Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 84.2%

Time: 20.1s

Alternatives: 9

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: 33.7% 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: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {\ell}^{2}\\ t_4 := t_3 + t_3\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.7 \cdot 10^{-171}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_4}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 4 \cdot 10^{+19}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{t_4}{{x}^{2}} + \left(2 \cdot \frac{{t_m}^{2}}{x} + \left(2 \cdot \frac{{t_m}^{2}}{{x}^{3}} + \left(t_2 + \left(\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{{x}^{3}}\right)\right)\right)\right)\right) + \left(\frac{t_3}{x} + \frac{t_3}{{x}^{3}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l 2.0)))
        (t_4 (+ t_3 t_3)))
   (*
    t_s
    (if (<= t_m 3.7e-171)
      (*
       t_m
       (/
        (sqrt 2.0)
        (+ (* 0.5 (/ t_4 (* t_m (* (sqrt 2.0) x)))) (* t_m (sqrt 2.0)))))
      (if (<= t_m 4e+19)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (+
             (/ t_4 (pow x 2.0))
             (+
              (* 2.0 (/ (pow t_m 2.0) x))
              (+
               (* 2.0 (/ (pow t_m 2.0) (pow x 3.0)))
               (+ t_2 (+ (/ (pow l 2.0) x) (/ (pow l 2.0) (pow x 3.0)))))))
            (+ (/ t_3 x) (/ t_3 (pow x 3.0)))))))
        (sqrt (/ (+ -1.0 x) (+ x 1.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 3.7e-171) {
		tmp = t_m * (sqrt(2.0) / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 4e+19) {
		tmp = t_m * (sqrt(2.0) / sqrt((((t_4 / pow(x, 2.0)) + ((2.0 * (pow(t_m, 2.0) / x)) + ((2.0 * (pow(t_m, 2.0) / pow(x, 3.0))) + (t_2 + ((pow(l, 2.0) / x) + (pow(l, 2.0) / pow(x, 3.0))))))) + ((t_3 / x) + (t_3 / pow(x, 3.0))))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 + t_3
    if (t_m <= 3.7d-171) then
        tmp = t_m * (sqrt(2.0d0) / ((0.5d0 * (t_4 / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 4d+19) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((t_4 / (x ** 2.0d0)) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + ((2.0d0 * ((t_m ** 2.0d0) / (x ** 3.0d0))) + (t_2 + (((l ** 2.0d0) / x) + ((l ** 2.0d0) / (x ** 3.0d0))))))) + ((t_3 / x) + (t_3 / (x ** 3.0d0))))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 3.7e-171) {
		tmp = t_m * (Math.sqrt(2.0) / ((0.5 * (t_4 / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 4e+19) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((t_4 / Math.pow(x, 2.0)) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + ((2.0 * (Math.pow(t_m, 2.0) / Math.pow(x, 3.0))) + (t_2 + ((Math.pow(l, 2.0) / x) + (Math.pow(l, 2.0) / Math.pow(x, 3.0))))))) + ((t_3 / x) + (t_3 / Math.pow(x, 3.0))))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l, 2.0)
	t_4 = t_3 + t_3
	tmp = 0
	if t_m <= 3.7e-171:
		tmp = t_m * (math.sqrt(2.0) / ((0.5 * (t_4 / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 4e+19:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((t_4 / math.pow(x, 2.0)) + ((2.0 * (math.pow(t_m, 2.0) / x)) + ((2.0 * (math.pow(t_m, 2.0) / math.pow(x, 3.0))) + (t_2 + ((math.pow(l, 2.0) / x) + (math.pow(l, 2.0) / math.pow(x, 3.0))))))) + ((t_3 / x) + (t_3 / math.pow(x, 3.0))))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	t_4 = Float64(t_3 + t_3)
	tmp = 0.0
	if (t_m <= 3.7e-171)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 4e+19)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(t_4 / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / (x ^ 3.0))) + Float64(t_2 + Float64(Float64((l ^ 2.0) / x) + Float64((l ^ 2.0) / (x ^ 3.0))))))) + Float64(Float64(t_3 / x) + Float64(t_3 / (x ^ 3.0)))))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l ^ 2.0);
	t_4 = t_3 + t_3;
	tmp = 0.0;
	if (t_m <= 3.7e-171)
		tmp = t_m * (sqrt(2.0) / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 4e+19)
		tmp = t_m * (sqrt(2.0) / sqrt((((t_4 / (x ^ 2.0)) + ((2.0 * ((t_m ^ 2.0) / x)) + ((2.0 * ((t_m ^ 2.0) / (x ^ 3.0))) + (t_2 + (((l ^ 2.0) / x) + ((l ^ 2.0) / (x ^ 3.0))))))) + ((t_3 / x) + (t_3 / (x ^ 3.0))))));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := 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, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + t$95$3), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.7e-171], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+19], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(t$95$4 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / x), $MachinePrecision] + N[(t$95$3 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.70000000000000012e-171

   1. Initial program 22.7%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in x around inf 14.4%

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

   if 3.70000000000000012e-171 < t < 4e19

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in x around -inf 83.3%

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

   if 4e19 < t

   1. Initial program 30.9%

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

   3. Simplified 30.9%

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

   4. Taylor expanded in l around 0 94.0%

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

      1. +-commutative 94.0%

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

      2. sub-neg 94.0%

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

      3. metadata-eval 94.0%

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

      4. +-commutative 94.0%

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

   6. Simplified 94.0%

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

      1. associate-/r* 94.0%

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

      2. sqrt-undiv 94.0%

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

      3. metadata-eval 94.0%

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

      4. metadata-eval 94.0%

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

      5. metadata-eval 94.0%

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

      6. +-commutative 94.0%

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

      7. metadata-eval 94.0%

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

      8. sub-neg 94.0%

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

      9. sqrt-div 94.0%

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

      10. sub-neg 94.0%

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

      11. metadata-eval 94.0%

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

      12. +-commutative 94.0%

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

      13. clear-num 94.0%

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

      14. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-171}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+19}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(2 \cdot {t}^{2} + \left(\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{{x}^{3}}\right)\right)\right)\right)\right) + \left(\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{{x}^{3}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 2: 84.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {\ell}^{2}\\ t_4 := t_3 + t_3\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.6 \cdot 10^{-171}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_4}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\frac{t_3}{x} + \left(\frac{t_4}{{x}^{2}} + \left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{\ell}^{2}}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l 2.0)))
        (t_4 (+ t_3 t_3)))
   (*
    t_s
    (if (<= t_m 4.6e-171)
      (*
       t_m
       (/
        (sqrt 2.0)
        (+ (* 0.5 (/ t_4 (* t_m (* (sqrt 2.0) x)))) (* t_m (sqrt 2.0)))))
      (if (<= t_m 1.25e+20)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (/ t_3 x)
            (+
             (/ t_4 (pow x 2.0))
             (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x))))))))
        (sqrt (/ (+ -1.0 x) (+ x 1.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 4.6e-171) {
		tmp = t_m * (sqrt(2.0) / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 1.25e+20) {
		tmp = t_m * (sqrt(2.0) / sqrt(((t_3 / x) + ((t_4 / pow(x, 2.0)) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x)))))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 + t_3
    if (t_m <= 4.6d-171) then
        tmp = t_m * (sqrt(2.0d0) / ((0.5d0 * (t_4 / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 1.25d+20) then
        tmp = t_m * (sqrt(2.0d0) / sqrt(((t_3 / x) + ((t_4 / (x ** 2.0d0)) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x)))))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 4.6e-171) {
		tmp = t_m * (Math.sqrt(2.0) / ((0.5 * (t_4 / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 1.25e+20) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt(((t_3 / x) + ((t_4 / Math.pow(x, 2.0)) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x)))))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l, 2.0)
	t_4 = t_3 + t_3
	tmp = 0
	if t_m <= 4.6e-171:
		tmp = t_m * (math.sqrt(2.0) / ((0.5 * (t_4 / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 1.25e+20:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt(((t_3 / x) + ((t_4 / math.pow(x, 2.0)) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x)))))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	t_4 = Float64(t_3 + t_3)
	tmp = 0.0
	if (t_m <= 4.6e-171)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 1.25e+20)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(t_3 / x) + Float64(Float64(t_4 / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))))))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l ^ 2.0);
	t_4 = t_3 + t_3;
	tmp = 0.0;
	if (t_m <= 4.6e-171)
		tmp = t_m * (sqrt(2.0) / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 1.25e+20)
		tmp = t_m * (sqrt(2.0) / sqrt(((t_3 / x) + ((t_4 / (x ^ 2.0)) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x)))))));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := 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, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + t$95$3), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.6e-171], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+20], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(t$95$3 / x), $MachinePrecision] + N[(N[(t$95$4 / N[Power[x, 2.0], $MachinePrecision]), $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, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.59999999999999956e-171

   1. Initial program 22.7%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in x around inf 14.4%

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

   if 4.59999999999999956e-171 < t < 1.25e20

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in x around -inf 83.5%

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

   if 1.25e20 < t

   1. Initial program 30.9%

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

   3. Simplified 30.9%

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

   4. Taylor expanded in l around 0 94.0%

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

      1. +-commutative 94.0%

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

      2. sub-neg 94.0%

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

      3. metadata-eval 94.0%

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

      4. +-commutative 94.0%

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

   6. Simplified 94.0%

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

      1. associate-/r* 94.0%

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

      2. sqrt-undiv 94.0%

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

      3. metadata-eval 94.0%

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

      4. metadata-eval 94.0%

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

      5. metadata-eval 94.0%

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

      6. +-commutative 94.0%

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

      7. metadata-eval 94.0%

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

      8. sub-neg 94.0%

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

      9. sqrt-div 94.0%

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

      10. sub-neg 94.0%

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

      11. metadata-eval 94.0%

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

      12. +-commutative 94.0%

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

      13. clear-num 94.0%

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

      14. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-171}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 3: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := t_2 + {\ell}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.9 \cdot 10^{-171}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{t_3 + t_3}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\frac{t_3}{x} + \left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))) (t_3 (+ t_2 (pow l 2.0))))
   (*
    t_s
    (if (<= t_m 2.9e-171)
      (*
       t_m
       (/
        (sqrt 2.0)
        (+
         (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 1.6e+20)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (/ t_3 x)
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x)))))))
        (sqrt (/ (+ -1.0 x) (+ x 1.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double tmp;
	if (t_m <= 2.9e-171) {
		tmp = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 1.6e+20) {
		tmp = t_m * (sqrt(2.0) / sqrt(((t_3 / x) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    if (t_m <= 2.9d-171) then
        tmp = t_m * (sqrt(2.0d0) / ((0.5d0 * ((t_3 + t_3) / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 1.6d+20) then
        tmp = t_m * (sqrt(2.0d0) / sqrt(((t_3 / x) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l, 2.0);
	double tmp;
	if (t_m <= 2.9e-171) {
		tmp = t_m * (Math.sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 1.6e+20) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt(((t_3 / x) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l, 2.0)
	tmp = 0
	if t_m <= 2.9e-171:
		tmp = t_m * (math.sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 1.6e+20:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt(((t_3 / x) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.9e-171)
		tmp = Float64(t_m * Float64(sqrt(2.0) / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 1.6e+20)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(t_3 / x) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x)))))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.9e-171)
		tmp = t_m * (sqrt(2.0) / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 1.6e+20)
		tmp = t_m * (sqrt(2.0) / sqrt(((t_3 / x) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))))));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := 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, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-171], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+20], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(t$95$3 / 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, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.8999999999999999e-171

   1. Initial program 22.7%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in x around inf 14.4%

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

   if 2.8999999999999999e-171 < t < 1.6e20

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in x around inf 82.7%

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

   if 1.6e20 < t

   1. Initial program 30.9%

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

   3. Simplified 30.9%

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

   4. Taylor expanded in l around 0 94.0%

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

      1. +-commutative 94.0%

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

      2. sub-neg 94.0%

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

      3. metadata-eval 94.0%

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

      4. +-commutative 94.0%

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

   6. Simplified 94.0%

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

      1. associate-/r* 94.0%

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

      2. sqrt-undiv 94.0%

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

      3. metadata-eval 94.0%

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

      4. metadata-eval 94.0%

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

      5. metadata-eval 94.0%

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

      6. +-commutative 94.0%

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

      7. metadata-eval 94.0%

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

      8. sub-neg 94.0%

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

      9. sqrt-div 94.0%

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

      10. sub-neg 94.0%

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

      11. metadata-eval 94.0%

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

      12. +-commutative 94.0%

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

      13. clear-num 94.0%

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

      14. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-171}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 4: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t_m \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t_m \cdot t_m\right)\right) - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell \cdot \sqrt{\frac{1}{-1 + x} + \left(\frac{1}{x} + \left({x}^{-2} + {x}^{-3}\right)\right)}}{t_m}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        (* t_m (sqrt 2.0))
        (sqrt
         (-
          (* (/ (+ x 1.0) (+ -1.0 x)) (+ (* l l) (* 2.0 (* t_m t_m))))
          (* l l))))
       INFINITY)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (/
     (sqrt 2.0)
     (/
      (*
       l
       (sqrt
        (+ (/ 1.0 (+ -1.0 x)) (+ (/ 1.0 x) (+ (pow x -2.0) (pow x -3.0))))))
      t_m)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (((t_m * sqrt(2.0)) / sqrt(((((x + 1.0) / (-1.0 + x)) * ((l * l) + (2.0 * (t_m * t_m)))) - (l * l)))) <= ((double) INFINITY)) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = sqrt(2.0) / ((l * sqrt(((1.0 / (-1.0 + x)) + ((1.0 / x) + (pow(x, -2.0) + pow(x, -3.0)))))) / t_m);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (((t_m * Math.sqrt(2.0)) / Math.sqrt(((((x + 1.0) / (-1.0 + x)) * ((l * l) + (2.0 * (t_m * t_m)))) - (l * l)))) <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = Math.sqrt(2.0) / ((l * Math.sqrt(((1.0 / (-1.0 + x)) + ((1.0 / x) + (Math.pow(x, -2.0) + Math.pow(x, -3.0)))))) / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if ((t_m * math.sqrt(2.0)) / math.sqrt(((((x + 1.0) / (-1.0 + x)) * ((l * l) + (2.0 * (t_m * t_m)))) - (l * l)))) <= math.inf:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	else:
		tmp = math.sqrt(2.0) / ((l * math.sqrt(((1.0 / (-1.0 + x)) + ((1.0 / x) + (math.pow(x, -2.0) + math.pow(x, -3.0)))))) / t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(-1.0 + x)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t_m * t_m)))) - Float64(l * l)))) <= Inf)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(2.0) / Float64(Float64(l * sqrt(Float64(Float64(1.0 / Float64(-1.0 + x)) + Float64(Float64(1.0 / x) + Float64((x ^ -2.0) + (x ^ -3.0)))))) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (((t_m * sqrt(2.0)) / sqrt(((((x + 1.0) / (-1.0 + x)) * ((l * l) + (2.0 * (t_m * t_m)))) - (l * l)))) <= Inf)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	else
		tmp = sqrt(2.0) / ((l * sqrt(((1.0 / (-1.0 + x)) + ((1.0 / x) + ((x ^ -2.0) + (x ^ -3.0)))))) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], Infinity], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(l * N[Sqrt[N[(N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(N[Power[x, -2.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 36.6%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in l around 0 42.2%

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

      1. +-commutative 42.2%

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

      2. sub-neg 42.2%

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

      3. metadata-eval 42.2%

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

      4. +-commutative 42.2%

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

   6. Simplified 42.2%

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

      1. associate-/r* 42.2%

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

      2. sqrt-undiv 42.2%

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

      3. metadata-eval 42.2%

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

      4. metadata-eval 42.2%

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

      5. metadata-eval 42.2%

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

      6. +-commutative 42.2%

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

      7. metadata-eval 42.2%

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

      8. sub-neg 42.2%

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

      9. sqrt-div 42.2%

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

      10. sub-neg 42.2%

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

      11. metadata-eval 42.2%

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

      12. +-commutative 42.2%

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

      13. clear-num 42.2%

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

      14. +-commutative 42.2%

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

   8. Applied egg-rr 42.2%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in l around inf 5.4%

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

      1. *-commutative 5.4%

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

      2. associate--l+ 24.0%

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

      3. sub-neg 24.0%

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

      4. metadata-eval 24.0%

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

      5. +-commutative 24.0%

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

      6. sub-neg 24.0%

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

      7. metadata-eval 24.0%

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

      8. +-commutative 24.0%

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

   6. Simplified 24.0%

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

   7. Taylor expanded in x around inf 39.5%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{1}{-1 + x} + \color{blue}{\left(\frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{\ell}{t}} \]
   8. Step-by-step derivation

      1. associate-*r/ 42.9%

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

      2. +-commutative 42.9%

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

      3. pow-flip 42.9%

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

      4. metadata-eval 42.9%

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

      5. pow-flip 42.9%

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

      6. metadata-eval 42.9%

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

   9. Applied egg-rr 42.9%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell \cdot \sqrt{\frac{1}{-1 + x} + \left(\frac{1}{x} + \left({x}^{-2} + {x}^{-3}\right)\right)}}{t}}\\ \end{array} \]

Alternative 5: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t_m \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t_m \cdot t_m\right)\right) - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \left(\sqrt{\frac{1}{\frac{1}{-1 + x} + \frac{1}{x}}} \cdot \frac{\sqrt{2}}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        (* t_m (sqrt 2.0))
        (sqrt
         (-
          (* (/ (+ x 1.0) (+ -1.0 x)) (+ (* l l) (* 2.0 (* t_m t_m))))
          (* l l))))
       INFINITY)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (*
     t_m
     (* (sqrt (/ 1.0 (+ (/ 1.0 (+ -1.0 x)) (/ 1.0 x)))) (/ (sqrt 2.0) l))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (((t_m * sqrt(2.0)) / sqrt(((((x + 1.0) / (-1.0 + x)) * ((l * l) + (2.0 * (t_m * t_m)))) - (l * l)))) <= ((double) INFINITY)) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt((1.0 / ((1.0 / (-1.0 + x)) + (1.0 / x)))) * (sqrt(2.0) / l));
	}
	return t_s * tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 36.6%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in l around 0 42.2%

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

      1. +-commutative 42.2%

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

      2. sub-neg 42.2%

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

      3. metadata-eval 42.2%

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

      4. +-commutative 42.2%

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

   6. Simplified 42.2%

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

      1. associate-/r* 42.2%

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

      2. sqrt-undiv 42.2%

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

      3. metadata-eval 42.2%

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

      4. metadata-eval 42.2%

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

      5. metadata-eval 42.2%

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

      6. +-commutative 42.2%

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

      7. metadata-eval 42.2%

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

      8. sub-neg 42.2%

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

      9. sqrt-div 42.2%

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

      10. sub-neg 42.2%

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

      11. metadata-eval 42.2%

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

      12. +-commutative 42.2%

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

      13. clear-num 42.2%

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

      14. +-commutative 42.2%

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

   8. Applied egg-rr 42.2%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in l around inf 5.8%

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

      1. *-commutative 5.8%

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

      2. associate--l+ 24.4%

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

      3. sub-neg 24.4%

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

      4. metadata-eval 24.4%

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

      5. +-commutative 24.4%

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

      6. sub-neg 24.4%

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

      7. metadata-eval 24.4%

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

      8. +-commutative 24.4%

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

   6. Simplified 24.4%

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

   7. Taylor expanded in x around inf 42.8%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{-1 + x} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{\frac{1}{\frac{1}{-1 + x} + \frac{1}{x}}} \cdot \frac{\sqrt{2}}{\ell}\right)\\ \end{array} \]

Alternative 6: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+251} \lor \neg \left(\ell \leq 3.35 \cdot 10^{+285}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{1}{-1 + x} + \frac{1}{x}} \cdot \frac{\ell}{t_m}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 2.2e+173)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (if (or (<= l 6.8e+251) (not (<= l 3.35e+285)))
      (/ (sqrt 2.0) (* (sqrt (+ (/ 1.0 (+ -1.0 x)) (/ 1.0 x))) (/ l t_m)))
      1.0))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 2.2e+173) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else if ((l <= 6.8e+251) || !(l <= 3.35e+285)) {
		tmp = sqrt(2.0) / (sqrt(((1.0 / (-1.0 + x)) + (1.0 / x))) * (l / t_m));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 2.2d+173) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else if ((l <= 6.8d+251) .or. (.not. (l <= 3.35d+285))) then
        tmp = sqrt(2.0d0) / (sqrt(((1.0d0 / ((-1.0d0) + x)) + (1.0d0 / x))) * (l / t_m))
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 2.2e+173) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if ((l <= 6.8e+251) || !(l <= 3.35e+285)) {
		tmp = Math.sqrt(2.0) / (Math.sqrt(((1.0 / (-1.0 + x)) + (1.0 / x))) * (l / t_m));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 2.2e+173:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif (l <= 6.8e+251) or not (l <= 3.35e+285):
		tmp = math.sqrt(2.0) / (math.sqrt(((1.0 / (-1.0 + x)) + (1.0 / x))) * (l / t_m))
	else:
		tmp = 1.0
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 2.2e+173)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	elseif ((l <= 6.8e+251) || !(l <= 3.35e+285))
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(1.0 / Float64(-1.0 + x)) + Float64(1.0 / x))) * Float64(l / t_m)));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 2.2e+173)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	elseif ((l <= 6.8e+251) || ~((l <= 3.35e+285)))
		tmp = sqrt(2.0) / (sqrt(((1.0 / (-1.0 + x)) + (1.0 / x))) * (l / t_m));
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 2.2e+173], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, 6.8e+251], N[Not[LessEqual[l, 3.35e+285]], $MachinePrecision]], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 2.2e173

   1. Initial program 31.6%

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

   3. Simplified 31.6%

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

   4. Taylor expanded in l around 0 39.2%

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

      1. +-commutative 39.2%

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

      2. sub-neg 39.2%

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

      3. metadata-eval 39.2%

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

      4. +-commutative 39.2%

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

   6. Simplified 39.2%

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

      1. associate-/r* 39.2%

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

      2. sqrt-undiv 39.2%

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

      3. metadata-eval 39.2%

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

      4. metadata-eval 39.2%

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

      5. metadata-eval 39.2%

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

      6. +-commutative 39.2%

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

      7. metadata-eval 39.2%

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

      8. sub-neg 39.2%

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

      9. sqrt-div 39.2%

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

      10. sub-neg 39.2%

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

      11. metadata-eval 39.2%

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

      12. +-commutative 39.2%

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

      13. clear-num 39.2%

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

      14. +-commutative 39.2%

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

   8. Applied egg-rr 39.2%

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

   if 2.2e173 < l < 6.80000000000000023e251 or 3.3499999999999998e285 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in l around inf 2.3%

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

      1. *-commutative 2.3%

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

      2. associate--l+ 31.6%

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

      3. sub-neg 31.6%

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

      4. metadata-eval 31.6%

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

      5. +-commutative 31.6%

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

      6. sub-neg 31.6%

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

      7. metadata-eval 31.6%

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

      8. +-commutative 31.6%

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

   6. Simplified 31.6%

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

   7. Taylor expanded in x around inf 92.2%

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

   if 6.80000000000000023e251 < l < 3.3499999999999998e285

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in l around 0 41.2%

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

      1. +-commutative 41.2%

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

      2. sub-neg 41.2%

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

      3. metadata-eval 41.2%

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

      4. +-commutative 41.2%

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

   6. Simplified 41.2%

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

   7. Taylor expanded in x around inf 41.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+251} \lor \neg \left(\ell \leq 3.35 \cdot 10^{+285}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{1}{-1 + x} + \frac{1}{x}} \cdot \frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 76.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

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

3. Simplified 29.4%

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

4. Taylor expanded in l around 0 37.8%

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

   1. +-commutative 37.8%

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

   2. sub-neg 37.8%

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

   3. metadata-eval 37.8%

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

   4. +-commutative 37.8%

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

6. Simplified 37.8%

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

   1. associate-/r* 37.8%

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

   2. sqrt-undiv 37.8%

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

   3. metadata-eval 37.8%

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

   4. metadata-eval 37.8%

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

   5. metadata-eval 37.8%

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

   6. +-commutative 37.8%

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

   7. metadata-eval 37.8%

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

   8. sub-neg 37.8%

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

   9. sqrt-div 37.8%

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

   10. sub-neg 37.8%

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

   11. metadata-eval 37.8%

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

   12. +-commutative 37.8%

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

   13. clear-num 37.8%

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

   14. +-commutative 37.8%

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

8. Applied egg-rr 37.8%

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

9. Final simplification 37.8%

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

Alternative 8: 76.3% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.3%

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

3. Simplified 29.4%

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

4. Taylor expanded in l around 0 37.8%

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

   1. +-commutative 37.8%

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

   2. sub-neg 37.8%

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

   3. metadata-eval 37.8%

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

   4. +-commutative 37.8%

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

6. Simplified 37.8%

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

7. Taylor expanded in x around inf 37.1%

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

8. Final simplification 37.1%

   \[\leadsto 1 + \frac{-1}{x} \]

Alternative 9: 75.7% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 29.3%

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

3. Simplified 29.4%

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

4. Taylor expanded in l around 0 37.8%

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

   1. +-commutative 37.8%

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

   2. sub-neg 37.8%

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

   3. metadata-eval 37.8%

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

   4. +-commutative 37.8%

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

6. Simplified 37.8%

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

7. Taylor expanded in x around inf 36.7%

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

8. Final simplification 36.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023337 
(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)))))