Toniolo and Linder, Equation (7)

Percentage Accurate: 33.1% → 84.1%

Time: 21.4s

Alternatives: 13

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_m \cdot \sqrt{2}\\ t_4 := t\_2 + {l\_m}^{2}\\ t_5 := t\_4 + t\_4\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(l\_m, t\_3\right) \cdot \sqrt{\frac{1 + x}{x + -1}}, l\_m\right)}{t\_3}}\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \frac{t\_5 + \frac{\left(t\_5 + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_4}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3 (* t_m (sqrt 2.0)))
        (t_4 (+ t_2 (pow l_m 2.0)))
        (t_5 (+ t_4 t_4)))
   (*
    t_s
    (if (<= t_m 9.2e-237)
      (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
      (if (<= t_m 1.15e-203)
        (/
         1.0
         (/
          (hypot (* (hypot l_m t_3) (sqrt (/ (+ 1.0 x) (+ x -1.0)))) l_m)
          t_3))
        (if (<= t_m 1.45e+48)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              t_2
              (/
               (+
                t_5
                (/
                 (+
                  (+ t_5 (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l_m 2.0) x)))
                  (/ t_4 x))
                 x))
               x)))))
          (sqrt (/ (+ x -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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_m * sqrt(2.0);
	double t_4 = t_2 + pow(l_m, 2.0);
	double t_5 = t_4 + t_4;
	double tmp;
	if (t_m <= 9.2e-237) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t_m <= 1.15e-203) {
		tmp = 1.0 / (hypot((hypot(l_m, t_3) * sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	} else if (t_m <= 1.45e+48) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((t_5 + (((t_5 + ((2.0 * (pow(t_m, 2.0) / x)) + (pow(l_m, 2.0) / x))) + (t_4 / x)) / x)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_m * Math.sqrt(2.0);
	double t_4 = t_2 + Math.pow(l_m, 2.0);
	double t_5 = t_4 + t_4;
	double tmp;
	if (t_m <= 9.2e-237) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else if (t_m <= 1.15e-203) {
		tmp = 1.0 / (Math.hypot((Math.hypot(l_m, t_3) * Math.sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	} else if (t_m <= 1.45e+48) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + ((t_5 + (((t_5 + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (Math.pow(l_m, 2.0) / x))) + (t_4 / x)) / x)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_m * math.sqrt(2.0)
	t_4 = t_2 + math.pow(l_m, 2.0)
	t_5 = t_4 + t_4
	tmp = 0
	if t_m <= 9.2e-237:
		tmp = math.sqrt(2.0) * (t_m / (l_m * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	elif t_m <= 1.15e-203:
		tmp = 1.0 / (math.hypot((math.hypot(l_m, t_3) * math.sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3)
	elif t_m <= 1.45e+48:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + ((t_5 + (((t_5 + ((2.0 * (math.pow(t_m, 2.0) / x)) + (math.pow(l_m, 2.0) / x))) + (t_4 / x)) / x)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_m * sqrt(2.0))
	t_4 = Float64(t_2 + (l_m ^ 2.0))
	t_5 = Float64(t_4 + t_4)
	tmp = 0.0
	if (t_m <= 9.2e-237)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t_m <= 1.15e-203)
		tmp = Float64(1.0 / Float64(hypot(Float64(hypot(l_m, t_3) * sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))), l_m) / t_3));
	elseif (t_m <= 1.45e+48)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(t_5 + Float64(Float64(Float64(t_5 + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l_m ^ 2.0) / x))) + Float64(t_4 / x)) / x)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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 = 2.0 * (t_m ^ 2.0);
	t_3 = t_m * sqrt(2.0);
	t_4 = t_2 + (l_m ^ 2.0);
	t_5 = t_4 + t_4;
	tmp = 0.0;
	if (t_m <= 9.2e-237)
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	elseif (t_m <= 1.15e-203)
		tmp = 1.0 / (hypot((hypot(l_m, t_3) * sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	elseif (t_m <= 1.45e+48)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((t_5 + (((t_5 + ((2.0 * ((t_m ^ 2.0) / x)) + ((l_m ^ 2.0) / x))) + (t_4 / x)) / x)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + t$95$4), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.2e-237], 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, 1.15e-203], N[(1.0 / N[(N[Sqrt[N[(N[Sqrt[l$95$m ^ 2 + t$95$3 ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + l$95$m ^ 2], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e+48], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(t$95$5 + N[(N[(N[(t$95$5 + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 9.20000000000000046e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   8. Simplified 18.1%

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

   if 9.20000000000000046e-237 < t < 1.14999999999999996e-203

   1. Initial program 3.0%

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

   3. Simplified 3.0%

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

   6. Applied egg-rr 84.4%

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

   if 1.14999999999999996e-203 < t < 1.4499999999999999e48

   1. Initial program 45.6%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in x around -inf 77.6%

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

   if 1.4499999999999999e48 < t

   1. Initial program 34.5%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in l around 0 98.1%

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

      1. associate-*l* 98.1%

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

      2. +-commutative 98.1%

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

      3. sub-neg 98.1%

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

      4. metadata-eval 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in t around 0 98.3%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}, \ell\right)}{t \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \frac{\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{\left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {l\_m}^{2}\\ t_4 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(l\_m, t\_4\right) \cdot \sqrt{\frac{1 + x}{x + -1}}, l\_m\right)}{t\_4}}\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \frac{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{l\_m}^{2}}{x}\right) + \left(\left(t\_3 + t\_3\right) + \frac{t\_3}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l_m 2.0)))
        (t_4 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 8e-237)
      (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
      (if (<= t_m 1.2e-205)
        (/
         1.0
         (/
          (hypot (* (hypot l_m t_4) (sqrt (/ (+ 1.0 x) (+ x -1.0)))) l_m)
          t_4))
        (if (<= t_m 1.8e+48)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              t_2
              (/
               (+
                (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l_m 2.0) x))
                (+ (+ t_3 t_3) (/ t_3 x)))
               x)))))
          (sqrt (/ (+ x -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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double t_4 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 8e-237) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t_m <= 1.2e-205) {
		tmp = 1.0 / (hypot((hypot(l_m, t_4) * sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_4);
	} else if (t_m <= 1.8e+48) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * (pow(t_m, 2.0) / x)) + (pow(l_m, 2.0) / x)) + ((t_3 + t_3) + (t_3 / x))) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l_m, 2.0);
	double t_4 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 8e-237) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else if (t_m <= 1.2e-205) {
		tmp = 1.0 / (Math.hypot((Math.hypot(l_m, t_4) * Math.sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_4);
	} else if (t_m <= 1.8e+48) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + ((((2.0 * (Math.pow(t_m, 2.0) / x)) + (Math.pow(l_m, 2.0) / x)) + ((t_3 + t_3) + (t_3 / x))) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l_m, 2.0)
	t_4 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 8e-237:
		tmp = math.sqrt(2.0) * (t_m / (l_m * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	elif t_m <= 1.2e-205:
		tmp = 1.0 / (math.hypot((math.hypot(l_m, t_4) * math.sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_4)
	elif t_m <= 1.8e+48:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + ((((2.0 * (math.pow(t_m, 2.0) / x)) + (math.pow(l_m, 2.0) / x)) + ((t_3 + t_3) + (t_3 / x))) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	t_4 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 8e-237)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t_m <= 1.2e-205)
		tmp = Float64(1.0 / Float64(hypot(Float64(hypot(l_m, t_4) * sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))), l_m) / t_4));
	elseif (t_m <= 1.8e+48)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l_m ^ 2.0) / x)) + Float64(Float64(t_3 + t_3) + Float64(t_3 / x))) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l_m ^ 2.0);
	t_4 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 8e-237)
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	elseif (t_m <= 1.2e-205)
		tmp = 1.0 / (hypot((hypot(l_m, t_4) * sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_4);
	elseif (t_m <= 1.8e+48)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * ((t_m ^ 2.0) / x)) + ((l_m ^ 2.0) / x)) + ((t_3 + t_3) + (t_3 / x))) / x))));
	else
		tmp = sqrt(((x + -1.0) / (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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8e-237], 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, 1.2e-205], N[(1.0 / N[(N[Sqrt[N[(N[Sqrt[l$95$m ^ 2 + t$95$4 ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + l$95$m ^ 2], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e+48], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 + t$95$3), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 7.9999999999999999e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   8. Simplified 18.1%

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

   if 7.9999999999999999e-237 < t < 1.2000000000000001e-205

   1. Initial program 3.0%

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

   3. Simplified 3.0%

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

   6. Applied egg-rr 84.4%

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

   if 1.2000000000000001e-205 < t < 1.79999999999999992e48

   1. Initial program 45.6%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in x around -inf 77.1%

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

   if 1.79999999999999992e48 < t

   1. Initial program 34.5%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in l around 0 98.1%

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

      1. associate-*l* 98.1%

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

      2. +-commutative 98.1%

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

      3. sub-neg 98.1%

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

      4. metadata-eval 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in t around 0 98.3%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}, \ell\right)}{t \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{-202}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(l\_m, t\_3\right) \cdot \sqrt{\frac{1 + x}{x + -1}}, l\_m\right)}{t\_3}}\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{t\_2 + {l\_m}^{2}}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 9.5e-237)
      (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
      (if (<= t_m 4.8e-202)
        (/
         1.0
         (/
          (hypot (* (hypot l_m t_3) (sqrt (/ (+ 1.0 x) (+ x -1.0)))) l_m)
          t_3))
        (if (<= t_m 2.1e+48)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              (/ (+ t_2 (pow l_m 2.0)) x)
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))))))
          (sqrt (/ (+ x -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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t_m <= 4.8e-202) {
		tmp = 1.0 / (hypot((hypot(l_m, t_3) * sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	} else if (t_m <= 2.1e+48) {
		tmp = sqrt(2.0) * (t_m / sqrt((((t_2 + pow(l_m, 2.0)) / x) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))))));
	} else {
		tmp = sqrt(((x + -1.0) / (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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else if (t_m <= 4.8e-202) {
		tmp = 1.0 / (Math.hypot((Math.hypot(l_m, t_3) * Math.sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	} else if (t_m <= 2.1e+48) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((t_2 + Math.pow(l_m, 2.0)) / x) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 9.5e-237:
		tmp = math.sqrt(2.0) * (t_m / (l_m * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	elif t_m <= 4.8e-202:
		tmp = 1.0 / (math.hypot((math.hypot(l_m, t_3) * math.sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3)
	elif t_m <= 2.1e+48:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((t_2 + math.pow(l_m, 2.0)) / x) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 9.5e-237)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t_m <= 4.8e-202)
		tmp = Float64(1.0 / Float64(hypot(Float64(hypot(l_m, t_3) * sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))), l_m) / t_3));
	elseif (t_m <= 2.1e+48)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(t_2 + (l_m ^ 2.0)) / x) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x)))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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 = 2.0 * (t_m ^ 2.0);
	t_3 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 9.5e-237)
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	elseif (t_m <= 4.8e-202)
		tmp = 1.0 / (hypot((hypot(l_m, t_3) * sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	elseif (t_m <= 2.1e+48)
		tmp = sqrt(2.0) * (t_m / sqrt((((t_2 + (l_m ^ 2.0)) / x) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))))));
	else
		tmp = sqrt(((x + -1.0) / (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[(2.0 * N[Power[t$95$m, 2.0], $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, 9.5e-237], 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, 4.8e-202], N[(1.0 / N[(N[Sqrt[N[(N[Sqrt[l$95$m ^ 2 + t$95$3 ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + l$95$m ^ 2], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e+48], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 9.4999999999999998e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   8. Simplified 18.1%

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

   if 9.4999999999999998e-237 < t < 4.8000000000000002e-202

   1. Initial program 3.0%

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

   3. Simplified 3.0%

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

   6. Applied egg-rr 84.4%

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

   if 4.8000000000000002e-202 < t < 2.0999999999999998e48

   1. Initial program 45.6%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in x around inf 75.8%

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

   if 2.0999999999999998e48 < t

   1. Initial program 34.5%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in l around 0 98.1%

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

      1. associate-*l* 98.1%

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

      2. +-commutative 98.1%

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

      3. sub-neg 98.1%

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

      4. metadata-eval 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in t around 0 98.3%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-202}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}, \ell\right)}{t \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{l\_m}^{2}}{x}\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(l\_m, t\_3\right) \cdot \sqrt{\frac{1 + x}{x + -1}}, l\_m\right)}{t\_3}}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(2 \cdot {t\_m}^{2} + t\_2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (pow l_m 2.0) x)) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 9.5e-237)
      (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
      (if (<= t_m 1.9e-205)
        (/
         1.0
         (/
          (hypot (* (hypot l_m t_3) (sqrt (/ (+ 1.0 x) (+ x -1.0)))) l_m)
          t_3))
        (if (<= t_m 1.6e+48)
          (*
           (sqrt 2.0)
           (/
            t_m
            (sqrt
             (+
              t_2
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ (* 2.0 (pow t_m 2.0)) t_2))))))
          (sqrt (/ (+ x -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) {
	double t_2 = pow(l_m, 2.0) / x;
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t_m <= 1.9e-205) {
		tmp = 1.0 / (hypot((hypot(l_m, t_3) * sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	} else if (t_m <= 1.6e+48) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((2.0 * (pow(t_m, 2.0) / x)) + ((2.0 * pow(t_m, 2.0)) + t_2)))));
	} else {
		tmp = sqrt(((x + -1.0) / (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 t_2 = Math.pow(l_m, 2.0) / x;
	double t_3 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else if (t_m <= 1.9e-205) {
		tmp = 1.0 / (Math.hypot((Math.hypot(l_m, t_3) * Math.sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	} else if (t_m <= 1.6e+48) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + ((2.0 * (Math.pow(t_m, 2.0) / x)) + ((2.0 * Math.pow(t_m, 2.0)) + t_2)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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.pow(l_m, 2.0) / x
	t_3 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 9.5e-237:
		tmp = math.sqrt(2.0) * (t_m / (l_m * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	elif t_m <= 1.9e-205:
		tmp = 1.0 / (math.hypot((math.hypot(l_m, t_3) * math.sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3)
	elif t_m <= 1.6e+48:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + ((2.0 * (math.pow(t_m, 2.0) / x)) + ((2.0 * math.pow(t_m, 2.0)) + t_2)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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((l_m ^ 2.0) / x)
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 9.5e-237)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t_m <= 1.9e-205)
		tmp = Float64(1.0 / Float64(hypot(Float64(hypot(l_m, t_3) * sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0)))), l_m) / t_3));
	elseif (t_m <= 1.6e+48)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(Float64(2.0 * (t_m ^ 2.0)) + t_2))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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 = (l_m ^ 2.0) / x;
	t_3 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 9.5e-237)
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	elseif (t_m <= 1.9e-205)
		tmp = 1.0 / (hypot((hypot(l_m, t_3) * sqrt(((1.0 + x) / (x + -1.0)))), l_m) / t_3);
	elseif (t_m <= 1.6e+48)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((2.0 * ((t_m ^ 2.0) / x)) + ((2.0 * (t_m ^ 2.0)) + t_2)))));
	else
		tmp = sqrt(((x + -1.0) / (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[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-237], 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, 1.9e-205], N[(1.0 / N[(N[Sqrt[N[(N[Sqrt[l$95$m ^ 2 + t$95$3 ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + l$95$m ^ 2], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+48], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 9.4999999999999998e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   8. Simplified 18.1%

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

   if 9.4999999999999998e-237 < t < 1.89999999999999996e-205

   1. Initial program 3.0%

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

   3. Simplified 3.0%

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

   6. Applied egg-rr 84.4%

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

   if 1.89999999999999996e-205 < t < 1.6000000000000001e48

   1. Initial program 45.6%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in x around inf 75.8%

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

   6. Taylor expanded in t around 0 74.9%

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

   if 1.6000000000000001e48 < t

   1. Initial program 34.5%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in l around 0 98.1%

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

      1. associate-*l* 98.1%

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

      2. +-commutative 98.1%

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

      3. sub-neg 98.1%

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

      4. metadata-eval 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in t around 0 98.3%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + -1}}, \ell\right)}{t \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-237)
    (* (sqrt 2.0) (/ t_m (* l_m (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
    (sqrt (/ (+ x -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) {
	double tmp;
	if (t_m <= 3.3e-237) {
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else {
		tmp = sqrt(((x + -1.0) / (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 <= 3.3d-237) then
        tmp = sqrt(2.0d0) * (t_m / (l_m * (sqrt(2.0d0) * sqrt((1.0d0 / x)))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (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 <= 3.3e-237) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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-237:
		tmp = math.sqrt(2.0) * (t_m / (l_m * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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-237)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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-237)
		tmp = sqrt(2.0) * (t_m / (l_m * (sqrt(2.0) * sqrt((1.0 / x)))));
	else
		tmp = sqrt(((x + -1.0) / (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-237], 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[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.3000000000000001e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   8. Simplified 18.1%

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

   if 3.3000000000000001e-237 < t

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in l around 0 77.1%

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

      1. associate-*l* 77.1%

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

      2. +-commutative 77.1%

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

      3. sub-neg 77.1%

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

      4. metadata-eval 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_m \cdot \left(\frac{1}{l\_m} \cdot \frac{1}{\sqrt{\frac{2}{x}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.2e-237)
    (* (sqrt 2.0) (* t_m (* (/ 1.0 l_m) (/ 1.0 (sqrt (/ 2.0 x))))))
    (sqrt (/ (+ x -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) {
	double tmp;
	if (t_m <= 3.2e-237) {
		tmp = sqrt(2.0) * (t_m * ((1.0 / l_m) * (1.0 / sqrt((2.0 / x)))));
	} else {
		tmp = sqrt(((x + -1.0) / (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 <= 3.2d-237) then
        tmp = sqrt(2.0d0) * (t_m * ((1.0d0 / l_m) * (1.0d0 / sqrt((2.0d0 / x)))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (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 <= 3.2e-237) {
		tmp = Math.sqrt(2.0) * (t_m * ((1.0 / l_m) * (1.0 / Math.sqrt((2.0 / x)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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 <= 3.2e-237:
		tmp = math.sqrt(2.0) * (t_m * ((1.0 / l_m) * (1.0 / math.sqrt((2.0 / x)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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 <= 3.2e-237)
		tmp = Float64(sqrt(2.0) * Float64(t_m * Float64(Float64(1.0 / l_m) * Float64(1.0 / sqrt(Float64(2.0 / x))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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 <= 3.2e-237)
		tmp = sqrt(2.0) * (t_m * ((1.0 / l_m) * (1.0 / sqrt((2.0 / x)))));
	else
		tmp = sqrt(((x + -1.0) / (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, 3.2e-237], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.2e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

      1. div-inv 18.1%

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

      2. associate-*l* 18.1%

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

      3. sqrt-prod 18.0%

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

      4. div-inv 18.0%

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

   8. Applied egg-rr 18.0%

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

      1. inv-pow 18.0%

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

      2. unpow-prod-down 18.1%

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

      3. inv-pow 18.1%

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

   10. Applied egg-rr 18.1%

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

       1. unpow-1 18.1%

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

   12. Simplified 18.1%

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

   if 3.2e-237 < t

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in l around 0 77.1%

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

      1. associate-*l* 77.1%

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

      2. +-commutative 77.1%

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

      3. sub-neg 77.1%

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

      4. metadata-eval 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \left(\frac{1}{\ell} \cdot \frac{1}{\sqrt{\frac{2}{x}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{l\_m \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.5e-237)
    (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (/ 2.0 x))))
    (sqrt (/ (+ x -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) {
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt((2.0 / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (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 <= 9.5d-237) then
        tmp = (t_m * sqrt(2.0d0)) / (l_m * sqrt((2.0d0 / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (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 <= 9.5e-237) {
		tmp = (t_m * Math.sqrt(2.0)) / (l_m * Math.sqrt((2.0 / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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 <= 9.5e-237:
		tmp = (t_m * math.sqrt(2.0)) / (l_m * math.sqrt((2.0 / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (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 <= 9.5e-237)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(Float64(2.0 / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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 <= 9.5e-237)
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt((2.0 / x)));
	else
		tmp = sqrt(((x + -1.0) / (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, 9.5e-237], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.4999999999999998e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

      1. associate-*r/ 18.1%

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

      2. associate-*l* 18.1%

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

      3. sqrt-prod 18.1%

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

      4. div-inv 18.1%

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

   8. Applied egg-rr 18.1%

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

   if 9.4999999999999998e-237 < t

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in l around 0 77.1%

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

      1. associate-*l* 77.1%

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

      2. +-commutative 77.1%

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

      3. sub-neg 77.1%

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

      4. metadata-eval 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.9e-237)
    (* (sqrt 2.0) (/ t_m (* l_m (sqrt (/ 2.0 x)))))
    (sqrt (/ (+ x -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) {
	double tmp;
	if (t_m <= 3.9e-237) {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt((2.0 / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (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 <= 3.9d-237) then
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt((2.0d0 / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (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 <= 3.9e-237) {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt((2.0 / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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 <= 3.9e-237:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt((2.0 / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (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 <= 3.9e-237)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(2.0 / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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 <= 3.9e-237)
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt((2.0 / x))));
	else
		tmp = sqrt(((x + -1.0) / (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, 3.9e-237], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.8999999999999998e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

      1. *-un-lft-identity 18.1%

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

      2. associate-*l* 18.1%

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

      3. sqrt-prod 18.1%

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

      4. div-inv 18.1%

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

   8. Applied egg-rr 18.1%

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

      1. *-lft-identity 18.1%

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

   10. Simplified 18.1%

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

   if 3.8999999999999998e-237 < t

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in l around 0 77.1%

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

      1. associate-*l* 77.1%

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

      2. +-commutative 77.1%

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

      3. sub-neg 77.1%

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

      4. metadata-eval 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.9 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-237}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-237)
    (* (/ t_m l_m) (sqrt x))
    (sqrt (/ (+ x -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) {
	double tmp;
	if (t_m <= 8e-237) {
		tmp = (t_m / l_m) * sqrt(x);
	} else {
		tmp = sqrt(((x + -1.0) / (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 <= 8d-237) then
        tmp = (t_m / l_m) * sqrt(x)
    else
        tmp = sqrt(((x + (-1.0d0)) / (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 <= 8e-237) {
		tmp = (t_m / l_m) * Math.sqrt(x);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (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 <= 8e-237:
		tmp = (t_m / l_m) * math.sqrt(x)
	else:
		tmp = math.sqrt(((x + -1.0) / (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 <= 8e-237)
		tmp = Float64(Float64(t_m / l_m) * sqrt(x));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / 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 <= 8e-237)
		tmp = (t_m / l_m) * sqrt(x);
	else
		tmp = sqrt(((x + -1.0) / (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, 8e-237], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.9999999999999999e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

   7. Taylor expanded in t around 0 14.9%

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

   if 7.9999999999999999e-237 < t

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in l around 0 77.1%

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

      1. associate-*l* 77.1%

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

      2. +-commutative 77.1%

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

      3. sub-neg 77.1%

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

      4. metadata-eval 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-237}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-237}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 - x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.7e-237)
    (* (/ t_m l_m) (sqrt x))
    (+ 1.0 (/ (/ (- 0.5 x) x) 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 <= 3.7e-237) {
		tmp = (t_m / l_m) * sqrt(x);
	} else {
		tmp = 1.0 + (((0.5 - x) / x) / 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 <= 3.7d-237) then
        tmp = (t_m / l_m) * sqrt(x)
    else
        tmp = 1.0d0 + (((0.5d0 - x) / x) / 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 <= 3.7e-237) {
		tmp = (t_m / l_m) * Math.sqrt(x);
	} else {
		tmp = 1.0 + (((0.5 - x) / x) / 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 <= 3.7e-237:
		tmp = (t_m / l_m) * math.sqrt(x)
	else:
		tmp = 1.0 + (((0.5 - x) / x) / 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 <= 3.7e-237)
		tmp = Float64(Float64(t_m / l_m) * sqrt(x));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(0.5 - x) / x) / 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 <= 3.7e-237)
		tmp = (t_m / l_m) * sqrt(x);
	else
		tmp = 1.0 + (((0.5 - x) / x) / 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, 3.7e-237], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(0.5 - x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.7000000000000001e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in x around inf 52.8%

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

   6. Taylor expanded in l around inf 18.1%

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

   7. Taylor expanded in t around 0 14.9%

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

   if 3.7000000000000001e-237 < t

   1. Initial program 38.3%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in l around 0 77.1%

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

      1. associate-*l* 77.1%

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

      2. +-commutative 77.1%

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

      3. sub-neg 77.1%

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

      4. metadata-eval 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   10. Simplified 77.2%

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

   11. Taylor expanded in x around 0 77.2%

       \[\leadsto 1 - \frac{\color{blue}{\frac{x - 0.5}{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-237}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{0.5 - x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.3% accurate, 25.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{\frac{0.5 - x}{x}}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ 1.0 (/ (/ (- 0.5 x) x) 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 + (((0.5 - x) / x) / 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 + (((0.5d0 - x) / x) / 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 + (((0.5 - x) / x) / 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 + (((0.5 - x) / x) / 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(Float64(Float64(0.5 - x) / x) / 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 + (((0.5 - x) / x) / 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[(N[(N[(0.5 - x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in l around 0 36.4%

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

   1. associate-*l* 36.4%

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

   2. +-commutative 36.4%

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

   3. sub-neg 36.4%

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

   4. metadata-eval 36.4%

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

   5. +-commutative 36.4%

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

7. Simplified 36.4%

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

8. Taylor expanded in x around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unsub-neg 0.0%

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

10. Simplified 36.4%

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

11. Taylor expanded in x around 0 36.4%

    \[\leadsto 1 - \frac{\color{blue}{\frac{x - 0.5}{x}}}{x} \]

12. Final simplification 36.4%

    \[\leadsto 1 + \frac{\frac{0.5 - x}{x}}{x} \]
13. Add Preprocessing

Alternative 12: 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 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in l around 0 36.4%

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

   1. associate-*l* 36.4%

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

   2. +-commutative 36.4%

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

   3. sub-neg 36.4%

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

   4. metadata-eval 36.4%

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

   5. +-commutative 36.4%

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

7. Simplified 36.4%

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

8. Taylor expanded in x around inf 36.4%

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

9. Final simplification 36.4%

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

Alternative 13: 75.5% 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 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in l around 0 36.4%

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

   1. associate-*l* 36.4%

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

   2. +-commutative 36.4%

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

   3. sub-neg 36.4%

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

   4. metadata-eval 36.4%

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

   5. +-commutative 36.4%

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

7. Simplified 36.4%

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

8. Taylor expanded in x around inf 36.1%

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

Reproduce

herbie shell --seed 2024132 
(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)))))