Toniolo and Linder, Equation (7)

Percentage Accurate: 32.6% → 83.6%

Time: 25.6s

Alternatives: 12

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, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := \frac{1 + x}{-1 + x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{l\_m \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{-0.5}\right)}{t\_2}}\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(2 \cdot {t\_m}^{2} + {l\_m}^{2}\right)}{t\_m \cdot \left(x \cdot \sqrt{2}\right)}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 6.6 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(2, {t\_m}^{2} \cdot t\_3, {l\_m}^{2} \cdot \left(\frac{1}{x} + \frac{1}{-1 + x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot t\_3}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))) (t_3 (/ (+ 1.0 x) (+ -1.0 x))))
   (*
    t_s
    (if (<= t_m 2.9e-251)
      (/ 1.0 (/ (* l_m (hypot (pow (+ -1.0 x) -0.5) (pow x -0.5))) t_2))
      (if (<= t_m 1.01e-201)
        (*
         (sqrt 2.0)
         (/
          t_m
          (fma
           0.5
           (/
            (* 2.0 (+ (* 2.0 (pow t_m 2.0)) (pow l_m 2.0)))
            (* t_m (* x (sqrt 2.0))))
           t_2)))
        (if (<= t_m 6.5e-188)
          (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
          (if (<= t_m 6.6e-168)
            1.0
            (if (<= t_m 4.2e-43)
              (*
               (sqrt 2.0)
               (/
                t_m
                (sqrt
                 (fma
                  2.0
                  (* (pow t_m 2.0) t_3)
                  (* (pow l_m 2.0) (+ (/ 1.0 x) (/ 1.0 (+ -1.0 x))))))))
              (* (sqrt 2.0) (/ 1.0 (sqrt (* 2.0 t_3))))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = (1.0 + x) / (-1.0 + x);
	double tmp;
	if (t_m <= 2.9e-251) {
		tmp = 1.0 / ((l_m * hypot(pow((-1.0 + x), -0.5), pow(x, -0.5))) / t_2);
	} else if (t_m <= 1.01e-201) {
		tmp = sqrt(2.0) * (t_m / fma(0.5, ((2.0 * ((2.0 * pow(t_m, 2.0)) + pow(l_m, 2.0))) / (t_m * (x * sqrt(2.0)))), t_2));
	} else if (t_m <= 6.5e-188) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t_m <= 6.6e-168) {
		tmp = 1.0;
	} else if (t_m <= 4.2e-43) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) * t_3), (pow(l_m, 2.0) * ((1.0 / x) + (1.0 / (-1.0 + x)))))));
	} else {
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * t_3)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(Float64(1.0 + x) / Float64(-1.0 + x))
	tmp = 0.0
	if (t_m <= 2.9e-251)
		tmp = Float64(1.0 / Float64(Float64(l_m * hypot((Float64(-1.0 + x) ^ -0.5), (x ^ -0.5))) / t_2));
	elseif (t_m <= 1.01e-201)
		tmp = Float64(sqrt(2.0) * Float64(t_m / fma(0.5, Float64(Float64(2.0 * Float64(Float64(2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0))) / Float64(t_m * Float64(x * sqrt(2.0)))), t_2)));
	elseif (t_m <= 6.5e-188)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t_m <= 6.6e-168)
		tmp = 1.0;
	elseif (t_m <= 4.2e-43)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) * t_3), Float64((l_m ^ 2.0) * Float64(Float64(1.0 / x) + Float64(1.0 / Float64(-1.0 + x))))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 / sqrt(Float64(2.0 * t_3))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-251], N[(1.0 / N[(N[(l$95$m * N[Sqrt[N[Power[N[(-1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] ^ 2 + N[Power[x, -0.5], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.01e-201], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(0.5 * N[(N[(2.0 * N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e-188], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.6e-168], 1.0, If[LessEqual[t$95$m, 4.2e-43], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 6 regimes

2. if t < 2.9000000000000001e-251

   1. Initial program 35.4%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in l around inf 3.2%

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

      1. associate--l+ 6.4%

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

      2. sub-neg 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. sub-neg 6.4%

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

      6. metadata-eval 6.4%

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

      7. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 8.3%

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

      1. associate-*r/ 8.2%

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

      2. *-commutative 8.2%

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

      3. expm1-log1p-u 7.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}} \]

      4. clear-num 7.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}} \]

      5. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}}} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      6. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}} + \color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      7. hypot-define 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \color{blue}{\mathsf{hypot}\left(\sqrt{\frac{1}{-1 + x}}, \sqrt{\frac{1}{x}}\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      8. inv-pow 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\sqrt{\color{blue}{{\left(-1 + x\right)}^{-1}}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      9. sqrt-pow1 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\color{blue}{{\left(-1 + x\right)}^{\left(\frac{-1}{2}\right)}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      10. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      11. inv-pow 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \sqrt{\color{blue}{{x}^{-1}}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      12. sqrt-pow1 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      13. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{\color{blue}{-0.5}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      14. expm1-log1p-u 8.2%

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

   10. Applied egg-rr 8.2%

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

   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%

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

   5. Taylor expanded in l around 0 9.9%

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

      1. fma-define 9.9%

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

      2. +-commutative 9.9%

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

      3. associate-*r/ 9.9%

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

      4. sub-neg 9.9%

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

      5. metadata-eval 9.9%

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

      6. +-commutative 9.9%

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

      7. associate--l+ 28.1%

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

      8. sub-neg 28.1%

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

      9. metadata-eval 28.1%

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

      10. +-commutative 28.1%

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

      11. sub-neg 28.1%

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

      12. metadata-eval 28.1%

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

      13. +-commutative 28.1%

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

   7. Simplified 28.1%

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

   8. Taylor expanded in x around inf 40.1%

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

   9. Taylor expanded in x around inf 85.3%

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

       1. fma-define 85.3%

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

       2. distribute-lft-out 85.3%

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

       3. cancel-sign-sub-inv 85.3%

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

       4. metadata-eval 85.3%

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

       5. distribute-rgt1-in 85.3%

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

       6. metadata-eval 85.3%

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

   11. Simplified 85.3%

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

   if 1.00999999999999997e-201 < t < 6.4999999999999998e-188

   1. Initial program 2.4%

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

   3. Simplified 2.4%

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

   5. Taylor expanded in l around 0 2.4%

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

      1. fma-define 2.4%

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

      2. +-commutative 2.4%

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

      3. associate-*r/ 2.4%

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

      4. sub-neg 2.4%

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

      5. metadata-eval 2.4%

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

      6. +-commutative 2.4%

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

      7. associate--l+ 20.5%

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

      8. sub-neg 20.5%

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

      9. metadata-eval 20.5%

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

      10. +-commutative 20.5%

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

      11. sub-neg 20.5%

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

      12. metadata-eval 20.5%

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

      13. +-commutative 20.5%

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

   7. Simplified 20.5%

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

   8. Taylor expanded in x around inf 99.0%

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

   9. Taylor expanded in t around 0 75.2%

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

       1. associate-*l* 75.2%

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

   11. Simplified 75.2%

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

   if 6.4999999999999998e-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%

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

   5. Taylor expanded in l around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. 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%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in l around 0 52.4%

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

      1. fma-define 52.4%

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

      2. +-commutative 52.4%

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

      3. associate-*r/ 52.4%

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

      4. sub-neg 52.4%

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

      5. metadata-eval 52.4%

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

      6. +-commutative 52.4%

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

      7. associate--l+ 61.7%

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

      8. sub-neg 61.7%

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

      9. metadata-eval 61.7%

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

      10. +-commutative 61.7%

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

      11. sub-neg 61.7%

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

      12. metadata-eval 61.7%

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

      13. +-commutative 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around inf 81.3%

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

   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 41.1%

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

   5. Taylor expanded in l around 0 93.6%

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

      1. +-commutative 93.6%

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

      2. sub-neg 93.6%

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

      3. metadata-eval 93.6%

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

      4. +-commutative 93.6%

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

   7. Simplified 93.6%

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

      1. associate-*r/ 93.8%

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

      2. associate-*l* 93.8%

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

      3. sqrt-unprod 93.8%

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

      4. +-commutative 93.8%

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

   9. Applied egg-rr 93.8%

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

       1. associate-/l* 93.5%

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

       2. associate-/r* 93.8%

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

       3. *-inverses 93.8%

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

   11. Simplified 93.8%

       \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{-0.5}\right)}{t \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{1 + x}{-1 + x}, {\ell}^{2} \cdot \left(\frac{1}{x} + \frac{1}{-1 + x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.4% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (+ 1.0 x) (+ -1.0 x))))
   (*
    t_s
    (if (<= t_m 3.3e-251)
      (/
       1.0
       (/
        (* l_m (hypot (pow (+ -1.0 x) -0.5) (pow x -0.5)))
        (* t_m (sqrt 2.0))))
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 9.5e-161)
          (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
          (if (<= t_m 4.2e-43)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (fma
                2.0
                (* (pow t_m 2.0) t_2)
                (* (pow l_m 2.0) (+ (/ 1.0 x) (/ 1.0 (+ -1.0 x))))))))
            (* (sqrt 2.0) (/ 1.0 (sqrt (* 2.0 t_2)))))))))))

C

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

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(1.0 + x) / Float64(-1.0 + x))
	tmp = 0.0
	if (t_m <= 3.3e-251)
		tmp = Float64(1.0 / Float64(Float64(l_m * hypot((Float64(-1.0 + x) ^ -0.5), (x ^ -0.5))) / Float64(t_m * sqrt(2.0))));
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 9.5e-161)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t_m <= 4.2e-43)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) * t_2), Float64((l_m ^ 2.0) * Float64(Float64(1.0 / x) + Float64(1.0 / Float64(-1.0 + x))))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 / sqrt(Float64(2.0 * t_2))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.3e-251], N[(1.0 / N[(N[(l$95$m * N[Sqrt[N[Power[N[(-1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] ^ 2 + N[Power[x, -0.5], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.01e-201], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e-161], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e-43], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 3.3e-251

   1. Initial program 35.4%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in l around inf 3.2%

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

      1. associate--l+ 6.4%

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

      2. sub-neg 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. sub-neg 6.4%

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

      6. metadata-eval 6.4%

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

      7. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 8.3%

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

      1. associate-*r/ 8.2%

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

      2. *-commutative 8.2%

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

      3. expm1-log1p-u 7.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}} \]

      4. clear-num 7.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}} \]

      5. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}}} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      6. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}} + \color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      7. hypot-define 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \color{blue}{\mathsf{hypot}\left(\sqrt{\frac{1}{-1 + x}}, \sqrt{\frac{1}{x}}\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      8. inv-pow 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\sqrt{\color{blue}{{\left(-1 + x\right)}^{-1}}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      9. sqrt-pow1 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\color{blue}{{\left(-1 + x\right)}^{\left(\frac{-1}{2}\right)}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      10. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      11. inv-pow 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \sqrt{\color{blue}{{x}^{-1}}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      12. sqrt-pow1 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      13. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{\color{blue}{-0.5}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      14. expm1-log1p-u 8.2%

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

   10. Applied egg-rr 8.2%

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

   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%

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

   5. Taylor expanded in l around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. sub-neg 61.5%

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

      3. metadata-eval 61.5%

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

      4. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 61.7%

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

   if 1.00999999999999997e-201 < t < 9.4999999999999996e-161

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   5. Taylor expanded in l around 0 2.6%

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

      1. fma-define 2.6%

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

      2. +-commutative 2.6%

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

      3. associate-*r/ 2.6%

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

      4. sub-neg 2.6%

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

      5. metadata-eval 2.6%

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

      6. +-commutative 2.6%

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

      7. associate--l+ 13.9%

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

      8. sub-neg 13.9%

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

      9. metadata-eval 13.9%

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

      10. +-commutative 13.9%

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

      11. sub-neg 13.9%

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

      12. metadata-eval 13.9%

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

      13. +-commutative 13.9%

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

   7. Simplified 13.9%

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

   8. Taylor expanded in x around inf 63.2%

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

   9. Taylor expanded in t around 0 50.3%

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

       1. associate-*l* 50.3%

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

   11. Simplified 50.3%

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

   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%

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

   5. Taylor expanded in l around 0 54.9%

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

      1. fma-define 54.9%

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

      2. +-commutative 54.9%

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

      3. associate-*r/ 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

      6. +-commutative 55.0%

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

      7. associate--l+ 63.8%

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

      8. sub-neg 63.8%

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

      9. metadata-eval 63.8%

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

      10. +-commutative 63.8%

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

      11. sub-neg 63.8%

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

      12. metadata-eval 63.8%

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

      13. +-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in x around inf 80.4%

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

   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 41.1%

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

   5. Taylor expanded in l around 0 93.6%

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

      1. +-commutative 93.6%

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

      2. sub-neg 93.6%

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

      3. metadata-eval 93.6%

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

      4. +-commutative 93.6%

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

   7. Simplified 93.6%

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

      1. associate-*r/ 93.8%

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

      2. associate-*l* 93.8%

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

      3. sqrt-unprod 93.8%

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

      4. +-commutative 93.8%

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

   9. Applied egg-rr 93.8%

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

       1. associate-/l* 93.5%

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

       2. associate-/r* 93.8%

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

       3. *-inverses 93.8%

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

   11. Simplified 93.8%

       \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.3%

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

Alternative 3: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1 + x}{-1 + x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{l\_m \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{-0.5}\right)}{t\_m \cdot \sqrt{2}}}\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(2, {t\_m}^{2} \cdot t\_2, 2 \cdot \frac{{l\_m}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot t\_2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (+ 1.0 x) (+ -1.0 x))))
   (*
    t_s
    (if (<= t_m 2.7e-251)
      (/
       1.0
       (/
        (* l_m (hypot (pow (+ -1.0 x) -0.5) (pow x -0.5)))
        (* t_m (sqrt 2.0))))
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 1.2e-160)
          (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
          (if (<= t_m 4.2e-43)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (fma 2.0 (* (pow t_m 2.0) t_2) (* 2.0 (/ (pow l_m 2.0) x))))))
            (* (sqrt 2.0) (/ 1.0 (sqrt (* 2.0 t_2)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (1.0 + x) / (-1.0 + x);
	double tmp;
	if (t_m <= 2.7e-251) {
		tmp = 1.0 / ((l_m * hypot(pow((-1.0 + x), -0.5), pow(x, -0.5))) / (t_m * sqrt(2.0)));
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.2e-160) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t_m <= 4.2e-43) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) * t_2), (2.0 * (pow(l_m, 2.0) / x)))));
	} else {
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * t_2)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(1.0 + x) / Float64(-1.0 + x))
	tmp = 0.0
	if (t_m <= 2.7e-251)
		tmp = Float64(1.0 / Float64(Float64(l_m * hypot((Float64(-1.0 + x) ^ -0.5), (x ^ -0.5))) / Float64(t_m * sqrt(2.0))));
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 1.2e-160)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t_m <= 4.2e-43)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) * t_2), Float64(2.0 * Float64((l_m ^ 2.0) / x))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 / sqrt(Float64(2.0 * t_2))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.7e-251], N[(1.0 / N[(N[(l$95$m * N[Sqrt[N[Power[N[(-1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] ^ 2 + N[Power[x, -0.5], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.01e-201], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e-160], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e-43], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 2.7000000000000001e-251

   1. Initial program 35.4%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in l around inf 3.2%

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

      1. associate--l+ 6.4%

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

      2. sub-neg 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. sub-neg 6.4%

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

      6. metadata-eval 6.4%

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

      7. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 8.3%

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

      1. associate-*r/ 8.2%

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

      2. *-commutative 8.2%

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

      3. expm1-log1p-u 7.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}} \]

      4. clear-num 7.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}} \]

      5. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}}} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      6. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}} + \color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      7. hypot-define 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \color{blue}{\mathsf{hypot}\left(\sqrt{\frac{1}{-1 + x}}, \sqrt{\frac{1}{x}}\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      8. inv-pow 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\sqrt{\color{blue}{{\left(-1 + x\right)}^{-1}}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      9. sqrt-pow1 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\color{blue}{{\left(-1 + x\right)}^{\left(\frac{-1}{2}\right)}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      10. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      11. inv-pow 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \sqrt{\color{blue}{{x}^{-1}}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      12. sqrt-pow1 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      13. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{\color{blue}{-0.5}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      14. expm1-log1p-u 8.2%

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

   10. Applied egg-rr 8.2%

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

   if 2.7000000000000001e-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%

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

   5. Taylor expanded in l around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. sub-neg 61.5%

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

      3. metadata-eval 61.5%

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

      4. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 61.7%

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

   if 1.00999999999999997e-201 < t < 1.19999999999999995e-160

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   5. Taylor expanded in l around 0 2.6%

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

      1. fma-define 2.6%

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

      2. +-commutative 2.6%

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

      3. associate-*r/ 2.6%

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

      4. sub-neg 2.6%

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

      5. metadata-eval 2.6%

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

      6. +-commutative 2.6%

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

      7. associate--l+ 13.9%

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

      8. sub-neg 13.9%

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

      9. metadata-eval 13.9%

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

      10. +-commutative 13.9%

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

      11. sub-neg 13.9%

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

      12. metadata-eval 13.9%

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

      13. +-commutative 13.9%

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

   7. Simplified 13.9%

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

   8. Taylor expanded in x around inf 63.2%

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

   9. Taylor expanded in t around 0 50.3%

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

       1. associate-*l* 50.3%

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

   11. Simplified 50.3%

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

   if 1.19999999999999995e-160 < 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%

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

   5. Taylor expanded in l around 0 54.9%

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

      1. fma-define 54.9%

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

      2. +-commutative 54.9%

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

      3. associate-*r/ 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

      6. +-commutative 55.0%

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

      7. associate--l+ 63.8%

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

      8. sub-neg 63.8%

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

      9. metadata-eval 63.8%

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

      10. +-commutative 63.8%

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

      11. sub-neg 63.8%

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

      12. metadata-eval 63.8%

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

      13. +-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in x around inf 80.2%

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

   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 41.1%

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

   5. Taylor expanded in l around 0 93.6%

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

      1. +-commutative 93.6%

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

      2. sub-neg 93.6%

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

      3. metadata-eval 93.6%

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

      4. +-commutative 93.6%

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

   7. Simplified 93.6%

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

      1. associate-*r/ 93.8%

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

      2. associate-*l* 93.8%

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

      3. sqrt-unprod 93.8%

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

      4. +-commutative 93.8%

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

   9. Applied egg-rr 93.8%

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

       1. associate-/l* 93.5%

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

       2. associate-/r* 93.8%

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

       3. *-inverses 93.8%

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

   11. Simplified 93.8%

       \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{-0.5}\right)}{t \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{1 + x}{-1 + x}, 2 \cdot \frac{{\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1 + x}{-1 + x}\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{l\_m \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{-0.5}\right)}{t\_3}}\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_3}{\sqrt{2 \cdot \left(\frac{{l\_m}^{2}}{x} + {t\_m}^{2} \cdot t\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot t\_2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (+ 1.0 x) (+ -1.0 x))) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 3.3e-251)
      (/ 1.0 (/ (* l_m (hypot (pow (+ -1.0 x) -0.5) (pow x -0.5))) t_3))
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 1.2e-160)
          (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
          (if (<= t_m 4.2e-43)
            (/
             t_3
             (sqrt (* 2.0 (+ (/ (pow l_m 2.0) x) (* (pow t_m 2.0) t_2)))))
            (* (sqrt 2.0) (/ 1.0 (sqrt (* 2.0 t_2)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (1.0 + x) / (-1.0 + x);
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 3.3e-251) {
		tmp = 1.0 / ((l_m * hypot(pow((-1.0 + x), -0.5), pow(x, -0.5))) / t_3);
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.2e-160) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t_m <= 4.2e-43) {
		tmp = t_3 / sqrt((2.0 * ((pow(l_m, 2.0) / x) + (pow(t_m, 2.0) * t_2))));
	} else {
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * t_2)));
	}
	return t_s * tmp;
}

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 = (1.0 + x) / (-1.0 + x);
	double t_3 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 3.3e-251) {
		tmp = 1.0 / ((l_m * Math.hypot(Math.pow((-1.0 + x), -0.5), Math.pow(x, -0.5))) / t_3);
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 1.2e-160) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else if (t_m <= 4.2e-43) {
		tmp = t_3 / Math.sqrt((2.0 * ((Math.pow(l_m, 2.0) / x) + (Math.pow(t_m, 2.0) * t_2))));
	} else {
		tmp = Math.sqrt(2.0) * (1.0 / Math.sqrt((2.0 * t_2)));
	}
	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 = (1.0 + x) / (-1.0 + x)
	t_3 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 3.3e-251:
		tmp = 1.0 / ((l_m * math.hypot(math.pow((-1.0 + x), -0.5), math.pow(x, -0.5))) / t_3)
	elif t_m <= 1.01e-201:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 1.2e-160:
		tmp = math.sqrt(2.0) * (t_m / (l_m * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	elif t_m <= 4.2e-43:
		tmp = t_3 / math.sqrt((2.0 * ((math.pow(l_m, 2.0) / x) + (math.pow(t_m, 2.0) * t_2))))
	else:
		tmp = math.sqrt(2.0) * (1.0 / math.sqrt((2.0 * t_2)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(1.0 + x) / Float64(-1.0 + x))
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 3.3e-251)
		tmp = Float64(1.0 / Float64(Float64(l_m * hypot((Float64(-1.0 + x) ^ -0.5), (x ^ -0.5))) / t_3));
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 1.2e-160)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t_m <= 4.2e-43)
		tmp = Float64(t_3 / sqrt(Float64(2.0 * Float64(Float64((l_m ^ 2.0) / x) + Float64((t_m ^ 2.0) * t_2)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 / sqrt(Float64(2.0 * t_2))));
	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 = (1.0 + x) / (-1.0 + x);
	t_3 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 3.3e-251)
		tmp = 1.0 / ((l_m * hypot(((-1.0 + x) ^ -0.5), (x ^ -0.5))) / t_3);
	elseif (t_m <= 1.01e-201)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 1.2e-160)
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	elseif (t_m <= 4.2e-43)
		tmp = t_3 / sqrt((2.0 * (((l_m ^ 2.0) / x) + ((t_m ^ 2.0) * t_2))));
	else
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * t_2)));
	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[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.3e-251], N[(1.0 / N[(N[(l$95$m * N[Sqrt[N[Power[N[(-1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] ^ 2 + N[Power[x, -0.5], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.01e-201], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e-160], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e-43], N[(t$95$3 / N[Sqrt[N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[t$95$m, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if t < 3.3e-251

   1. Initial program 35.4%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in l around inf 3.2%

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

      1. associate--l+ 6.4%

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

      2. sub-neg 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. sub-neg 6.4%

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

      6. metadata-eval 6.4%

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

      7. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 8.3%

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

      1. associate-*r/ 8.2%

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

      2. *-commutative 8.2%

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

      3. expm1-log1p-u 7.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}} \]

      4. clear-num 7.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}} \]

      5. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}}} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      6. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}} + \color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      7. hypot-define 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \color{blue}{\mathsf{hypot}\left(\sqrt{\frac{1}{-1 + x}}, \sqrt{\frac{1}{x}}\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      8. inv-pow 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\sqrt{\color{blue}{{\left(-1 + x\right)}^{-1}}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      9. sqrt-pow1 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\color{blue}{{\left(-1 + x\right)}^{\left(\frac{-1}{2}\right)}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      10. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      11. inv-pow 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \sqrt{\color{blue}{{x}^{-1}}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      12. sqrt-pow1 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      13. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{\color{blue}{-0.5}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      14. expm1-log1p-u 8.2%

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

   10. Applied egg-rr 8.2%

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

   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%

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

   5. Taylor expanded in l around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. sub-neg 61.5%

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

      3. metadata-eval 61.5%

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

      4. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 61.7%

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

   if 1.00999999999999997e-201 < t < 1.19999999999999995e-160

   1. Initial program 2.6%

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

   3. Simplified 2.6%

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

   5. Taylor expanded in l around 0 2.6%

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

      1. fma-define 2.6%

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

      2. +-commutative 2.6%

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

      3. associate-*r/ 2.6%

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

      4. sub-neg 2.6%

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

      5. metadata-eval 2.6%

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

      6. +-commutative 2.6%

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

      7. associate--l+ 13.9%

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

      8. sub-neg 13.9%

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

      9. metadata-eval 13.9%

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

      10. +-commutative 13.9%

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

      11. sub-neg 13.9%

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

      12. metadata-eval 13.9%

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

      13. +-commutative 13.9%

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

   7. Simplified 13.9%

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

   8. Taylor expanded in x around inf 63.2%

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

   9. Taylor expanded in t around 0 50.3%

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

       1. associate-*l* 50.3%

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

   11. Simplified 50.3%

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

   if 1.19999999999999995e-160 < 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%

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

   5. Taylor expanded in l around 0 54.9%

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

      1. fma-define 54.9%

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

      2. +-commutative 54.9%

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

      3. associate-*r/ 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

      6. +-commutative 55.0%

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

      7. associate--l+ 63.8%

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

      8. sub-neg 63.8%

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

      9. metadata-eval 63.8%

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

      10. +-commutative 63.8%

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

      11. sub-neg 63.8%

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

      12. metadata-eval 63.8%

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

      13. +-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in x around inf 80.2%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. fma-undefine 80.3%

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

      4. distribute-lft-out 80.3%

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

      5. +-commutative 80.3%

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

   10. Applied egg-rr 80.3%

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

   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 41.1%

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

   5. Taylor expanded in l around 0 93.6%

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

      1. +-commutative 93.6%

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

      2. sub-neg 93.6%

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

      3. metadata-eval 93.6%

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

      4. +-commutative 93.6%

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

   7. Simplified 93.6%

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

      1. associate-*r/ 93.8%

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

      2. associate-*l* 93.8%

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

      3. sqrt-unprod 93.8%

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

      4. +-commutative 93.8%

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

   9. Applied egg-rr 93.8%

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

       1. associate-/l* 93.5%

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

       2. associate-/r* 93.8%

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

       3. *-inverses 93.8%

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

   11. Simplified 93.8%

       \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{-0.5}\right)}{t \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \frac{1 + x}{-1 + x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{l\_m \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{-0.5}\right)}{t\_m \cdot \sqrt{2}}}\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 4.9 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-251)
    (/
     1.0
     (/ (* l_m (hypot (pow (+ -1.0 x) -0.5) (pow x -0.5))) (* t_m (sqrt 2.0))))
    (if (<= t_m 1.01e-201)
      (+ 1.0 (/ -1.0 x))
      (if (<= t_m 4.9e-188)
        (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
        (* (sqrt 2.0) (/ 1.0 (sqrt (* 2.0 (/ (+ 1.0 x) (+ -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) {
	double tmp;
	if (t_m <= 3.3e-251) {
		tmp = 1.0 / ((l_m * hypot(pow((-1.0 + x), -0.5), pow(x, -0.5))) / (t_m * sqrt(2.0)));
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 4.9e-188) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else {
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	}
	return t_s * tmp;
}

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 tmp;
	if (t_m <= 3.3e-251) {
		tmp = 1.0 / ((l_m * Math.hypot(Math.pow((-1.0 + x), -0.5), Math.pow(x, -0.5))) / (t_m * Math.sqrt(2.0)));
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 4.9e-188) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else {
		tmp = Math.sqrt(2.0) * (1.0 / Math.sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	}
	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):
	tmp = 0
	if t_m <= 3.3e-251:
		tmp = 1.0 / ((l_m * math.hypot(math.pow((-1.0 + x), -0.5), math.pow(x, -0.5))) / (t_m * math.sqrt(2.0)))
	elif t_m <= 1.01e-201:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 4.9e-188:
		tmp = math.sqrt(2.0) * (t_m / (l_m * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	else:
		tmp = math.sqrt(2.0) * (1.0 / math.sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 3.3e-251)
		tmp = Float64(1.0 / Float64(Float64(l_m * hypot((Float64(-1.0 + x) ^ -0.5), (x ^ -0.5))) / Float64(t_m * sqrt(2.0))));
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 4.9e-188)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 / sqrt(Float64(2.0 * Float64(Float64(1.0 + x) / Float64(-1.0 + x))))));
	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)
	tmp = 0.0;
	if (t_m <= 3.3e-251)
		tmp = 1.0 / ((l_m * hypot(((-1.0 + x) ^ -0.5), (x ^ -0.5))) / (t_m * sqrt(2.0)));
	elseif (t_m <= 1.01e-201)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 4.9e-188)
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	else
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e-251], N[(1.0 / N[(N[(l$95$m * N[Sqrt[N[Power[N[(-1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] ^ 2 + N[Power[x, -0.5], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.01e-201], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.9e-188], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.3e-251

   1. Initial program 35.4%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in l around inf 3.2%

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

      1. associate--l+ 6.4%

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

      2. sub-neg 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. sub-neg 6.4%

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

      6. metadata-eval 6.4%

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

      7. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 8.3%

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

      1. associate-*r/ 8.2%

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

      2. *-commutative 8.2%

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

      3. expm1-log1p-u 7.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}} \]

      4. clear-num 7.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\ell \cdot \sqrt{\frac{1}{-1 + x} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}}} \]

      5. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}}} + \frac{1}{x}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      6. add-sqr-sqrt 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \sqrt{\sqrt{\frac{1}{-1 + x}} \cdot \sqrt{\frac{1}{-1 + x}} + \color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      7. hypot-define 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \color{blue}{\mathsf{hypot}\left(\sqrt{\frac{1}{-1 + x}}, \sqrt{\frac{1}{x}}\right)}}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      8. inv-pow 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\sqrt{\color{blue}{{\left(-1 + x\right)}^{-1}}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      9. sqrt-pow1 7.4%

         \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left(\color{blue}{{\left(-1 + x\right)}^{\left(\frac{-1}{2}\right)}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      10. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      11. inv-pow 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \sqrt{\color{blue}{{x}^{-1}}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      12. sqrt-pow1 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      13. metadata-eval 7.4%

          \[\leadsto \frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{\color{blue}{-0.5}}\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}} \]

      14. expm1-log1p-u 8.2%

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

   10. Applied egg-rr 8.2%

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

   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%

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

   5. Taylor expanded in l around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. sub-neg 61.5%

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

      3. metadata-eval 61.5%

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

      4. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 61.7%

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

   if 1.00999999999999997e-201 < t < 4.90000000000000004e-188

   1. Initial program 2.4%

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

   3. Simplified 2.4%

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

   5. Taylor expanded in l around 0 2.4%

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

      1. fma-define 2.4%

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

      2. +-commutative 2.4%

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

      3. associate-*r/ 2.4%

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

      4. sub-neg 2.4%

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

      5. metadata-eval 2.4%

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

      6. +-commutative 2.4%

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

      7. associate--l+ 20.5%

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

      8. sub-neg 20.5%

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

      9. metadata-eval 20.5%

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

      10. +-commutative 20.5%

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

      11. sub-neg 20.5%

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

      12. metadata-eval 20.5%

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

      13. +-commutative 20.5%

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

   7. Simplified 20.5%

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

   8. Taylor expanded in x around inf 99.0%

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

   9. Taylor expanded in t around 0 75.2%

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

       1. associate-*l* 75.2%

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

   11. Simplified 75.2%

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

   if 4.90000000000000004e-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 39.6%

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

   5. Taylor expanded in l around 0 86.9%

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

      1. +-commutative 86.9%

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

      2. sub-neg 86.9%

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

      3. metadata-eval 86.9%

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

      4. +-commutative 86.9%

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

   7. Simplified 86.9%

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

      1. associate-*r/ 87.0%

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

      2. associate-*l* 87.0%

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

      3. sqrt-unprod 87.0%

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

      4. +-commutative 87.0%

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

   9. Applied egg-rr 87.0%

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

       1. associate-/l* 86.8%

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

       2. associate-/r* 87.0%

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

       3. *-inverses 87.0%

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

   11. Simplified 87.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{\ell \cdot \mathsf{hypot}\left({\left(-1 + x\right)}^{-0.5}, {x}^{-0.5}\right)}{t \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.15 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.15e-251)
    (* (sqrt 2.0) (/ t_m (* l_m (sqrt (+ (/ 1.0 x) (/ 1.0 (+ -1.0 x)))))))
    (if (<= t_m 1.01e-201)
      (+ 1.0 (/ -1.0 x))
      (if (<= t_m 2.7e-187)
        (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
        (* (sqrt 2.0) (/ 1.0 (sqrt (* 2.0 (/ (+ 1.0 x) (+ -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) {
	double tmp;
	if (t_m <= 3.15e-251) {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 2.7e-187) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else {
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	}
	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) :: tmp
    if (t_m <= 3.15d-251) then
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt(((1.0d0 / x) + (1.0d0 / ((-1.0d0) + x))))))
    else if (t_m <= 1.01d-201) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 2.7d-187) then
        tmp = sqrt(2.0d0) * (t_m / (l_m * (sqrt(2.0d0) * sqrt((1.0d0 / x)))))
    else
        tmp = sqrt(2.0d0) * (1.0d0 / sqrt((2.0d0 * ((1.0d0 + x) / ((-1.0d0) + x)))))
    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 tmp;
	if (t_m <= 3.15e-251) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 2.7e-187) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else {
		tmp = Math.sqrt(2.0) * (1.0 / Math.sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	}
	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):
	tmp = 0
	if t_m <= 3.15e-251:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))))
	elif t_m <= 1.01e-201:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 2.7e-187:
		tmp = math.sqrt(2.0) * (t_m / (l_m * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	else:
		tmp = math.sqrt(2.0) * (1.0 / math.sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 3.15e-251)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(-1.0 + x)))))));
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 2.7e-187)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 / sqrt(Float64(2.0 * Float64(Float64(1.0 + x) / Float64(-1.0 + x))))));
	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)
	tmp = 0.0;
	if (t_m <= 3.15e-251)
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	elseif (t_m <= 1.01e-201)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 2.7e-187)
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	else
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.15e-251], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.01e-201], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.7e-187], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.1499999999999999e-251

   1. Initial program 35.4%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in l around inf 3.2%

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

      1. associate--l+ 6.4%

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

      2. sub-neg 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. sub-neg 6.4%

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

      6. metadata-eval 6.4%

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

      7. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 8.3%

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

   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%

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

   5. Taylor expanded in l around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. sub-neg 61.5%

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

      3. metadata-eval 61.5%

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

      4. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 61.7%

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

   if 1.00999999999999997e-201 < t < 2.7000000000000001e-187

   1. Initial program 2.4%

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

   3. Simplified 2.4%

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

   5. Taylor expanded in l around 0 2.4%

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

      1. fma-define 2.4%

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

      2. +-commutative 2.4%

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

      3. associate-*r/ 2.4%

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

      4. sub-neg 2.4%

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

      5. metadata-eval 2.4%

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

      6. +-commutative 2.4%

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

      7. associate--l+ 20.5%

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

      8. sub-neg 20.5%

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

      9. metadata-eval 20.5%

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

      10. +-commutative 20.5%

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

      11. sub-neg 20.5%

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

      12. metadata-eval 20.5%

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

      13. +-commutative 20.5%

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

   7. Simplified 20.5%

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

   8. Taylor expanded in x around inf 99.0%

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

   9. Taylor expanded in t around 0 75.2%

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

       1. associate-*l* 75.2%

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

   11. Simplified 75.2%

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

   if 2.7000000000000001e-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 39.6%

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

   5. Taylor expanded in l around 0 86.9%

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

      1. +-commutative 86.9%

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

      2. sub-neg 86.9%

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

      3. metadata-eval 86.9%

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

      4. +-commutative 86.9%

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

   7. Simplified 86.9%

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

      1. associate-*r/ 87.0%

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

      2. associate-*l* 87.0%

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

      3. sqrt-unprod 87.0%

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

      4. +-commutative 87.0%

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

   9. Applied egg-rr 87.0%

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

       1. associate-/l* 86.8%

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

       2. associate-/r* 87.0%

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

       3. *-inverses 87.0%

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

   11. Simplified 87.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.15 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-251}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{-188}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2
         (*
          (sqrt 2.0)
          (/ t_m (* l_m (sqrt (+ (/ 1.0 x) (/ 1.0 (+ -1.0 x)))))))))
   (*
    t_s
    (if (<= t_m 3e-251)
      t_2
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 9.5e-188)
          t_2
          (* (sqrt 2.0) (/ 1.0 (sqrt (* 2.0 (/ (+ 1.0 x) (+ -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) {
	double t_2 = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	double tmp;
	if (t_m <= 3e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 9.5e-188) {
		tmp = t_2;
	} else {
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	}
	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 = sqrt(2.0d0) * (t_m / (l_m * sqrt(((1.0d0 / x) + (1.0d0 / ((-1.0d0) + x))))))
    if (t_m <= 3d-251) then
        tmp = t_2
    else if (t_m <= 1.01d-201) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 9.5d-188) then
        tmp = t_2
    else
        tmp = sqrt(2.0d0) * (1.0d0 / sqrt((2.0d0 * ((1.0d0 + x) / ((-1.0d0) + x)))))
    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 = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	double tmp;
	if (t_m <= 3e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 9.5e-188) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(2.0) * (1.0 / Math.sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	}
	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 = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))))
	tmp = 0
	if t_m <= 3e-251:
		tmp = t_2
	elif t_m <= 1.01e-201:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 9.5e-188:
		tmp = t_2
	else:
		tmp = math.sqrt(2.0) * (1.0 / math.sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(-1.0 + x)))))))
	tmp = 0.0
	if (t_m <= 3e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 9.5e-188)
		tmp = t_2;
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 / sqrt(Float64(2.0 * Float64(Float64(1.0 + x) / Float64(-1.0 + x))))));
	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 = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	tmp = 0.0;
	if (t_m <= 3e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 9.5e-188)
		tmp = t_2;
	else
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	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[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e-188], t$95$2, N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9999999999999999e-251 or 1.00999999999999997e-201 < t < 9.50000000000000063e-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%

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

   5. Taylor expanded in l around inf 3.2%

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

      1. associate--l+ 6.7%

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

      2. sub-neg 6.7%

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

      3. metadata-eval 6.7%

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

      4. +-commutative 6.7%

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

      5. sub-neg 6.7%

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

      6. metadata-eval 6.7%

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

      7. +-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around inf 10.4%

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

   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%

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

   5. Taylor expanded in l around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. sub-neg 61.5%

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

      3. metadata-eval 61.5%

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

      4. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 61.7%

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

   if 9.50000000000000063e-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 39.6%

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

   5. Taylor expanded in l around 0 86.9%

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

      1. +-commutative 86.9%

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

      2. sub-neg 86.9%

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

      3. metadata-eval 86.9%

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

      4. +-commutative 86.9%

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

   7. Simplified 86.9%

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

      1. associate-*r/ 87.0%

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

      2. associate-*l* 87.0%

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

      3. sqrt-unprod 87.0%

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

      4. +-commutative 87.0%

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

   9. Applied egg-rr 87.0%

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

       1. associate-/l* 86.8%

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

       2. associate-/r* 87.0%

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

       3. *-inverses 87.0%

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

   11. Simplified 87.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t\_m}{l\_m}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{-188}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) (* (sqrt (* x 0.5)) (/ t_m l_m)))))
   (*
    t_s
    (if (<= t_m 2.9e-251)
      t_2
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 4.8e-188)
          t_2
          (* (sqrt 2.0) (/ 1.0 (sqrt (* 2.0 (/ (+ 1.0 x) (+ -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) {
	double t_2 = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / l_m));
	double tmp;
	if (t_m <= 2.9e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 4.8e-188) {
		tmp = t_2;
	} else {
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	}
	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 = sqrt(2.0d0) * (sqrt((x * 0.5d0)) * (t_m / l_m))
    if (t_m <= 2.9d-251) then
        tmp = t_2
    else if (t_m <= 1.01d-201) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 4.8d-188) then
        tmp = t_2
    else
        tmp = sqrt(2.0d0) * (1.0d0 / sqrt((2.0d0 * ((1.0d0 + x) / ((-1.0d0) + x)))))
    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 = Math.sqrt(2.0) * (Math.sqrt((x * 0.5)) * (t_m / l_m));
	double tmp;
	if (t_m <= 2.9e-251) {
		tmp = t_2;
	} else if (t_m <= 1.01e-201) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 4.8e-188) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(2.0) * (1.0 / Math.sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	}
	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 = math.sqrt(2.0) * (math.sqrt((x * 0.5)) * (t_m / l_m))
	tmp = 0
	if t_m <= 2.9e-251:
		tmp = t_2
	elif t_m <= 1.01e-201:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 4.8e-188:
		tmp = t_2
	else:
		tmp = math.sqrt(2.0) * (1.0 / math.sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * Float64(sqrt(Float64(x * 0.5)) * Float64(t_m / l_m)))
	tmp = 0.0
	if (t_m <= 2.9e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 4.8e-188)
		tmp = t_2;
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 / sqrt(Float64(2.0 * Float64(Float64(1.0 + x) / Float64(-1.0 + x))))));
	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 = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / l_m));
	tmp = 0.0;
	if (t_m <= 2.9e-251)
		tmp = t_2;
	elseif (t_m <= 1.01e-201)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 4.8e-188)
		tmp = t_2;
	else
		tmp = sqrt(2.0) * (1.0 / sqrt((2.0 * ((1.0 + x) / (-1.0 + x)))));
	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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-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, 4.8e-188], t$95$2, N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9000000000000001e-251 or 1.00999999999999997e-201 < t < 4.8e-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%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
   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. *-commutative 10.4%

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

   10. Simplified 10.4%

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

   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%

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

   5. Taylor expanded in l around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. sub-neg 61.5%

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

      3. metadata-eval 61.5%

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

      4. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 61.7%

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

   if 4.8e-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 39.6%

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

   5. Taylor expanded in l around 0 86.9%

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

      1. +-commutative 86.9%

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

      2. sub-neg 86.9%

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

      3. metadata-eval 86.9%

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

      4. +-commutative 86.9%

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

   7. Simplified 86.9%

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

      1. associate-*r/ 87.0%

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

      2. associate-*l* 87.0%

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

      3. sqrt-unprod 87.0%

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

      4. +-commutative 87.0%

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

   9. Applied egg-rr 87.0%

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

       1. associate-/l* 86.8%

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

       2. associate-/r* 87.0%

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

       3. *-inverses 87.0%

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

   11. Simplified 87.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{1}{\sqrt{2 \cdot \frac{1 + x}{-1 + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t\_m}{l\_m}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 1.75 \cdot 10^{-187}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) (* (sqrt (* x 0.5)) (/ t_m l_m)))))
   (*
    t_s
    (if (<= t_m 3.3e-251)
      t_2
      (if (<= t_m 1.01e-201)
        (+ 1.0 (/ -1.0 x))
        (if (<= t_m 1.75e-187) t_2 (sqrt (/ (+ -1.0 x) (+ 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) {
	double t_2 = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / 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 / x);
	} else if (t_m <= 1.75e-187) {
		tmp = t_2;
	} else {
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	}
	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 = sqrt(2.0d0) * (sqrt((x * 0.5d0)) * (t_m / 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) / x)
    else if (t_m <= 1.75d-187) then
        tmp = t_2
    else
        tmp = sqrt((((-1.0d0) + x) / (1.0d0 + x)))
    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 = Math.sqrt(2.0) * (Math.sqrt((x * 0.5)) * (t_m / 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 / x);
	} else if (t_m <= 1.75e-187) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (1.0 + x)));
	}
	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 = math.sqrt(2.0) * (math.sqrt((x * 0.5)) * (t_m / l_m))
	tmp = 0
	if t_m <= 3.3e-251:
		tmp = t_2
	elif t_m <= 1.01e-201:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 1.75e-187:
		tmp = t_2
	else:
		tmp = math.sqrt(((-1.0 + x) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * Float64(sqrt(Float64(x * 0.5)) * Float64(t_m / 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(-1.0 / x));
	elseif (t_m <= 1.75e-187)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + x)));
	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 = sqrt(2.0) * (sqrt((x * 0.5)) * (t_m / 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 / x);
	elseif (t_m <= 1.75e-187)
		tmp = t_2;
	else
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $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[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.75e-187], t$95$2, N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.3e-251 or 1.00999999999999997e-201 < t < 1.74999999999999989e-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%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
   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. *-commutative 10.4%

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

   10. Simplified 10.4%

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

   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%

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

   5. Taylor expanded in l around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. sub-neg 61.5%

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

      3. metadata-eval 61.5%

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

      4. +-commutative 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 61.7%

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

   if 1.74999999999999989e-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 39.6%

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

   5. Taylor expanded in l around 0 86.9%

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

      1. +-commutative 86.9%

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

      2. sub-neg 86.9%

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

      3. metadata-eval 86.9%

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

      4. +-commutative 86.9%

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

   7. Simplified 86.9%

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

   8. 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 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 1.01 \cdot 10^{-201}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.7% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (sqrt (/ (+ -1.0 x) (+ 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 * sqrt(((-1.0 + x) / (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 * sqrt((((-1.0d0) + x) / (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 * Math.sqrt(((-1.0 + x) / (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 * math.sqrt(((-1.0 + x) / (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 * sqrt(Float64(Float64(-1.0 + x) / 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 * sqrt(((-1.0 + x) / (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[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $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 35.1%

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

5. Taylor expanded in l around 0 45.5%

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

   1. +-commutative 45.5%

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

   2. sub-neg 45.5%

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

   3. metadata-eval 45.5%

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

   4. +-commutative 45.5%

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

7. Simplified 45.5%

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

8. Taylor expanded in t around 0 45.6%

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

9. Final simplification 45.6%

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

Alternative 11: 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 35.1%

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

5. Taylor expanded in l around 0 45.5%

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

   1. +-commutative 45.5%

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

   2. sub-neg 45.5%

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

   3. metadata-eval 45.5%

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

   4. +-commutative 45.5%

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

7. Simplified 45.5%

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

8. Taylor expanded in x around inf 45.4%

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

9. Final simplification 45.4%

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

Alternative 12: 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 35.1%

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

5. Taylor expanded in l around 0 45.5%

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

   1. +-commutative 45.5%

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

   2. sub-neg 45.5%

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

   3. metadata-eval 45.5%

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

   4. +-commutative 45.5%

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

7. Simplified 45.5%

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

8. Taylor expanded in x around inf 45.1%

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

9. Final simplification 45.1%

   \[\leadsto 1 \]
10. 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)))))