Toniolo and Linder, Equation (7)

Percentage Accurate: 32.6% → 83.6%

Time: 25.0s

Alternatives: 10

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: 32.6% 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: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t_4 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(2, \frac{\frac{t\_m}{\sqrt{2}}}{x}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{{l\_m}^{2}}{x \cdot t\_4}\right)\right)}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 6.7 \cdot 10^{-188}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 6.6 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{{l\_m}^{2} + t\_2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_m (/ (sqrt x) l_m)))
        (t_4 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 2.9e-251)
      t_3
      (if (<= t_m 1.01e-201)
        (/
         (sqrt 2.0)
         (/
          (fma
           2.0
           (/ (/ t_m (sqrt 2.0)) x)
           (fma t_m (sqrt 2.0) (/ (pow l_m 2.0) (* x t_4))))
          t_m))
        (if (<= t_m 6.7e-188)
          t_3
          (if (<= t_m 6.6e-168)
            1.0
            (if (<= t_m 4.2e-43)
              (/
               t_4
               (sqrt
                (+
                 (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
                 (/ (+ (pow l_m 2.0) t_2) x))))
              (sqrt (/ (+ x -1.0) (+ 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_m * (sqrt(x) / l_m);
	double t_4 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 2.9e-251) {
		tmp = t_3;
	} else if (t_m <= 1.01e-201) {
		tmp = sqrt(2.0) / (fma(2.0, ((t_m / sqrt(2.0)) / x), fma(t_m, sqrt(2.0), (pow(l_m, 2.0) / (x * t_4)))) / t_m);
	} else if (t_m <= 6.7e-188) {
		tmp = t_3;
	} else if (t_m <= 6.6e-168) {
		tmp = 1.0;
	} else if (t_m <= 4.2e-43) {
		tmp = t_4 / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((pow(l_m, 2.0) + t_2) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (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_m * Float64(sqrt(x) / l_m))
	t_4 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 2.9e-251)
		tmp = t_3;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(sqrt(2.0) / Float64(fma(2.0, Float64(Float64(t_m / sqrt(2.0)) / x), fma(t_m, sqrt(2.0), Float64((l_m ^ 2.0) / Float64(x * t_4)))) / t_m));
	elseif (t_m <= 6.7e-188)
		tmp = t_3;
	elseif (t_m <= 6.6e-168)
		tmp = 1.0;
	elseif (t_m <= 4.2e-43)
		tmp = Float64(t_4 / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64((l_m ^ 2.0) + t_2) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-251], t$95$3, If[LessEqual[t$95$m, 1.01e-201], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(2.0 * N[(N[(t$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(x * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.7e-188], t$95$3, If[LessEqual[t$95$m, 6.6e-168], 1.0, If[LessEqual[t$95$m, 4.2e-43], N[(t$95$4 / 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[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if t < 2.9000000000000001e-251 or 1.00999999999999997e-201 < t < 6.69999999999999988e-188

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 6.7%

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

      3. sub-neg 6.7%

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

      4. metadata-eval 6.7%

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

      5. +-commutative 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around inf 10.4%

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

      1. expm1-log1p-u 9.4%

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

      2. expm1-udef 4.0%

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

      3. associate-*r* 4.0%

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

      4. sqrt-unprod 4.0%

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

      5. *-commutative 4.0%

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

   10. Applied egg-rr 4.0%

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

       1. expm1-def 9.4%

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

       2. expm1-log1p 10.4%

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

       3. associate-*r/ 10.4%

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

       4. associate-*l/ 10.4%

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

       5. *-commutative 10.4%

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

       6. associate-*r* 10.4%

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

       7. *-commutative 10.4%

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

       8. associate-*l* 10.4%

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

       9. metadata-eval 10.4%

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

   12. Simplified 10.4%

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

   if 2.9000000000000001e-251 < t < 1.00999999999999997e-201

   1. Initial program 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in x around inf 85.2%

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

      1. expm1-log1p-u 85.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \frac{\sqrt{2}}{2 \cdot \frac{t}{x \cdot \sqrt{2}} + \left(t \cdot \sqrt{2} + \frac{{\ell}^{2}}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)}\right)\right)} \]

      2. expm1-udef 85.4%

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

   7. Applied egg-rr 85.6%

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

      1. expm1-def 85.6%

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

      2. expm1-log1p 85.4%

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

      3. associate-*r/ 85.5%

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

      4. *-commutative 85.5%

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

      5. associate-/l* 85.6%

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

   9. Simplified 85.6%

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

   if 6.69999999999999988e-188 < t < 6.6000000000000003e-168

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in t around inf 99.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if 6.6000000000000003e-168 < t < 4.2000000000000001e-43

   1. Initial program 37.6%

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

   3. Taylor expanded in x around inf 81.1%

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

   if 4.2000000000000001e-43 < t

   1. Initial program 41.2%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in t around inf 93.4%

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

   6. Taylor expanded in t around 0 93.8%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(2, \frac{\frac{t}{\sqrt{2}}}{x}, \mathsf{fma}\left(t, \sqrt{2}, \frac{{\ell}^{2}}{x \cdot \left(t \cdot \sqrt{2}\right)}\right)\right)}{t}}\\ \mathbf{elif}\;t \leq 6.7 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \sqrt{2}\\ t_3 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t_4 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-251}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(2, \frac{t\_m}{t\_2}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{{l\_m}^{2}}{t\_m \cdot t\_2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{-188}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{t\_4 + 2 \cdot \frac{{l\_m}^{2} + t\_4}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* x (sqrt 2.0)))
        (t_3 (* t_m (/ (sqrt x) l_m)))
        (t_4 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 2.1e-251)
      t_3
      (if (<= t_m 1.01e-201)
        (*
         t_m
         (/
          (sqrt 2.0)
          (fma
           2.0
           (/ t_m t_2)
           (fma t_m (sqrt 2.0) (/ (pow l_m 2.0) (* t_m t_2))))))
        (if (<= t_m 8.6e-188)
          t_3
          (if (<= t_m 1.05e-166)
            1.0
            (if (<= t_m 4.2e-43)
              (*
               t_m
               (/
                (sqrt 2.0)
                (sqrt (+ t_4 (* 2.0 (/ (+ (pow l_m 2.0) t_4) x))))))
              (sqrt (/ (+ x -1.0) (+ 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 = x * sqrt(2.0);
	double t_3 = t_m * (sqrt(x) / l_m);
	double t_4 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.1e-251) {
		tmp = t_3;
	} else if (t_m <= 1.01e-201) {
		tmp = t_m * (sqrt(2.0) / fma(2.0, (t_m / t_2), fma(t_m, sqrt(2.0), (pow(l_m, 2.0) / (t_m * t_2)))));
	} else if (t_m <= 8.6e-188) {
		tmp = t_3;
	} else if (t_m <= 1.05e-166) {
		tmp = 1.0;
	} else if (t_m <= 4.2e-43) {
		tmp = t_m * (sqrt(2.0) / sqrt((t_4 + (2.0 * ((pow(l_m, 2.0) + t_4) / x)))));
	} else {
		tmp = sqrt(((x + -1.0) / (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(x * sqrt(2.0))
	t_3 = Float64(t_m * Float64(sqrt(x) / l_m))
	t_4 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.1e-251)
		tmp = t_3;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(t_m * Float64(sqrt(2.0) / fma(2.0, Float64(t_m / t_2), fma(t_m, sqrt(2.0), Float64((l_m ^ 2.0) / Float64(t_m * t_2))))));
	elseif (t_m <= 8.6e-188)
		tmp = t_3;
	elseif (t_m <= 1.05e-166)
		tmp = 1.0;
	elseif (t_m <= 4.2e-43)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(t_4 + Float64(2.0 * Float64(Float64((l_m ^ 2.0) + t_4) / x))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.1e-251], t$95$3, If[LessEqual[t$95$m, 1.01e-201], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(2.0 * N[(t$95$m / t$95$2), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e-188], t$95$3, If[LessEqual[t$95$m, 1.05e-166], 1.0, If[LessEqual[t$95$m, 4.2e-43], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$4 + N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$4), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if t < 2.09999999999999982e-251 or 1.00999999999999997e-201 < t < 8.59999999999999975e-188

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 6.7%

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

      3. sub-neg 6.7%

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

      4. metadata-eval 6.7%

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

      5. +-commutative 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around inf 10.4%

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

      1. expm1-log1p-u 9.4%

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

      2. expm1-udef 4.0%

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

      3. associate-*r* 4.0%

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

      4. sqrt-unprod 4.0%

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

      5. *-commutative 4.0%

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

   10. Applied egg-rr 4.0%

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

       1. expm1-def 9.4%

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

       2. expm1-log1p 10.4%

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

       3. associate-*r/ 10.4%

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

       4. associate-*l/ 10.4%

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

       5. *-commutative 10.4%

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

       6. associate-*r* 10.4%

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

       7. *-commutative 10.4%

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

       8. associate-*l* 10.4%

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

       9. metadata-eval 10.4%

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

   12. Simplified 10.4%

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

   if 2.09999999999999982e-251 < t < 1.00999999999999997e-201

   1. Initial program 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in x around inf 85.2%

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

      1. fma-def 85.2%

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

      2. fma-def 85.4%

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

   7. Simplified 85.4%

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

   if 8.59999999999999975e-188 < t < 1.05e-166

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in t around inf 99.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if 1.05e-166 < t < 4.2000000000000001e-43

   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 37.7%

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

   5. Taylor expanded in x around inf 81.0%

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

   if 4.2000000000000001e-43 < t

   1. Initial program 41.2%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in t around inf 93.4%

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

   6. Taylor expanded in t around 0 93.8%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(2, \frac{t}{x \cdot \sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{{\ell}^{2}}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-251}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(2, \frac{\frac{t\_m}{\sqrt{2}}}{x}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{{l\_m}^{2}}{x \cdot \left(t\_m \cdot \sqrt{2}\right)}\right)\right)}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{-187}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{t\_3 + 2 \cdot \frac{{l\_m}^{2} + t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m (/ (sqrt x) l_m))) (t_3 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 3e-251)
      t_2
      (if (<= t_m 1.01e-201)
        (/
         (sqrt 2.0)
         (/
          (fma
           2.0
           (/ (/ t_m (sqrt 2.0)) x)
           (fma t_m (sqrt 2.0) (/ (pow l_m 2.0) (* x (* t_m (sqrt 2.0))))))
          t_m))
        (if (<= t_m 3.4e-187)
          t_2
          (if (<= t_m 1.9e-166)
            1.0
            (if (<= t_m 4.2e-43)
              (*
               t_m
               (/
                (sqrt 2.0)
                (sqrt (+ t_3 (* 2.0 (/ (+ (pow l_m 2.0) t_3) x))))))
              (sqrt (/ (+ x -1.0) (+ 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 = t_m * (sqrt(x) / l_m);
	double t_3 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 3e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = sqrt(2.0) / (fma(2.0, ((t_m / sqrt(2.0)) / x), fma(t_m, sqrt(2.0), (pow(l_m, 2.0) / (x * (t_m * sqrt(2.0)))))) / t_m);
	} else if (t_m <= 3.4e-187) {
		tmp = t_2;
	} else if (t_m <= 1.9e-166) {
		tmp = 1.0;
	} else if (t_m <= 4.2e-43) {
		tmp = t_m * (sqrt(2.0) / sqrt((t_3 + (2.0 * ((pow(l_m, 2.0) + t_3) / x)))));
	} else {
		tmp = sqrt(((x + -1.0) / (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(t_m * Float64(sqrt(x) / l_m))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 3e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(sqrt(2.0) / Float64(fma(2.0, Float64(Float64(t_m / sqrt(2.0)) / x), fma(t_m, sqrt(2.0), Float64((l_m ^ 2.0) / Float64(x * Float64(t_m * sqrt(2.0)))))) / t_m));
	elseif (t_m <= 3.4e-187)
		tmp = t_2;
	elseif (t_m <= 1.9e-166)
		tmp = 1.0;
	elseif (t_m <= 4.2e-43)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(t_3 + Float64(2.0 * Float64(Float64((l_m ^ 2.0) + t_3) / x))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3e-251], t$95$2, If[LessEqual[t$95$m, 1.01e-201], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(2.0 * N[(N[(t$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(x * N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e-187], t$95$2, If[LessEqual[t$95$m, 1.9e-166], 1.0, If[LessEqual[t$95$m, 4.2e-43], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$3), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if t < 2.9999999999999999e-251 or 1.00999999999999997e-201 < t < 3.4000000000000001e-187

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 6.7%

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

      3. sub-neg 6.7%

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

      4. metadata-eval 6.7%

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

      5. +-commutative 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around inf 10.4%

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

      1. expm1-log1p-u 9.4%

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

      2. expm1-udef 4.0%

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

      3. associate-*r* 4.0%

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

      4. sqrt-unprod 4.0%

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

      5. *-commutative 4.0%

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

   10. Applied egg-rr 4.0%

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

       1. expm1-def 9.4%

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

       2. expm1-log1p 10.4%

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

       3. associate-*r/ 10.4%

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

       4. associate-*l/ 10.4%

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

       5. *-commutative 10.4%

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

       6. associate-*r* 10.4%

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

       7. *-commutative 10.4%

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

       8. associate-*l* 10.4%

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

       9. metadata-eval 10.4%

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

   12. Simplified 10.4%

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

   if 2.9999999999999999e-251 < t < 1.00999999999999997e-201

   1. Initial program 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in x around inf 85.2%

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

      1. expm1-log1p-u 85.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \frac{\sqrt{2}}{2 \cdot \frac{t}{x \cdot \sqrt{2}} + \left(t \cdot \sqrt{2} + \frac{{\ell}^{2}}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)}\right)\right)} \]

      2. expm1-udef 85.4%

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

   7. Applied egg-rr 85.6%

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

      1. expm1-def 85.6%

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

      2. expm1-log1p 85.4%

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

      3. associate-*r/ 85.5%

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

      4. *-commutative 85.5%

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

      5. associate-/l* 85.6%

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

   9. Simplified 85.6%

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

   if 3.4000000000000001e-187 < t < 1.89999999999999991e-166

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in t around inf 99.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if 1.89999999999999991e-166 < t < 4.2000000000000001e-43

   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 37.7%

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

   5. Taylor expanded in x around inf 81.0%

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

   if 4.2000000000000001e-43 < t

   1. Initial program 41.2%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in t around inf 93.4%

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

   6. Taylor expanded in t around 0 93.8%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(2, \frac{\frac{t}{\sqrt{2}}}{x}, \mathsf{fma}\left(t, \sqrt{2}, \frac{{\ell}^{2}}{x \cdot \left(t \cdot \sqrt{2}\right)}\right)\right)}{t}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-187}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \sqrt{2}\\ t_3 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t_4 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.15 \cdot 10^{-251}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{2 \cdot \frac{t\_m}{t\_2} + \left(t\_m \cdot \sqrt{2} + \frac{{l\_m}^{2}}{t\_m \cdot t\_2}\right)}\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{-188}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{t\_4 + 2 \cdot \frac{{l\_m}^{2} + t\_4}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* x (sqrt 2.0)))
        (t_3 (* t_m (/ (sqrt x) l_m)))
        (t_4 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 3.15e-251)
      t_3
      (if (<= t_m 1.01e-201)
        (*
         t_m
         (/
          (sqrt 2.0)
          (+
           (* 2.0 (/ t_m t_2))
           (+ (* t_m (sqrt 2.0)) (/ (pow l_m 2.0) (* t_m t_2))))))
        (if (<= t_m 9.2e-188)
          t_3
          (if (<= t_m 1.4e-168)
            1.0
            (if (<= t_m 4.2e-43)
              (*
               t_m
               (/
                (sqrt 2.0)
                (sqrt (+ t_4 (* 2.0 (/ (+ (pow l_m 2.0) t_4) x))))))
              (sqrt (/ (+ x -1.0) (+ 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 = x * sqrt(2.0);
	double t_3 = t_m * (sqrt(x) / l_m);
	double t_4 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 3.15e-251) {
		tmp = t_3;
	} else if (t_m <= 1.01e-201) {
		tmp = t_m * (sqrt(2.0) / ((2.0 * (t_m / t_2)) + ((t_m * sqrt(2.0)) + (pow(l_m, 2.0) / (t_m * t_2)))));
	} else if (t_m <= 9.2e-188) {
		tmp = t_3;
	} else if (t_m <= 1.4e-168) {
		tmp = 1.0;
	} else if (t_m <= 4.2e-43) {
		tmp = t_m * (sqrt(2.0) / sqrt((t_4 + (2.0 * ((pow(l_m, 2.0) + t_4) / x)))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = x * sqrt(2.0d0)
    t_3 = t_m * (sqrt(x) / l_m)
    t_4 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 3.15d-251) then
        tmp = t_3
    else if (t_m <= 1.01d-201) then
        tmp = t_m * (sqrt(2.0d0) / ((2.0d0 * (t_m / t_2)) + ((t_m * sqrt(2.0d0)) + ((l_m ** 2.0d0) / (t_m * t_2)))))
    else if (t_m <= 9.2d-188) then
        tmp = t_3
    else if (t_m <= 1.4d-168) then
        tmp = 1.0d0
    else if (t_m <= 4.2d-43) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((t_4 + (2.0d0 * (((l_m ** 2.0d0) + t_4) / x)))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
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) {
	double t_2 = x * Math.sqrt(2.0);
	double t_3 = t_m * (Math.sqrt(x) / l_m);
	double t_4 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 3.15e-251) {
		tmp = t_3;
	} else if (t_m <= 1.01e-201) {
		tmp = t_m * (Math.sqrt(2.0) / ((2.0 * (t_m / t_2)) + ((t_m * Math.sqrt(2.0)) + (Math.pow(l_m, 2.0) / (t_m * t_2)))));
	} else if (t_m <= 9.2e-188) {
		tmp = t_3;
	} else if (t_m <= 1.4e-168) {
		tmp = 1.0;
	} else if (t_m <= 4.2e-43) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((t_4 + (2.0 * ((Math.pow(l_m, 2.0) + t_4) / x)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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):
	t_2 = x * math.sqrt(2.0)
	t_3 = t_m * (math.sqrt(x) / l_m)
	t_4 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 3.15e-251:
		tmp = t_3
	elif t_m <= 1.01e-201:
		tmp = t_m * (math.sqrt(2.0) / ((2.0 * (t_m / t_2)) + ((t_m * math.sqrt(2.0)) + (math.pow(l_m, 2.0) / (t_m * t_2)))))
	elif t_m <= 9.2e-188:
		tmp = t_3
	elif t_m <= 1.4e-168:
		tmp = 1.0
	elif t_m <= 4.2e-43:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((t_4 + (2.0 * ((math.pow(l_m, 2.0) + t_4) / x)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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(x * sqrt(2.0))
	t_3 = Float64(t_m * Float64(sqrt(x) / l_m))
	t_4 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.15e-251)
		tmp = t_3;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(2.0 * Float64(t_m / t_2)) + Float64(Float64(t_m * sqrt(2.0)) + Float64((l_m ^ 2.0) / Float64(t_m * t_2))))));
	elseif (t_m <= 9.2e-188)
		tmp = t_3;
	elseif (t_m <= 1.4e-168)
		tmp = 1.0;
	elseif (t_m <= 4.2e-43)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(t_4 + Float64(2.0 * Float64(Float64((l_m ^ 2.0) + t_4) / x))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = x * sqrt(2.0);
	t_3 = t_m * (sqrt(x) / l_m);
	t_4 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.15e-251)
		tmp = t_3;
	elseif (t_m <= 1.01e-201)
		tmp = t_m * (sqrt(2.0) / ((2.0 * (t_m / t_2)) + ((t_m * sqrt(2.0)) + ((l_m ^ 2.0) / (t_m * t_2)))));
	elseif (t_m <= 9.2e-188)
		tmp = t_3;
	elseif (t_m <= 1.4e-168)
		tmp = 1.0;
	elseif (t_m <= 4.2e-43)
		tmp = t_m * (sqrt(2.0) / sqrt((t_4 + (2.0 * (((l_m ^ 2.0) + t_4) / x)))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = 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[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.15e-251], t$95$3, If[LessEqual[t$95$m, 1.01e-201], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(2.0 * N[(t$95$m / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e-188], t$95$3, If[LessEqual[t$95$m, 1.4e-168], 1.0, If[LessEqual[t$95$m, 4.2e-43], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$4 + N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$4), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if t < 3.1499999999999999e-251 or 1.00999999999999997e-201 < t < 9.1999999999999999e-188

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 6.7%

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

      3. sub-neg 6.7%

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

      4. metadata-eval 6.7%

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

      5. +-commutative 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around inf 10.4%

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

      1. expm1-log1p-u 9.4%

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

      2. expm1-udef 4.0%

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

      3. associate-*r* 4.0%

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

      4. sqrt-unprod 4.0%

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

      5. *-commutative 4.0%

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

   10. Applied egg-rr 4.0%

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

       1. expm1-def 9.4%

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

       2. expm1-log1p 10.4%

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

       3. associate-*r/ 10.4%

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

       4. associate-*l/ 10.4%

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

       5. *-commutative 10.4%

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

       6. associate-*r* 10.4%

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

       7. *-commutative 10.4%

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

       8. associate-*l* 10.4%

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

       9. metadata-eval 10.4%

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

   12. Simplified 10.4%

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

   if 3.1499999999999999e-251 < t < 1.00999999999999997e-201

   1. Initial program 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in x around inf 85.2%

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

   if 9.1999999999999999e-188 < t < 1.4000000000000001e-168

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in t around inf 99.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if 1.4000000000000001e-168 < t < 4.2000000000000001e-43

   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 37.7%

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

   5. Taylor expanded in x around inf 81.0%

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

   if 4.2000000000000001e-43 < t

   1. Initial program 41.2%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in t around inf 93.4%

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

   6. Taylor expanded in t around 0 93.8%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.15 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{2 \cdot \frac{t}{x \cdot \sqrt{2}} + \left(t \cdot \sqrt{2} + \frac{{\ell}^{2}}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1 - \frac{0.5}{{\left(\frac{t\_m}{l\_m}\right)}^{2}}}{x}\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{-161}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{t\_2 + 2 \cdot \frac{{l\_m}^{2} + t\_2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_m (/ (sqrt x) l_m))))
   (*
    t_s
    (if (<= t_m 3.3e-251)
      t_3
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ (- -1.0 (/ 0.5 (pow (/ t_m l_m) 2.0))) x))
        (if (<= t_m 9.5e-161)
          t_3
          (if (<= t_m 4.2e-43)
            (*
             t_m
             (/ (sqrt 2.0) (sqrt (+ t_2 (* 2.0 (/ (+ (pow l_m 2.0) t_2) x))))))
            (sqrt (/ (+ x -1.0) (+ 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_m * (sqrt(x) / l_m);
	double tmp;
	if (t_m <= 3.3e-251) {
		tmp = t_3;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + ((-1.0 - (0.5 / pow((t_m / l_m), 2.0))) / x);
	} else if (t_m <= 9.5e-161) {
		tmp = t_3;
	} else if (t_m <= 4.2e-43) {
		tmp = t_m * (sqrt(2.0) / sqrt((t_2 + (2.0 * ((pow(l_m, 2.0) + t_2) / x)))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_m * (sqrt(x) / l_m)
    if (t_m <= 3.3d-251) then
        tmp = t_3
    else if (t_m <= 1.01d-201) then
        tmp = 1.0d0 + (((-1.0d0) - (0.5d0 / ((t_m / l_m) ** 2.0d0))) / x)
    else if (t_m <= 9.5d-161) then
        tmp = t_3
    else if (t_m <= 4.2d-43) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((t_2 + (2.0d0 * (((l_m ** 2.0d0) + t_2) / x)))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
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) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_m * (Math.sqrt(x) / l_m);
	double tmp;
	if (t_m <= 3.3e-251) {
		tmp = t_3;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + ((-1.0 - (0.5 / Math.pow((t_m / l_m), 2.0))) / x);
	} else if (t_m <= 9.5e-161) {
		tmp = t_3;
	} else if (t_m <= 4.2e-43) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((t_2 + (2.0 * ((Math.pow(l_m, 2.0) + t_2) / x)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_m * (math.sqrt(x) / l_m)
	tmp = 0
	if t_m <= 3.3e-251:
		tmp = t_3
	elif t_m <= 1.01e-201:
		tmp = 1.0 + ((-1.0 - (0.5 / math.pow((t_m / l_m), 2.0))) / x)
	elif t_m <= 9.5e-161:
		tmp = t_3
	elif t_m <= 4.2e-43:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((t_2 + (2.0 * ((math.pow(l_m, 2.0) + t_2) / x)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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_m * Float64(sqrt(x) / l_m))
	tmp = 0.0
	if (t_m <= 3.3e-251)
		tmp = t_3;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(0.5 / (Float64(t_m / l_m) ^ 2.0))) / x));
	elseif (t_m <= 9.5e-161)
		tmp = t_3;
	elseif (t_m <= 4.2e-43)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(2.0 * Float64(Float64((l_m ^ 2.0) + t_2) / x))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_m * (sqrt(x) / l_m);
	tmp = 0.0;
	if (t_m <= 3.3e-251)
		tmp = t_3;
	elseif (t_m <= 1.01e-201)
		tmp = 1.0 + ((-1.0 - (0.5 / ((t_m / l_m) ^ 2.0))) / x);
	elseif (t_m <= 9.5e-161)
		tmp = t_3;
	elseif (t_m <= 4.2e-43)
		tmp = t_m * (sqrt(2.0) / sqrt((t_2 + (2.0 * (((l_m ^ 2.0) + t_2) / x)))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = 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$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.3e-251], t$95$3, If[LessEqual[t$95$m, 1.01e-201], N[(1.0 + N[(N[(-1.0 - N[(0.5 / N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e-161], t$95$3, If[LessEqual[t$95$m, 4.2e-43], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.3e-251 or 1.00999999999999997e-201 < t < 9.4999999999999996e-161

   1. Initial program 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 6.7%

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

      3. sub-neg 6.7%

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

      4. metadata-eval 6.7%

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

      5. +-commutative 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around inf 10.9%

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

      1. expm1-log1p-u 9.9%

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

      2. expm1-udef 3.9%

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

      3. associate-*r* 3.9%

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

      4. sqrt-unprod 3.9%

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

      5. *-commutative 3.9%

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

   10. Applied egg-rr 3.9%

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

       1. expm1-def 9.9%

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

       2. expm1-log1p 10.9%

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

       3. associate-*r/ 10.9%

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

       4. associate-*l/ 10.9%

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

       5. *-commutative 10.9%

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

       6. associate-*r* 10.9%

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

       7. *-commutative 10.9%

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

       8. associate-*l* 10.9%

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

       9. metadata-eval 10.9%

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

   12. Simplified 10.9%

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

   if 3.3e-251 < t < 1.00999999999999997e-201

   1. Initial program 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in x around inf 85.2%

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

   6. Taylor expanded in x around inf 0.6%

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

      1. mul-1-neg 0.6%

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

      2. unsub-neg 0.6%

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

      3. unpow2 0.6%

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

      4. rem-square-sqrt 0.6%

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

      5. metadata-eval 0.6%

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

      6. metadata-eval 0.6%

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

      7. *-commutative 0.6%

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

      8. unpow2 0.6%

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

      9. rem-square-sqrt 0.6%

         \[\leadsto 1 - \frac{1 + \frac{{\ell}^{2}}{\color{blue}{2} \cdot {t}^{2}}}{x} \]

   8. Simplified 0.6%

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

      1. clear-num 0.6%

         \[\leadsto 1 - \frac{1 + \color{blue}{\frac{1}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}{x} \]

      2. inv-pow 0.6%

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

      3. *-un-lft-identity 0.6%

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

      4. times-frac 0.6%

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

      5. metadata-eval 0.6%

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

      6. unpow2 0.6%

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

      7. unpow2 0.6%

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

      8. frac-times 62.1%

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

      9. pow2 62.1%

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

   10. Applied egg-rr 62.1%

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

       1. unpow-1 62.1%

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

       2. associate-/r* 62.1%

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

       3. metadata-eval 62.1%

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

   12. Simplified 62.1%

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

   if 9.4999999999999996e-161 < t < 4.2000000000000001e-43

   1. Initial program 39.4%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in x around inf 80.2%

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

   if 4.2000000000000001e-43 < t

   1. Initial program 41.2%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in t around inf 93.4%

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

   6. Taylor expanded in t around 0 93.8%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1 - \frac{0.5}{{\left(\frac{t}{\ell}\right)}^{2}}}{x}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-251}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{-187}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m (/ (sqrt x) l_m))))
   (*
    t_s
    (if (<= t_m 3.1e-251)
      t_2
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 7.5e-187) t_2 (sqrt (/ (+ x -1.0) (+ 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 = t_m * (sqrt(x) / l_m);
	double tmp;
	if (t_m <= 3.1e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 7.5e-187) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (sqrt(x) / l_m)
    if (t_m <= 3.1d-251) then
        tmp = t_2
    else if (t_m <= 1.01d-201) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 7.5d-187) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
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) {
	double t_2 = t_m * (Math.sqrt(x) / l_m);
	double tmp;
	if (t_m <= 3.1e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 7.5e-187) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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):
	t_2 = t_m * (math.sqrt(x) / l_m)
	tmp = 0
	if t_m <= 3.1e-251:
		tmp = t_2
	elif t_m <= 1.01e-201:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 7.5e-187:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (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(t_m * Float64(sqrt(x) / l_m))
	tmp = 0.0
	if (t_m <= 3.1e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 7.5e-187)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * (sqrt(x) / l_m);
	tmp = 0.0;
	if (t_m <= 3.1e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 7.5e-187)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = 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[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.1e-251], t$95$2, If[LessEqual[t$95$m, 1.01e-201], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e-187], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.10000000000000003e-251 or 1.00999999999999997e-201 < t < 7.5000000000000004e-187

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 6.7%

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

      3. sub-neg 6.7%

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

      4. metadata-eval 6.7%

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

      5. +-commutative 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around inf 10.4%

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

      1. expm1-log1p-u 9.4%

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

      2. expm1-udef 4.0%

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

      3. associate-*r* 4.0%

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

      4. sqrt-unprod 4.0%

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

      5. *-commutative 4.0%

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

   10. Applied egg-rr 4.0%

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

       1. expm1-def 9.4%

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

       2. expm1-log1p 10.4%

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

       3. associate-*r/ 10.4%

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

       4. associate-*l/ 10.4%

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

       5. *-commutative 10.4%

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

       6. associate-*r* 10.4%

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

       7. *-commutative 10.4%

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

       8. associate-*l* 10.4%

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

       9. metadata-eval 10.4%

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

   12. Simplified 10.4%

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

   if 3.10000000000000003e-251 < t < 1.00999999999999997e-201

   1. Initial program 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in t around inf 61.4%

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

   6. Taylor expanded in x around inf 61.7%

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

   if 7.5000000000000004e-187 < t

   1. Initial program 39.6%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in t around inf 86.7%

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

   6. Taylor expanded in t around 0 87.0%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-187}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1 - \frac{0.5}{{\left(\frac{t\_m}{l\_m}\right)}^{2}}}{x}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{-188}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m (/ (sqrt x) l_m))))
   (*
    t_s
    (if (<= t_m 3.3e-251)
      t_2
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ (- -1.0 (/ 0.5 (pow (/ t_m l_m) 2.0))) x))
        (if (<= t_m 8.6e-188) t_2 (sqrt (/ (+ x -1.0) (+ 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 = t_m * (sqrt(x) / l_m);
	double tmp;
	if (t_m <= 3.3e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + ((-1.0 - (0.5 / pow((t_m / l_m), 2.0))) / x);
	} else if (t_m <= 8.6e-188) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (sqrt(x) / l_m)
    if (t_m <= 3.3d-251) then
        tmp = t_2
    else if (t_m <= 1.01d-201) then
        tmp = 1.0d0 + (((-1.0d0) - (0.5d0 / ((t_m / l_m) ** 2.0d0))) / x)
    else if (t_m <= 8.6d-188) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
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) {
	double t_2 = t_m * (Math.sqrt(x) / l_m);
	double tmp;
	if (t_m <= 3.3e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + ((-1.0 - (0.5 / Math.pow((t_m / l_m), 2.0))) / x);
	} else if (t_m <= 8.6e-188) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

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):
	t_2 = t_m * (math.sqrt(x) / l_m)
	tmp = 0
	if t_m <= 3.3e-251:
		tmp = t_2
	elif t_m <= 1.01e-201:
		tmp = 1.0 + ((-1.0 - (0.5 / math.pow((t_m / l_m), 2.0))) / x)
	elif t_m <= 8.6e-188:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (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(t_m * Float64(sqrt(x) / l_m))
	tmp = 0.0
	if (t_m <= 3.3e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(0.5 / (Float64(t_m / l_m) ^ 2.0))) / x));
	elseif (t_m <= 8.6e-188)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * (sqrt(x) / l_m);
	tmp = 0.0;
	if (t_m <= 3.3e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = 1.0 + ((-1.0 - (0.5 / ((t_m / l_m) ^ 2.0))) / x);
	elseif (t_m <= 8.6e-188)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = 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[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.3e-251], t$95$2, If[LessEqual[t$95$m, 1.01e-201], N[(1.0 + N[(N[(-1.0 - N[(0.5 / N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e-188], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.3e-251 or 1.00999999999999997e-201 < t < 8.59999999999999975e-188

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 6.7%

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

      3. sub-neg 6.7%

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

      4. metadata-eval 6.7%

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

      5. +-commutative 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around inf 10.4%

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

      1. expm1-log1p-u 9.4%

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

      2. expm1-udef 4.0%

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

      3. associate-*r* 4.0%

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

      4. sqrt-unprod 4.0%

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

      5. *-commutative 4.0%

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

   10. Applied egg-rr 4.0%

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

       1. expm1-def 9.4%

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

       2. expm1-log1p 10.4%

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

       3. associate-*r/ 10.4%

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

       4. associate-*l/ 10.4%

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

       5. *-commutative 10.4%

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

       6. associate-*r* 10.4%

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

       7. *-commutative 10.4%

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

       8. associate-*l* 10.4%

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

       9. metadata-eval 10.4%

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

   12. Simplified 10.4%

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

   if 3.3e-251 < t < 1.00999999999999997e-201

   1. Initial program 2.2%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in x around inf 85.2%

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

   6. Taylor expanded in x around inf 0.6%

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

      1. mul-1-neg 0.6%

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

      2. unsub-neg 0.6%

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

      3. unpow2 0.6%

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

      4. rem-square-sqrt 0.6%

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

      5. metadata-eval 0.6%

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

      6. metadata-eval 0.6%

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

      7. *-commutative 0.6%

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

      8. unpow2 0.6%

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

      9. rem-square-sqrt 0.6%

         \[\leadsto 1 - \frac{1 + \frac{{\ell}^{2}}{\color{blue}{2} \cdot {t}^{2}}}{x} \]

   8. Simplified 0.6%

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

      1. clear-num 0.6%

         \[\leadsto 1 - \frac{1 + \color{blue}{\frac{1}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}{x} \]

      2. inv-pow 0.6%

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

      3. *-un-lft-identity 0.6%

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

      4. times-frac 0.6%

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

      5. metadata-eval 0.6%

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

      6. unpow2 0.6%

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

      7. unpow2 0.6%

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

      8. frac-times 62.1%

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

      9. pow2 62.1%

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

   10. Applied egg-rr 62.1%

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

       1. unpow-1 62.1%

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

       2. associate-/r* 62.1%

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

       3. metadata-eval 62.1%

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

   12. Simplified 62.1%

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

   if 8.59999999999999975e-188 < t

   1. Initial program 39.6%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in t around inf 86.7%

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

   6. Taylor expanded in t around 0 87.0%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1 - \frac{0.5}{{\left(\frac{t}{\ell}\right)}^{2}}}{x}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-188}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 2.1× speedup

Math

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

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 (/ (+ x -1.0) (+ 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(((x + -1.0) / (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(((x + (-1.0d0)) / (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(((x + -1.0) / (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(((x + -1.0) / (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(x + -1.0) / 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(((x + -1.0) / (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[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

3. Simplified 35.2%

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

5. Taylor expanded in t around inf 45.4%

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

6. Taylor expanded in t around 0 45.6%

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

7. Final simplification 45.6%

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

Alternative 9: 76.1% 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 35.2%

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

3. Simplified 35.2%

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

5. Taylor expanded in t around inf 45.4%

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

6. Taylor expanded in x around inf 45.4%

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

7. Final simplification 45.4%

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

Alternative 10: 75.4% 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 35.2%

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

3. Simplified 35.2%

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

5. Taylor expanded in t around inf 45.4%

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

6. Taylor expanded in x around inf 45.1%

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

7. Final simplification 45.1%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))))