Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 85.2%

Time: 18.9s

Alternatives: 10

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% 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: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_3 + t\_3\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_4}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \frac{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(t\_4 + \frac{t\_3}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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-155)
      (*
       (sqrt 2.0)
       (/ t_m (+ (* 0.5 (/ t_4 (* t_m (* (sqrt 2.0) x)))) (* t_m (sqrt 2.0)))))
      (if (<= t_m 2.5e+50)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (+
            t_2
            (/
             (+
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l 2.0) x))
              (+ t_4 (/ t_3 x)))
             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-155) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 2.5e+50) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * (pow(t_m, 2.0) / x)) + (pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / 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-155) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * (t_4 / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 2.5d+50) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((t_2 + ((((2.0d0 * ((t_m ** 2.0d0) / x)) + ((l ** 2.0d0) / x)) + (t_4 + (t_3 / x))) / 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-155) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 2.5e+50) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + ((((2.0 * (Math.pow(t_m, 2.0) / x)) + (Math.pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / 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-155:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 2.5e+50:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + ((((2.0 * (math.pow(t_m, 2.0) / x)) + (math.pow(l, 2.0) / x)) + (t_4 + (t_3 / x))) / 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-155)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 2.5e+50)
		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 ^ 2.0) / x)) + Float64(t_4 + Float64(t_3 / x))) / 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-155)
		tmp = sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 2.5e+50)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * ((t_m ^ 2.0) / x)) + ((l ^ 2.0) / x)) + (t_4 + (t_3 / x))) / 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-155], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / 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, 2.5e+50], 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, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.60000000000000011e-155

   1. Initial program 28.3%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in x around inf 16.5%

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

   if 4.60000000000000011e-155 < t < 2.5e50

   1. Initial program 61.2%

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

   3. Simplified 61.0%

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

   5. Taylor expanded in x around -inf 92.3%

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

   if 2.5e50 < t

   1. Initial program 26.0%

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

   3. Simplified 26.0%

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

   5. Taylor expanded in t around inf 93.9%

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

   6. Taylor expanded in t around 0 96.2%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-254}:\\ \;\;\;\;{\left(\sqrt{t\_m \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x}\right)}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{-155} \lor \neg \left(t\_m \leq 2.5 \cdot 10^{+29}\right):\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{t\_2 + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 2.1e-254)
      (pow (sqrt (* t_m (* (/ (sqrt 2.0) l) (sqrt (* 0.5 x))))) 2.0)
      (if (or (<= t_m 4.6e-155) (not (<= t_m 2.5e+29)))
        (sqrt (/ (+ -1.0 x) (+ x 1.0)))
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (+
            (/ (+ t_2 (pow l 2.0)) x)
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x))))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.1e-254) {
		tmp = pow(sqrt((t_m * ((sqrt(2.0) / l) * sqrt((0.5 * x))))), 2.0);
	} else if ((t_m <= 4.6e-155) || !(t_m <= 2.5e+29)) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = sqrt(2.0) * (t_m / sqrt((((t_2 + pow(l, 2.0)) / x) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))))));
	}
	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) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 2.1d-254) then
        tmp = sqrt((t_m * ((sqrt(2.0d0) / l) * sqrt((0.5d0 * x))))) ** 2.0d0
    else if ((t_m <= 4.6d-155) .or. (.not. (t_m <= 2.5d+29))) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else
        tmp = sqrt(2.0d0) * (t_m / sqrt((((t_2 + (l ** 2.0d0)) / x) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))))))
    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 tmp;
	if (t_m <= 2.1e-254) {
		tmp = Math.pow(Math.sqrt((t_m * ((Math.sqrt(2.0) / l) * Math.sqrt((0.5 * x))))), 2.0);
	} else if ((t_m <= 4.6e-155) || !(t_m <= 2.5e+29)) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((t_2 + Math.pow(l, 2.0)) / x) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))))));
	}
	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)
	tmp = 0
	if t_m <= 2.1e-254:
		tmp = math.pow(math.sqrt((t_m * ((math.sqrt(2.0) / l) * math.sqrt((0.5 * x))))), 2.0)
	elif (t_m <= 4.6e-155) or not (t_m <= 2.5e+29):
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	else:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((t_2 + math.pow(l, 2.0)) / x) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))))))
	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))
	tmp = 0.0
	if (t_m <= 2.1e-254)
		tmp = sqrt(Float64(t_m * Float64(Float64(sqrt(2.0) / l) * sqrt(Float64(0.5 * x))))) ^ 2.0;
	elseif ((t_m <= 4.6e-155) || !(t_m <= 2.5e+29))
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(t_2 + (l ^ 2.0)) / x) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x)))))));
	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);
	tmp = 0.0;
	if (t_m <= 2.1e-254)
		tmp = sqrt((t_m * ((sqrt(2.0) / l) * sqrt((0.5 * x))))) ^ 2.0;
	elseif ((t_m <= 4.6e-155) || ~((t_m <= 2.5e+29)))
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	else
		tmp = sqrt(2.0) * (t_m / sqrt((((t_2 + (l ^ 2.0)) / x) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))))));
	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]}, N[(t$95$s * If[LessEqual[t$95$m, 2.1e-254], N[Power[N[Sqrt[N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], If[Or[LessEqual[t$95$m, 4.6e-155], N[Not[LessEqual[t$95$m, 2.5e+29]], $MachinePrecision]], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(t$95$2 + N[Power[l, 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, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.09999999999999997e-254

   1. Initial program 30.1%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate-/l* 2.8%

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

      2. associate--l+ 9.9%

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

      3. sub-neg 9.9%

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

      4. metadata-eval 9.9%

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

      5. +-commutative 9.9%

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

      6. sub-neg 9.9%

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

      7. metadata-eval 9.9%

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

      8. +-commutative 9.9%

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

   7. Simplified 9.9%

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

   8. Taylor expanded in x around inf 18.2%

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

      1. *-commutative 18.2%

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

   10. Simplified 18.2%

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

       1. add-sqr-sqrt 7.4%

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

       2. pow2 7.4%

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

       3. associate-*l* 8.0%

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

   12. Applied egg-rr 8.0%

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

   if 2.09999999999999997e-254 < t < 4.60000000000000011e-155 or 2.5e29 < t

   1. Initial program 26.7%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in t around inf 89.3%

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

   6. Taylor expanded in t around 0 91.1%

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

   if 4.60000000000000011e-155 < t < 2.5e29

   1. Initial program 58.9%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in x around inf 90.9%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-254}:\\ \;\;\;\;{\left(\sqrt{t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x}\right)}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-155} \lor \neg \left(t \leq 2.5 \cdot 10^{+29}\right):\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_3 + t\_3}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 5.9 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{t\_3}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) 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 4.6e-155)
      (*
       (sqrt 2.0)
       (/
        t_m
        (+
         (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 5.9e+29)
        (*
         (sqrt 2.0)
         (/
          t_m
          (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 <= 4.6e-155) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 5.9e+29) {
		tmp = sqrt(2.0) * (t_m / 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 <= 4.6d-155) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * ((t_3 + t_3) / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 5.9d+29) then
        tmp = sqrt(2.0d0) * (t_m / 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 <= 4.6e-155) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 5.9e+29) {
		tmp = Math.sqrt(2.0) * (t_m / 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 <= 4.6e-155:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 5.9e+29:
		tmp = math.sqrt(2.0) * (t_m / 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 <= 4.6e-155)
		tmp = Float64(sqrt(2.0) * Float64(t_m / 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 <= 5.9e+29)
		tmp = Float64(sqrt(2.0) * Float64(t_m / 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 <= 4.6e-155)
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 5.9e+29)
		tmp = sqrt(2.0) * (t_m / 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, 4.6e-155], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / 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, 5.9e+29], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / 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 < 4.60000000000000011e-155

   1. Initial program 28.3%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in x around inf 16.5%

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

   if 4.60000000000000011e-155 < t < 5.8999999999999999e29

   1. Initial program 58.9%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in x around inf 90.9%

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

   if 5.8999999999999999e29 < t

   1. Initial program 31.1%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in t around inf 94.1%

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

   6. Taylor expanded in t around 0 96.3%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_m \cdot {\left({2}^{0.25}\right)}^{2}}{t\_m \cdot \sqrt{2} + 0.5 \cdot \frac{2 \cdot \left({t\_m}^{2} + {t\_m}^{2}\right) + 2 \cdot {\ell}^{2}}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t\_m \leq 6.6 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{t\_2 + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 4.6e-155)
      (/
       (* t_m (pow (pow 2.0 0.25) 2.0))
       (+
        (* t_m (sqrt 2.0))
        (*
         0.5
         (/
          (+ (* 2.0 (+ (pow t_m 2.0) (pow t_m 2.0))) (* 2.0 (pow l 2.0)))
          (* t_m (* (sqrt 2.0) x))))))
      (if (<= t_m 6.6e+28)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (+
            (/ (+ t_2 (pow l 2.0)) 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 tmp;
	if (t_m <= 4.6e-155) {
		tmp = (t_m * pow(pow(2.0, 0.25), 2.0)) / ((t_m * sqrt(2.0)) + (0.5 * (((2.0 * (pow(t_m, 2.0) + pow(t_m, 2.0))) + (2.0 * pow(l, 2.0))) / (t_m * (sqrt(2.0) * x)))));
	} else if (t_m <= 6.6e+28) {
		tmp = sqrt(2.0) * (t_m / sqrt((((t_2 + pow(l, 2.0)) / 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) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 4.6d-155) then
        tmp = (t_m * ((2.0d0 ** 0.25d0) ** 2.0d0)) / ((t_m * sqrt(2.0d0)) + (0.5d0 * (((2.0d0 * ((t_m ** 2.0d0) + (t_m ** 2.0d0))) + (2.0d0 * (l ** 2.0d0))) / (t_m * (sqrt(2.0d0) * x)))))
    else if (t_m <= 6.6d+28) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((t_2 + (l ** 2.0d0)) / 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 tmp;
	if (t_m <= 4.6e-155) {
		tmp = (t_m * Math.pow(Math.pow(2.0, 0.25), 2.0)) / ((t_m * Math.sqrt(2.0)) + (0.5 * (((2.0 * (Math.pow(t_m, 2.0) + Math.pow(t_m, 2.0))) + (2.0 * Math.pow(l, 2.0))) / (t_m * (Math.sqrt(2.0) * x)))));
	} else if (t_m <= 6.6e+28) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((t_2 + Math.pow(l, 2.0)) / 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)
	tmp = 0
	if t_m <= 4.6e-155:
		tmp = (t_m * math.pow(math.pow(2.0, 0.25), 2.0)) / ((t_m * math.sqrt(2.0)) + (0.5 * (((2.0 * (math.pow(t_m, 2.0) + math.pow(t_m, 2.0))) + (2.0 * math.pow(l, 2.0))) / (t_m * (math.sqrt(2.0) * x)))))
	elif t_m <= 6.6e+28:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((t_2 + math.pow(l, 2.0)) / 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))
	tmp = 0.0
	if (t_m <= 4.6e-155)
		tmp = Float64(Float64(t_m * ((2.0 ^ 0.25) ^ 2.0)) / Float64(Float64(t_m * sqrt(2.0)) + Float64(0.5 * Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) + (t_m ^ 2.0))) + Float64(2.0 * (l ^ 2.0))) / Float64(t_m * Float64(sqrt(2.0) * x))))));
	elseif (t_m <= 6.6e+28)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(t_2 + (l ^ 2.0)) / 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);
	tmp = 0.0;
	if (t_m <= 4.6e-155)
		tmp = (t_m * ((2.0 ^ 0.25) ^ 2.0)) / ((t_m * sqrt(2.0)) + (0.5 * (((2.0 * ((t_m ^ 2.0) + (t_m ^ 2.0))) + (2.0 * (l ^ 2.0))) / (t_m * (sqrt(2.0) * x)))));
	elseif (t_m <= 6.6e+28)
		tmp = sqrt(2.0) * (t_m / sqrt((((t_2 + (l ^ 2.0)) / 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]}, N[(t$95$s * If[LessEqual[t$95$m, 4.6e-155], N[(N[(t$95$m * N[Power[N[Power[2.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.6e+28], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(t$95$2 + N[Power[l, 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, 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 < 4.60000000000000011e-155

   1. Initial program 28.3%

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

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

      3. pow1/2 28.2%

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

      4. sqrt-pow1 28.2%

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

      5. metadata-eval 28.2%

         \[\leadsto \frac{{\left({2}^{\color{blue}{0.25}}\right)}^{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 28.2%

      \[\leadsto \frac{\color{blue}{{\left({2}^{0.25}\right)}^{2}} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in l around 0 27.9%

      \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)}}} \]
   6. Step-by-step derivation

      1. fma-define 27.9%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}}} \]

      2. associate-/l* 34.5%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \color{blue}{{t}^{2} \cdot \frac{1 + x}{x - 1}}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}} \]

      3. sub-neg 34.5%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{1 + x}{\color{blue}{x + \left(-1\right)}}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}} \]

      4. metadata-eval 34.5%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{1 + x}{x + \color{blue}{-1}}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}} \]

      5. +-commutative 34.5%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{\color{blue}{x + 1}}{x + -1}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}} \]

      6. associate--l+ 42.3%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \color{blue}{\left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)}\right)}} \]

      7. sub-neg 42.3%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{\color{blue}{x + \left(-1\right)}} + \left(\frac{x}{x - 1} - 1\right)\right)\right)}} \]

      8. metadata-eval 42.3%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + \color{blue}{-1}} + \left(\frac{x}{x - 1} - 1\right)\right)\right)}} \]

      9. sub-neg 42.3%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + -1} + \left(\frac{x}{\color{blue}{x + \left(-1\right)}} - 1\right)\right)\right)}} \]

      10. metadata-eval 42.3%

          \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + -1} + \left(\frac{x}{x + \color{blue}{-1}} - 1\right)\right)\right)}} \]

   7. Simplified 42.3%

      \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + -1} + \left(\frac{x}{x + -1} - 1\right)\right)\right)}}} \]

   8. Taylor expanded in x around inf 16.3%

      \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\color{blue}{0.5 \cdot \frac{2 \cdot \left({t}^{2} - -1 \cdot {t}^{2}\right) + 2 \cdot {\ell}^{2}}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}} \]

   if 4.60000000000000011e-155 < t < 6.6e28

   1. Initial program 58.9%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in x around inf 90.9%

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

   if 6.6e28 < t

   1. Initial program 31.1%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in t around inf 94.1%

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

   6. Taylor expanded in t around 0 96.3%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{t \cdot {\left({2}^{0.25}\right)}^{2}}{t \cdot \sqrt{2} + 0.5 \cdot \frac{2 \cdot \left({t}^{2} + {t}^{2}\right) + 2 \cdot {\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.3 \cdot 10^{-253}:\\ \;\;\;\;{\left(\sqrt{t\_m \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x}\right)}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{-155} \lor \neg \left(t\_m \leq 3.55 \cdot 10^{+52}\right):\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot {\left({2}^{0.25}\right)}^{2}}{\sqrt{\mathsf{fma}\left(2, {t\_m}^{2} \cdot \frac{x + 1}{-1 + x}, 2 \cdot \frac{{\ell}^{2}}{x}\right)}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.3e-253)
    (pow (sqrt (* t_m (* (/ (sqrt 2.0) l) (sqrt (* 0.5 x))))) 2.0)
    (if (or (<= t_m 4.6e-155) (not (<= t_m 3.55e+52)))
      (sqrt (/ (+ -1.0 x) (+ x 1.0)))
      (/
       (* t_m (pow (pow 2.0 0.25) 2.0))
       (sqrt
        (fma
         2.0
         (* (pow t_m 2.0) (/ (+ x 1.0) (+ -1.0 x)))
         (* 2.0 (/ (pow l 2.0) x)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 4.3e-253) {
		tmp = pow(sqrt((t_m * ((sqrt(2.0) / l) * sqrt((0.5 * x))))), 2.0);
	} else if ((t_m <= 4.6e-155) || !(t_m <= 3.55e+52)) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = (t_m * pow(pow(2.0, 0.25), 2.0)) / sqrt(fma(2.0, (pow(t_m, 2.0) * ((x + 1.0) / (-1.0 + x))), (2.0 * (pow(l, 2.0) / x))));
	}
	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 (t_m <= 4.3e-253)
		tmp = sqrt(Float64(t_m * Float64(Float64(sqrt(2.0) / l) * sqrt(Float64(0.5 * x))))) ^ 2.0;
	elseif ((t_m <= 4.6e-155) || !(t_m <= 3.55e+52))
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(t_m * ((2.0 ^ 0.25) ^ 2.0)) / sqrt(fma(2.0, Float64((t_m ^ 2.0) * Float64(Float64(x + 1.0) / Float64(-1.0 + x))), Float64(2.0 * Float64((l ^ 2.0) / x)))));
	end
	return Float64(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[t$95$m, 4.3e-253], N[Power[N[Sqrt[N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], If[Or[LessEqual[t$95$m, 4.6e-155], N[Not[LessEqual[t$95$m, 3.55e+52]], $MachinePrecision]], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$m * N[Power[N[Power[2.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.3000000000000002e-253

   1. Initial program 30.1%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. associate-/l* 2.8%

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

      2. associate--l+ 9.9%

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

      3. sub-neg 9.9%

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

      4. metadata-eval 9.9%

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

      5. +-commutative 9.9%

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

      6. sub-neg 9.9%

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

      7. metadata-eval 9.9%

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

      8. +-commutative 9.9%

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

   7. Simplified 9.9%

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

   8. Taylor expanded in x around inf 18.2%

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

      1. *-commutative 18.2%

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

   10. Simplified 18.2%

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

       1. add-sqr-sqrt 7.4%

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

       2. pow2 7.4%

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

       3. associate-*l* 8.0%

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

   12. Applied egg-rr 8.0%

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

   if 4.3000000000000002e-253 < t < 4.60000000000000011e-155 or 3.54999999999999998e52 < t

   1. Initial program 22.5%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in t around inf 88.8%

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

   6. Taylor expanded in t around 0 90.6%

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

   if 4.60000000000000011e-155 < t < 3.54999999999999998e52

   1. Initial program 61.2%

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

      1. add-sqr-sqrt 60.7%

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

      2. pow2 60.7%

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

      3. pow1/2 60.7%

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

      4. sqrt-pow1 60.7%

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

      5. metadata-eval 60.7%

         \[\leadsto \frac{{\left({2}^{\color{blue}{0.25}}\right)}^{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   4. Applied egg-rr 60.7%

      \[\leadsto \frac{\color{blue}{{\left({2}^{0.25}\right)}^{2}} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   5. Taylor expanded in l around 0 71.1%

      \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1} + {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)}}} \]
   6. Step-by-step derivation

      1. fma-define 71.1%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}}} \]

      2. associate-/l* 75.0%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \color{blue}{{t}^{2} \cdot \frac{1 + x}{x - 1}}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}} \]

      3. sub-neg 75.0%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{1 + x}{\color{blue}{x + \left(-1\right)}}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}} \]

      4. metadata-eval 75.0%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{1 + x}{x + \color{blue}{-1}}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}} \]

      5. +-commutative 75.0%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{\color{blue}{x + 1}}{x + -1}, {\ell}^{2} \cdot \left(\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1\right)\right)}} \]

      6. associate--l+ 79.3%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \color{blue}{\left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)}\right)}} \]

      7. sub-neg 79.3%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{\color{blue}{x + \left(-1\right)}} + \left(\frac{x}{x - 1} - 1\right)\right)\right)}} \]

      8. metadata-eval 79.3%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + \color{blue}{-1}} + \left(\frac{x}{x - 1} - 1\right)\right)\right)}} \]

      9. sub-neg 79.3%

         \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + -1} + \left(\frac{x}{\color{blue}{x + \left(-1\right)}} - 1\right)\right)\right)}} \]

      10. metadata-eval 79.3%

          \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + -1} + \left(\frac{x}{x + \color{blue}{-1}} - 1\right)\right)\right)}} \]

   7. Simplified 79.3%

      \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + -1} + \left(\frac{x}{x + -1} - 1\right)\right)\right)}}} \]

   8. Taylor expanded in x around inf 91.5%

      \[\leadsto \frac{{\left({2}^{0.25}\right)}^{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, \color{blue}{2 \cdot \frac{{\ell}^{2}}{x}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-253}:\\ \;\;\;\;{\left(\sqrt{t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x}\right)}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-155} \lor \neg \left(t \leq 3.55 \cdot 10^{+52}\right):\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot {\left({2}^{0.25}\right)}^{2}}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{-1 + x}, 2 \cdot \frac{{\ell}^{2}}{x}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+200)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (* t_m (* (/ (sqrt 2.0) l) (sqrt (* 0.5 x)))))))

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 * l) <= 5e+200) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = t_m * ((sqrt(2.0) / l) * sqrt((0.5 * x)));
	}
	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 * l) <= 5d+200) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else
        tmp = t_m * ((sqrt(2.0d0) / l) * sqrt((0.5d0 * x)))
    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 * l) <= 5e+200) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = t_m * ((Math.sqrt(2.0) / l) * Math.sqrt((0.5 * x)));
	}
	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 * l) <= 5e+200:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	else:
		tmp = t_m * ((math.sqrt(2.0) / l) * math.sqrt((0.5 * x)))
	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(l * l) <= 5e+200)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l) * sqrt(Float64(0.5 * x))));
	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 * l) <= 5e+200)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	else
		tmp = t_m * ((sqrt(2.0) / l) * sqrt((0.5 * x)));
	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[(l * l), $MachinePrecision], 5e+200], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000019e200

   1. Initial program 42.5%

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

   3. Simplified 31.9%

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

   5. Taylor expanded in t around inf 43.7%

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

   6. Taylor expanded in t around 0 43.8%

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

   if 5.00000000000000019e200 < (*.f64 l l)

   1. Initial program 4.6%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in l around inf 4.3%

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

      1. associate-/l* 4.3%

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

      2. associate--l+ 24.8%

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

      3. sub-neg 24.8%

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

      4. metadata-eval 24.8%

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

      5. +-commutative 24.8%

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

      6. sub-neg 24.8%

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

      7. metadata-eval 24.8%

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

      8. +-commutative 24.8%

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

   7. Simplified 24.8%

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

   8. Taylor expanded in x around inf 45.9%

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

      1. *-commutative 45.9%

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

   10. Simplified 45.9%

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

       1. pow1 45.9%

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

       2. associate-*l* 49.1%

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

   12. Applied egg-rr 49.1%

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

       1. unpow1 49.1%

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

   14. Simplified 49.1%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% 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 #s(literal 1 binary64) 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 34.0%

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

3. Simplified 25.8%

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

5. Taylor expanded in t around inf 38.4%

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

6. Taylor expanded in t around 0 38.9%

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

7. Final simplification 38.9%

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

Alternative 8: 76.8% accurate, 25.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 34.0%

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

3. Simplified 25.8%

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

5. Taylor expanded in t around inf 38.4%

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

6. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{0.5 \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x \cdot {\left(\sqrt{-1}\right)}^{2}} - \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
7. 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%

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

8. Simplified 38.6%

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

9. Final simplification 38.6%

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

Alternative 9: 76.7% 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 #s(literal 1 binary64) 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 34.0%

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

3. Simplified 25.8%

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

5. Taylor expanded in t around inf 38.4%

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

6. Taylor expanded in x around inf 38.4%

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

7. Final simplification 38.4%

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

Alternative 10: 76.0% 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 #s(literal 1 binary64) 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 34.0%

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

3. Simplified 25.8%

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

5. Taylor expanded in t around inf 38.4%

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

6. Taylor expanded in x around inf 37.7%

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

Reproduce

herbie shell --seed 2024181 
(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)))))