.0, fma(Float64(t_m * t_m), 2.0, Float64(Float64(l * l) / x))) + Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) / x)))) * t_m);
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	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[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.4e-161], N[(t$95$2 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+66], N[(N[Sqrt[N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.39999999999999999e-161

   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 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 20.8%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 20.5%

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

   if 2.39999999999999999e-161 < t < 4.99999999999999991e66

   1. Initial program 47.2%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 62.9

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

   5. Applied rewrites 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

   10. Applied rewrites 89.5%

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

   if 4.99999999999999991e66 < t

   1. Initial program 19.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 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 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \mathsf{fma}\left(t \cdot t, 2, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 1.5e-91)
      (/ t_2 (fma (/ 0.5 (* x (sqrt 2.0))) (* (/ (* l l) t_m) 2.0) t_2))
      (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) 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 <= 1.5e-91) {
		tmp = t_2 / fma((0.5 / (x * sqrt(2.0))), (((l * l) / t_m) * 2.0), t_2);
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * 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 <= 1.5e-91)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(x * sqrt(2.0))), Float64(Float64(Float64(l * l) / t_m) * 2.0), t_2));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	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[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e-91], N[(t$95$2 / N[(N[(0.5 / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5000000000000001e-91

   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

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{1}{2} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}} \]
   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 24.1%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 23.7%

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

   if 1.5000000000000001e-91 < t

   1. Initial program 31.2%

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

   3. 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 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{x \cdot \sqrt{2}}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x - -1}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{x}{x - 1} - 1\right) + \frac{1}{x - 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e-218)
    (*
     (/ (* (sqrt 2.0) t_m) l)
     (sqrt (/ 1.0 (+ (- (/ x (- x 1.0)) 1.0) (/ 1.0 (- x 1.0))))))
    (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) {
	double tmp;
	if (t_m <= 4e-218) {
		tmp = ((sqrt(2.0) * t_m) / l) * sqrt((1.0 / (((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0)))));
	} else {
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 4d-218) then
        tmp = ((sqrt(2.0d0) * t_m) / l) * sqrt((1.0d0 / (((x / (x - 1.0d0)) - 1.0d0) + (1.0d0 / (x - 1.0d0)))))
    else
        tmp = sqrt(((1.0d0 - x) / ((-1.0d0) - x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 4e-218) {
		tmp = ((Math.sqrt(2.0) * t_m) / l) * Math.sqrt((1.0 / (((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0)))));
	} else {
		tmp = Math.sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 4e-218:
		tmp = ((math.sqrt(2.0) * t_m) / l) * math.sqrt((1.0 / (((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0)))))
	else:
		tmp = math.sqrt(((1.0 - x) / (-1.0 - x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 4e-218)
		tmp = Float64(Float64(Float64(sqrt(2.0) * t_m) / l) * sqrt(Float64(1.0 / Float64(Float64(Float64(x / Float64(x - 1.0)) - 1.0) + Float64(1.0 / Float64(x - 1.0))))));
	else
		tmp = sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 4e-218)
		tmp = ((sqrt(2.0) * t_m) / l) * sqrt((1.0 / (((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0)))));
	else
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-218], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(N[(x / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000001e-218

   1. Initial program 28.8%

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

   3. Taylor expanded in 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 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 13.6

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

   8. Applied rewrites 13.6%

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

   if 4.0000000000000001e-218 < t

   1. Initial program 27.6%

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

   3. 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 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 80.5%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{x}{x - 1} - 1\right) + \frac{1}{x - 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\left(\frac{x}{x - 1} - 1\right) + \frac{1}{x - 1}} \cdot \ell} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e-218)
    (*
     (/ (sqrt 2.0) (* (sqrt (+ (- (/ x (- x 1.0)) 1.0) (/ 1.0 (- x 1.0)))) l))
     t_m)
    (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) {
	double tmp;
	if (t_m <= 4e-218) {
		tmp = (sqrt(2.0) / (sqrt((((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0)))) * l)) * t_m;
	} else {
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 4d-218) then
        tmp = (sqrt(2.0d0) / (sqrt((((x / (x - 1.0d0)) - 1.0d0) + (1.0d0 / (x - 1.0d0)))) * l)) * t_m
    else
        tmp = sqrt(((1.0d0 - x) / ((-1.0d0) - x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 4e-218) {
		tmp = (Math.sqrt(2.0) / (Math.sqrt((((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0)))) * l)) * t_m;
	} else {
		tmp = Math.sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 4e-218:
		tmp = (math.sqrt(2.0) / (math.sqrt((((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0)))) * l)) * t_m
	else:
		tmp = math.sqrt(((1.0 - x) / (-1.0 - x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 4e-218)
		tmp = Float64(Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(Float64(x / Float64(x - 1.0)) - 1.0) + Float64(1.0 / Float64(x - 1.0)))) * l)) * t_m);
	else
		tmp = sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 4e-218)
		tmp = (sqrt(2.0) / (sqrt((((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0)))) * l)) * t_m;
	else
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-218], N[(N[(N[Sqrt[2.0], $MachinePrecision] / 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] * t$95$m), $MachinePrecision], N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000001e-218

   1. Initial program 28.8%

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

   3. Taylor expanded in 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 4.6

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

   5. Applied rewrites 4.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 4.6

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 13.5

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

   10. Applied rewrites 13.5%

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

   if 4.0000000000000001e-218 < t

   1. Initial program 27.6%

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

   3. 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 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 80.5%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\left(\frac{x}{x - 1} - 1\right) + \frac{1}{x - 1}} \cdot \ell} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\frac{2}{\left(\left(\frac{x}{x - 1} - 1\right) + \frac{1}{x - 1}\right) \cdot \left(\ell \cdot \ell\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-218)
    (*
     (sqrt (/ 2.0 (* (+ (- (/ x (- x 1.0)) 1.0) (/ 1.0 (- x 1.0))) (* l l))))
     t_m)
    (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) {
	double tmp;
	if (t_m <= 2.6e-218) {
		tmp = sqrt((2.0 / ((((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0))) * (l * l)))) * t_m;
	} else {
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.6d-218) then
        tmp = sqrt((2.0d0 / ((((x / (x - 1.0d0)) - 1.0d0) + (1.0d0 / (x - 1.0d0))) * (l * l)))) * t_m
    else
        tmp = sqrt(((1.0d0 - x) / ((-1.0d0) - x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 2.6e-218) {
		tmp = Math.sqrt((2.0 / ((((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0))) * (l * l)))) * t_m;
	} else {
		tmp = Math.sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 2.6e-218:
		tmp = math.sqrt((2.0 / ((((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0))) * (l * l)))) * t_m
	else:
		tmp = math.sqrt(((1.0 - x) / (-1.0 - x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 2.6e-218)
		tmp = Float64(sqrt(Float64(2.0 / Float64(Float64(Float64(Float64(x / Float64(x - 1.0)) - 1.0) + Float64(1.0 / Float64(x - 1.0))) * Float64(l * l)))) * t_m);
	else
		tmp = sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 2.6e-218)
		tmp = sqrt((2.0 / ((((x / (x - 1.0)) - 1.0) + (1.0 / (x - 1.0))) * (l * l)))) * t_m;
	else
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-218], N[(N[Sqrt[N[(2.0 / N[(N[(N[(N[(x / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.59999999999999983e-218

   1. Initial program 28.8%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 35.1

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

   5. Applied rewrites 35.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

   7. Applied rewrites 35.1%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 14.3

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

   10. Applied rewrites 14.3%

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

   if 2.59999999999999983e-218 < t

   1. Initial program 27.6%

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

   3. 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 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 80.5%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\frac{2}{\left(\left(\frac{x}{x - 1} - 1\right) + \frac{1}{x - 1}\right) \cdot \left(\ell \cdot \ell\right)}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\frac{\frac{\ell \cdot \ell}{t\_m}}{x \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-218)
    (/ (* (sqrt 2.0) t_m) (/ (/ (* l l) t_m) (* x (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) {
	double tmp;
	if (t_m <= 2.6e-218) {
		tmp = (sqrt(2.0) * t_m) / (((l * l) / t_m) / (x * sqrt(2.0)));
	} else {
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.6d-218) then
        tmp = (sqrt(2.0d0) * t_m) / (((l * l) / t_m) / (x * sqrt(2.0d0)))
    else
        tmp = sqrt(((1.0d0 - x) / ((-1.0d0) - x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 2.6e-218) {
		tmp = (Math.sqrt(2.0) * t_m) / (((l * l) / t_m) / (x * Math.sqrt(2.0)));
	} else {
		tmp = Math.sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 2.6e-218:
		tmp = (math.sqrt(2.0) * t_m) / (((l * l) / t_m) / (x * math.sqrt(2.0)))
	else:
		tmp = math.sqrt(((1.0 - x) / (-1.0 - x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 2.6e-218)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(Float64(Float64(l * l) / t_m) / Float64(x * sqrt(2.0))));
	else
		tmp = sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 2.6e-218)
		tmp = (sqrt(2.0) * t_m) / (((l * l) / t_m) / (x * sqrt(2.0)));
	else
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-218], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.59999999999999983e-218

   1. Initial program 28.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{1}{2} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}} \]
   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 15.3%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 12.7%

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

   if 2.59999999999999983e-218 < t

   1. Initial program 27.6%

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

   3. 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 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 80.5%

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

Alternative 8: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\right)}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-218)
    (* (/ t_m (sqrt (fma (- l) l (fma (* t_m t_m) 2.0 (* l l))))) (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) {
	double tmp;
	if (t_m <= 2.6e-218) {
		tmp = (t_m / sqrt(fma(-l, l, fma((t_m * t_m), 2.0, (l * l))))) * sqrt(2.0);
	} else {
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 2.6e-218)
		tmp = Float64(Float64(t_m / sqrt(fma(Float64(-l), l, fma(Float64(t_m * t_m), 2.0, Float64(l * l))))) * sqrt(2.0));
	else
		tmp = sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)));
	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, 2.6e-218], N[(N[(t$95$m / N[Sqrt[N[((-l) * l + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.59999999999999983e-218

   1. Initial program 28.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 39.7%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 39.1

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

   7. Applied rewrites 39.1%

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

   if 2.59999999999999983e-218 < t

   1. Initial program 27.6%

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

   3. 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 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 80.5%

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

Alternative 9: 74.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\frac{2}{\left(-\ell\right) \cdot \ell - \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-218)
    (* (sqrt (/ 2.0 (- (* (- l) l) (fma (* t_m t_m) 2.0 (* l l))))) t_m)
    (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) {
	double tmp;
	if (t_m <= 2.6e-218) {
		tmp = sqrt((2.0 / ((-l * l) - fma((t_m * t_m), 2.0, (l * l))))) * t_m;
	} else {
		tmp = sqrt(((1.0 - x) / (-1.0 - x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 2.6e-218)
		tmp = Float64(sqrt(Float64(2.0 / Float64(Float64(Float64(-l) * l) - fma(Float64(t_m * t_m), 2.0, Float64(l * l))))) * t_m);
	else
		tmp = sqrt(Float64(Float64(1.0 - x) / Float64(-1.0 - x)));
	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, 2.6e-218], N[(N[Sqrt[N[(2.0 / N[(N[((-l) * l), $MachinePrecision] - N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.59999999999999983e-218

   1. Initial program 28.8%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 35.1

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

   5. Applied rewrites 35.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

   7. Applied rewrites 35.1%

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

   8. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 11.3

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

   10. Applied rewrites 11.3%

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

   if 2.59999999999999983e-218 < t

   1. Initial program 27.6%

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

   3. 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 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 80.5%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\frac{2}{\left(-\ell\right) \cdot \ell - \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 - x}{-1 - x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.0% accurate, 3.0× speedup

Math

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

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)))))

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)));
}

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)))
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)));
}

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)))

Julia

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

Derivation

1. Initial program 28.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 36.1

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

5. Applied rewrites 36.1%

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

7. Applied rewrites 36.7%

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

Alternative 11: 75.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 36.1

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

5. Applied rewrites 36.1%

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

7. Applied rewrites 17.7%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 36.0%

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

Alternative 12: 74.7% 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 28.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 35.0

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

5. Applied rewrites 35.0%

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

7. Applied rewrites 35.5%

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

Reproduce

herbie shell --seed 2024295 
(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)))))