Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 82.9%

Time: 23.9s

Alternatives: 8

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {l_m}^{2}\\ t_4 := t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_3 + t_3}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.9 \cdot 10^{-254}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_m \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{t_m \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l_m}\\ \mathbf{elif}\;t_m \leq 4.4 \cdot 10^{-159}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_m \leq 3.5 \cdot 10^{-53}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l_m 2.0)))
        (t_4
         (*
          t_m
          (/
           (sqrt 2.0)
           (+
            (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x))))
            (* t_m (sqrt 2.0)))))))
   (*
    t_s
    (if (<= t_m 1.9e-254)
      t_4
      (if (<= t_m 3.2e-199)
        (/ (* t_m (sqrt (* 2.0 (fma x 0.5 -0.5)))) l_m)
        (if (<= t_m 4.4e-159)
          t_4
          (if (<= t_m 3.5e-53)
            (*
             t_m
             (/
              (sqrt 2.0)
              (sqrt
               (+
                (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
                (/ t_3 x)))))
            (sqrt (/ (+ -1.0 x) (+ x 1.0))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double t_4 = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	double tmp;
	if (t_m <= 1.9e-254) {
		tmp = t_4;
	} else if (t_m <= 3.2e-199) {
		tmp = (t_m * sqrt((2.0 * fma(x, 0.5, -0.5)))) / l_m;
	} else if (t_m <= 4.4e-159) {
		tmp = t_4;
	} else if (t_m <= 3.5e-53) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	t_4 = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))))
	tmp = 0.0
	if (t_m <= 1.9e-254)
		tmp = t_4;
	elseif (t_m <= 3.2e-199)
		tmp = Float64(Float64(t_m * sqrt(Float64(2.0 * fma(x, 0.5, -0.5)))) / l_m);
	elseif (t_m <= 4.4e-159)
		tmp = t_4;
	elseif (t_m <= 3.5e-53)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-254], t$95$4, If[LessEqual[t$95$m, 3.2e-199], N[(N[(t$95$m * N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 4.4e-159], t$95$4, If[LessEqual[t$95$m, 3.5e-53], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.9000000000000001e-254 or 3.1999999999999999e-199 < t < 4.4e-159

   1. Initial program 27.4%

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

   3. Simplified 27.4%

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

   5. Taylor expanded in x around inf 11.4%

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

   if 1.9000000000000001e-254 < t < 3.1999999999999999e-199

   1. Initial program 2.0%

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

   3. Simplified 2.0%

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

   5. Taylor expanded in l around inf 1.3%

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

      1. *-commutative 1.3%

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

      2. associate--l+ 26.6%

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

      3. sub-neg 26.6%

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

      4. metadata-eval 26.6%

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

      5. +-commutative 26.6%

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

      6. sub-neg 26.6%

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

      7. metadata-eval 26.6%

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

      8. +-commutative 26.6%

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

   7. Simplified 26.6%

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

   8. Taylor expanded in x around 0 52.6%

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

   9. Taylor expanded in t around 0 52.8%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-*l/ 59.6%

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

       2. *-commutative 59.6%

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

       3. fma-neg 59.6%

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

       4. metadata-eval 59.6%

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

       5. associate-*l* 59.6%

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

       6. sqrt-prod 59.6%

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

       7. associate-*l/ 52.5%

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

       8. *-commutative 52.5%

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

       9. associate-*r/ 59.6%

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

   11. Applied egg-rr 59.6%

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

   if 4.4e-159 < t < 3.49999999999999993e-53

   1. Initial program 37.6%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in x around inf 84.4%

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

   if 3.49999999999999993e-53 < t

   1. Initial program 46.9%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in t around inf 92.0%

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

      1. associate-*l* 92.0%

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

      2. +-commutative 92.0%

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

      3. sub-neg 92.0%

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

      4. metadata-eval 92.0%

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

      5. +-commutative 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in t around 0 92.2%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-254}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{t \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-159}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-53}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 9.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{t_m \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l_m}\\ \mathbf{elif}\;t_m \leq 10^{-159} \lor \neg \left(t_m \leq 1.95 \cdot 10^{-53}\right):\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_2 + {l_m}^{2}}{x}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 9.5e-196)
      (/ (* t_m (sqrt (* 2.0 (fma x 0.5 -0.5)))) l_m)
      (if (or (<= t_m 1e-159) (not (<= t_m 1.95e-53)))
        (sqrt (/ (+ -1.0 x) (+ x 1.0)))
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
            (/ (+ t_2 (pow l_m 2.0)) x))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 9.5e-196) {
		tmp = (t_m * sqrt((2.0 * fma(x, 0.5, -0.5)))) / l_m;
	} else if ((t_m <= 1e-159) || !(t_m <= 1.95e-53)) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 9.5e-196)
		tmp = Float64(Float64(t_m * sqrt(Float64(2.0 * fma(x, 0.5, -0.5)))) / l_m);
	elseif ((t_m <= 1e-159) || !(t_m <= 1.95e-53))
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-196], N[(N[(t$95$m * N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[Or[LessEqual[t$95$m, 1e-159], N[Not[LessEqual[t$95$m, 1.95e-53]], $MachinePrecision]], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 9.50000000000000032e-196

   1. Initial program 26.0%

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

   3. Simplified 26.0%

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

   5. Taylor expanded in l around inf 3.0%

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

      1. *-commutative 3.0%

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

      2. associate--l+ 8.4%

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

      3. sub-neg 8.4%

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

      4. metadata-eval 8.4%

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

      5. +-commutative 8.4%

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

      6. sub-neg 8.4%

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

      7. metadata-eval 8.4%

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

      8. +-commutative 8.4%

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

   7. Simplified 8.4%

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

   8. Taylor expanded in x around 0 14.3%

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

   9. Taylor expanded in t around 0 14.4%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-*l/ 17.2%

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

       2. *-commutative 17.2%

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

       3. fma-neg 17.2%

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

       4. metadata-eval 17.2%

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

       5. associate-*l* 17.2%

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

       6. sqrt-prod 17.2%

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

       7. associate-*l/ 14.4%

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

       8. *-commutative 14.4%

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

       9. associate-*r/ 17.2%

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

   11. Applied egg-rr 17.2%

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

   if 9.50000000000000032e-196 < t < 9.99999999999999989e-160 or 1.9500000000000001e-53 < t

   1. Initial program 44.5%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in t around inf 90.4%

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

      1. associate-*l* 90.5%

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

      2. +-commutative 90.5%

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

      3. sub-neg 90.5%

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

      4. metadata-eval 90.5%

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

      5. +-commutative 90.5%

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

   7. Simplified 90.5%

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

   8. Taylor expanded in t around 0 90.7%

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

   if 9.99999999999999989e-160 < t < 1.9500000000000001e-53

   1. Initial program 37.6%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in x around inf 84.4%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{t \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{elif}\;t \leq 10^{-159} \lor \neg \left(t \leq 1.95 \cdot 10^{-53}\right):\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t_m \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{t_2}{l_m}\\ \mathbf{elif}\;t_m \leq 3.6 \cdot 10^{-144} \lor \neg \left(t_m \leq 8.8 \cdot 10^{-125}\right):\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{l_m}{t_2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt (* 2.0 (fma x 0.5 -0.5))))))
   (*
    t_s
    (if (<= t_m 4.2e-199)
      (/ t_2 l_m)
      (if (or (<= t_m 3.6e-144) (not (<= t_m 8.8e-125)))
        (sqrt (/ (+ -1.0 x) (+ x 1.0)))
        (/ 1.0 (/ l_m t_2)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt((2.0 * fma(x, 0.5, -0.5)));
	double tmp;
	if (t_m <= 4.2e-199) {
		tmp = t_2 / l_m;
	} else if ((t_m <= 3.6e-144) || !(t_m <= 8.8e-125)) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = 1.0 / (l_m / t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(Float64(2.0 * fma(x, 0.5, -0.5))))
	tmp = 0.0
	if (t_m <= 4.2e-199)
		tmp = Float64(t_2 / l_m);
	elseif ((t_m <= 3.6e-144) || !(t_m <= 8.8e-125))
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(1.0 / Float64(l_m / t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-199], N[(t$95$2 / l$95$m), $MachinePrecision], If[Or[LessEqual[t$95$m, 3.6e-144], N[Not[LessEqual[t$95$m, 8.8e-125]], $MachinePrecision]], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.20000000000000004e-199

   1. Initial program 26.0%

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

   3. Simplified 26.0%

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

   5. Taylor expanded in l around inf 3.0%

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

      1. *-commutative 3.0%

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

      2. associate--l+ 8.4%

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

      3. sub-neg 8.4%

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

      4. metadata-eval 8.4%

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

      5. +-commutative 8.4%

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

      6. sub-neg 8.4%

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

      7. metadata-eval 8.4%

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

      8. +-commutative 8.4%

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

   7. Simplified 8.4%

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

   8. Taylor expanded in x around 0 14.3%

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

   9. Taylor expanded in t around 0 14.4%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-*l/ 17.2%

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

       2. *-commutative 17.2%

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

       3. fma-neg 17.2%

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

       4. metadata-eval 17.2%

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

       5. associate-*l* 17.2%

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

       6. sqrt-prod 17.2%

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

       7. associate-*l/ 14.4%

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

       8. *-commutative 14.4%

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

       9. associate-*r/ 17.2%

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

   11. Applied egg-rr 17.2%

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

   if 4.20000000000000004e-199 < t < 3.6e-144 or 8.79999999999999979e-125 < t

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in t around inf 86.6%

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

      1. associate-*l* 86.6%

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

      2. +-commutative 86.6%

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

      3. sub-neg 86.6%

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

      4. metadata-eval 86.6%

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

      5. +-commutative 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in t around 0 86.7%

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

   if 3.6e-144 < t < 8.79999999999999979e-125

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 9.3%

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

      3. sub-neg 9.3%

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

      4. metadata-eval 9.3%

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

      5. +-commutative 9.3%

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

      6. sub-neg 9.3%

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

      7. metadata-eval 9.3%

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

      8. +-commutative 9.3%

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

   7. Simplified 9.3%

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

   8. Taylor expanded in x around 0 35.9%

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

   9. Taylor expanded in t around 0 35.4%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-*l/ 35.4%

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

       2. *-commutative 35.4%

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

       3. fma-neg 35.4%

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

       4. metadata-eval 35.4%

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

       5. associate-*l* 35.9%

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

       6. sqrt-prod 35.9%

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

       7. associate-*l/ 35.9%

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

       8. *-commutative 35.9%

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

       9. associate-*r/ 35.9%

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

       10. clear-num 35.9%

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

   11. Applied egg-rr 35.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{\ell}{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{t \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-144} \lor \neg \left(t \leq 8.8 \cdot 10^{-125}\right):\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\ell}{t \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.2 \cdot 10^{-196} \lor \neg \left(t_m \leq 3.7 \cdot 10^{-144}\right) \land t_m \leq 8.8 \cdot 10^{-125}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 1.2e-196) (and (not (<= t_m 3.7e-144)) (<= t_m 8.8e-125)))
    (* t_m (/ (sqrt (* 2.0 (fma x 0.5 -0.5))) l_m))
    (sqrt (/ (+ -1.0 x) (+ x 1.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((t_m <= 1.2e-196) || (!(t_m <= 3.7e-144) && (t_m <= 8.8e-125))) {
		tmp = t_m * (sqrt((2.0 * fma(x, 0.5, -0.5))) / l_m);
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if ((t_m <= 1.2e-196) || (!(t_m <= 3.7e-144) && (t_m <= 8.8e-125)))
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 * fma(x, 0.5, -0.5))) / l_m));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 1.2e-196], And[N[Not[LessEqual[t$95$m, 3.7e-144]], $MachinePrecision], LessEqual[t$95$m, 8.8e-125]]], N[(t$95$m * N[(N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2000000000000001e-196 or 3.7000000000000003e-144 < t < 8.79999999999999979e-125

   1. Initial program 25.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in l around inf 3.0%

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

      1. *-commutative 3.0%

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

      2. associate--l+ 8.5%

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

      3. sub-neg 8.5%

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

      4. metadata-eval 8.5%

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

      5. +-commutative 8.5%

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

      6. sub-neg 8.5%

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

      7. metadata-eval 8.5%

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

      8. +-commutative 8.5%

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

   7. Simplified 8.5%

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

   8. Taylor expanded in x around 0 14.8%

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

   9. Taylor expanded in t around 0 14.8%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-*l/ 17.5%

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

       2. *-commutative 17.5%

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

       3. fma-neg 17.5%

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

       4. metadata-eval 17.5%

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

       5. associate-*l* 17.6%

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

       6. sqrt-prod 17.6%

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

       7. associate-*l/ 14.8%

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

       8. *-commutative 14.8%

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

       9. expm1-log1p-u 14.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)\right)} \]

       10. expm1-udef 6.5%

           \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)} - 1} \]

   11. Applied egg-rr 6.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 14.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)\right)} \]

       2. expm1-log1p 14.8%

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

       3. associate-*r/ 17.6%

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

       4. *-rgt-identity 17.6%

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

       5. *-commutative 17.6%

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

       6. associate-*r/ 17.5%

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

       7. associate-*l* 17.6%

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

       8. *-commutative 17.6%

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

       9. associate-*l/ 17.6%

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

       10. *-lft-identity 17.6%

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

   13. Simplified 17.6%

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

   if 1.2000000000000001e-196 < t < 3.7000000000000003e-144 or 8.79999999999999979e-125 < t

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in t around inf 86.6%

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

      1. associate-*l* 86.6%

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

      2. +-commutative 86.6%

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

      3. sub-neg 86.6%

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

      4. metadata-eval 86.6%

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

      5. +-commutative 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in t around 0 86.7%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-196} \lor \neg \left(t \leq 3.7 \cdot 10^{-144}\right) \land t \leq 8.8 \cdot 10^{-125}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.5 \cdot 10^{-198} \lor \neg \left(t_m \leq 3.6 \cdot 10^{-144}\right) \land t_m \leq 1.26 \cdot 10^{-121}:\\ \;\;\;\;\frac{t_m \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 2.5e-198) (and (not (<= t_m 3.6e-144)) (<= t_m 1.26e-121)))
    (/ (* t_m (sqrt (* 2.0 (fma x 0.5 -0.5)))) l_m)
    (sqrt (/ (+ -1.0 x) (+ x 1.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((t_m <= 2.5e-198) || (!(t_m <= 3.6e-144) && (t_m <= 1.26e-121))) {
		tmp = (t_m * sqrt((2.0 * fma(x, 0.5, -0.5)))) / l_m;
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if ((t_m <= 2.5e-198) || (!(t_m <= 3.6e-144) && (t_m <= 1.26e-121)))
		tmp = Float64(Float64(t_m * sqrt(Float64(2.0 * fma(x, 0.5, -0.5)))) / l_m);
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 2.5e-198], And[N[Not[LessEqual[t$95$m, 3.6e-144]], $MachinePrecision], LessEqual[t$95$m, 1.26e-121]]], N[(N[(t$95$m * N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e-198 or 3.6e-144 < t < 1.26e-121

   1. Initial program 25.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in l around inf 3.0%

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

      1. *-commutative 3.0%

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

      2. associate--l+ 8.5%

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

      3. sub-neg 8.5%

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

      4. metadata-eval 8.5%

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

      5. +-commutative 8.5%

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

      6. sub-neg 8.5%

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

      7. metadata-eval 8.5%

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

      8. +-commutative 8.5%

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

   7. Simplified 8.5%

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

   8. Taylor expanded in x around 0 14.8%

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

   9. Taylor expanded in t around 0 14.8%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-*l/ 17.5%

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

       2. *-commutative 17.5%

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

       3. fma-neg 17.5%

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

       4. metadata-eval 17.5%

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

       5. associate-*l* 17.6%

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

       6. sqrt-prod 17.6%

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

       7. associate-*l/ 14.8%

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

       8. *-commutative 14.8%

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

       9. associate-*r/ 17.6%

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

   11. Applied egg-rr 17.6%

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

   if 2.5e-198 < t < 3.6e-144 or 1.26e-121 < t

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in t around inf 86.6%

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

      1. associate-*l* 86.6%

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

      2. +-commutative 86.6%

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

      3. sub-neg 86.6%

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

      4. metadata-eval 86.6%

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

      5. +-commutative 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in t around 0 86.7%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-198} \lor \neg \left(t \leq 3.6 \cdot 10^{-144}\right) \land t \leq 1.26 \cdot 10^{-121}:\\ \;\;\;\;\frac{t \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (sqrt (/ (+ -1.0 x) (+ x 1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in t around inf 39.5%

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

   1. associate-*l* 39.5%

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

   2. +-commutative 39.5%

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

   3. sub-neg 39.5%

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

   4. metadata-eval 39.5%

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

   5. +-commutative 39.5%

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

7. Simplified 39.5%

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

8. Taylor expanded in t around 0 39.6%

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

9. Final simplification 39.6%

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

Alternative 7: 75.9% accurate, 45.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in t around inf 39.5%

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

   1. associate-*l* 39.5%

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

   2. +-commutative 39.5%

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

   3. sub-neg 39.5%

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

   4. metadata-eval 39.5%

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

   5. +-commutative 39.5%

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

7. Simplified 39.5%

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

8. Taylor expanded in x around inf 39.3%

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

9. Final simplification 39.3%

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

Alternative 8: 75.3% accurate, 225.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in t around inf 39.5%

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

   1. associate-*l* 39.5%

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

   2. +-commutative 39.5%

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

   3. sub-neg 39.5%

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

   4. metadata-eval 39.5%

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

   5. +-commutative 39.5%

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

7. Simplified 39.5%

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

8. Taylor expanded in x around inf 39.1%

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

9. Final simplification 39.1%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))))