Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 84.0%

Time: 21.1s

Alternatives: 13

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.9% 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: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot x\\ t_3 := 2 \cdot {\ell}^{2}\\ t_4 := 2 \cdot {t\_m}^{2}\\ t_5 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{t\_3 + \frac{t\_3 + \frac{t\_3}{x}}{x}}{x}}}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{t\_5}{2 \cdot \frac{t\_m}{t\_2} + \left(t\_5 + \frac{{\ell}^{2}}{t\_m \cdot t\_2}\right)}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;\frac{t\_5}{\sqrt{t\_4 + 2 \cdot \frac{t\_4 + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x \cdot \left(1 + \frac{1}{x}\right)}}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) x))
        (t_3 (* 2.0 (pow l 2.0)))
        (t_4 (* 2.0 (pow t_m 2.0)))
        (t_5 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 2e-227)
      (* (sqrt 2.0) (/ t_m (sqrt (/ (+ t_3 (/ (+ t_3 (/ t_3 x)) x)) x))))
      (if (<= t_m 8.5e-182)
        (/ t_5 (+ (* 2.0 (/ t_m t_2)) (+ t_5 (/ (pow l 2.0) (* t_m t_2)))))
        (if (<= t_m 2.9e+23)
          (/ t_5 (sqrt (+ t_4 (* 2.0 (/ (+ t_4 (pow l 2.0)) x)))))
          (sqrt (/ (+ -1.0 x) (* x (+ 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) {
	double t_2 = sqrt(2.0) * x;
	double t_3 = 2.0 * pow(l, 2.0);
	double t_4 = 2.0 * pow(t_m, 2.0);
	double t_5 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 2e-227) {
		tmp = sqrt(2.0) * (t_m / sqrt(((t_3 + ((t_3 + (t_3 / x)) / x)) / x)));
	} else if (t_m <= 8.5e-182) {
		tmp = t_5 / ((2.0 * (t_m / t_2)) + (t_5 + (pow(l, 2.0) / (t_m * t_2))));
	} else if (t_m <= 2.9e+23) {
		tmp = t_5 / sqrt((t_4 + (2.0 * ((t_4 + pow(l, 2.0)) / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * x
    t_3 = 2.0d0 * (l ** 2.0d0)
    t_4 = 2.0d0 * (t_m ** 2.0d0)
    t_5 = t_m * sqrt(2.0d0)
    if (t_m <= 2d-227) then
        tmp = sqrt(2.0d0) * (t_m / sqrt(((t_3 + ((t_3 + (t_3 / x)) / x)) / x)))
    else if (t_m <= 8.5d-182) then
        tmp = t_5 / ((2.0d0 * (t_m / t_2)) + (t_5 + ((l ** 2.0d0) / (t_m * t_2))))
    else if (t_m <= 2.9d+23) then
        tmp = t_5 / sqrt((t_4 + (2.0d0 * ((t_4 + (l ** 2.0d0)) / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x * (1.0d0 + (1.0d0 / 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 = Math.sqrt(2.0) * x;
	double t_3 = 2.0 * Math.pow(l, 2.0);
	double t_4 = 2.0 * Math.pow(t_m, 2.0);
	double t_5 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 2e-227) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt(((t_3 + ((t_3 + (t_3 / x)) / x)) / x)));
	} else if (t_m <= 8.5e-182) {
		tmp = t_5 / ((2.0 * (t_m / t_2)) + (t_5 + (Math.pow(l, 2.0) / (t_m * t_2))));
	} else if (t_m <= 2.9e+23) {
		tmp = t_5 / Math.sqrt((t_4 + (2.0 * ((t_4 + Math.pow(l, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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 = math.sqrt(2.0) * x
	t_3 = 2.0 * math.pow(l, 2.0)
	t_4 = 2.0 * math.pow(t_m, 2.0)
	t_5 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 2e-227:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt(((t_3 + ((t_3 + (t_3 / x)) / x)) / x)))
	elif t_m <= 8.5e-182:
		tmp = t_5 / ((2.0 * (t_m / t_2)) + (t_5 + (math.pow(l, 2.0) / (t_m * t_2))))
	elif t_m <= 2.9e+23:
		tmp = t_5 / math.sqrt((t_4 + (2.0 * ((t_4 + math.pow(l, 2.0)) / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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(sqrt(2.0) * x)
	t_3 = Float64(2.0 * (l ^ 2.0))
	t_4 = Float64(2.0 * (t_m ^ 2.0))
	t_5 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 2e-227)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(t_3 + Float64(Float64(t_3 + Float64(t_3 / x)) / x)) / x))));
	elseif (t_m <= 8.5e-182)
		tmp = Float64(t_5 / Float64(Float64(2.0 * Float64(t_m / t_2)) + Float64(t_5 + Float64((l ^ 2.0) / Float64(t_m * t_2)))));
	elseif (t_m <= 2.9e+23)
		tmp = Float64(t_5 / sqrt(Float64(t_4 + Float64(2.0 * Float64(Float64(t_4 + (l ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x * Float64(1.0 + Float64(1.0 / 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 = sqrt(2.0) * x;
	t_3 = 2.0 * (l ^ 2.0);
	t_4 = 2.0 * (t_m ^ 2.0);
	t_5 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 2e-227)
		tmp = sqrt(2.0) * (t_m / sqrt(((t_3 + ((t_3 + (t_3 / x)) / x)) / x)));
	elseif (t_m <= 8.5e-182)
		tmp = t_5 / ((2.0 * (t_m / t_2)) + (t_5 + ((l ^ 2.0) / (t_m * t_2))));
	elseif (t_m <= 2.9e+23)
		tmp = t_5 / sqrt((t_4 + (2.0 * ((t_4 + (l ^ 2.0)) / x))));
	else
		tmp = sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2e-227], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(t$95$3 + N[(N[(t$95$3 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e-182], N[(t$95$5 / N[(N[(2.0 * N[(t$95$m / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+23], N[(t$95$5 / N[Sqrt[N[(t$95$4 + N[(2.0 * N[(N[(t$95$4 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.99999999999999989e-227

   1. Initial program 27.3%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in t around 0 3.1%

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

   6. Taylor expanded in x around -inf 21.3%

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

   8. Simplified 20.9%

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

   if 1.99999999999999989e-227 < t < 8.5000000000000001e-182

   1. Initial program 1.6%

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

      1. flip-- 1.6%

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

      2. associate-/r/ 1.6%

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

      3. metadata-eval 1.6%

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

      4. fma-neg 1.6%

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

      5. metadata-eval 1.6%

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

   4. Applied egg-rr 1.6%

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

   5. Taylor expanded in x around inf 82.7%

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

   if 8.5000000000000001e-182 < t < 2.90000000000000013e23

   1. Initial program 49.3%

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

      1. flip-- 27.9%

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

      2. associate-/r/ 27.9%

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

      3. metadata-eval 27.9%

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

      4. fma-neg 27.9%

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

      5. metadata-eval 27.9%

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

   4. Applied egg-rr 27.9%

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

   5. Taylor expanded in x around inf 88.1%

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

   if 2.90000000000000013e23 < t

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in t around inf 95.8%

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

   6. Taylor expanded in t around 0 95.9%

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

   7. Taylor expanded in x around inf 95.9%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {\ell}^{2} + \frac{2 \cdot {\ell}^{2} + \frac{2 \cdot {\ell}^{2}}{x}}{x}}{x}}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{2 \cdot \frac{t}{\sqrt{2} \cdot x} + \left(t \cdot \sqrt{2} + \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x \cdot \left(1 + \frac{1}{x}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_3 + t\_3\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.65 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \frac{t\_4 + \frac{\left(t\_4 + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{t\_3}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x \cdot \left(1 + \frac{1}{x}\right)}}\\ \end{array} \end{array} \end{array} \]

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_3 (+ t_2 (pow l 2.0)))
        (t_4 (+ t_3 t_3)))
   (*
    t_s
    (if (<= t_m 2.65e+23)
      (*
       (sqrt 2.0)
       (/
        t_m
        (sqrt
         (+
          t_2
          (/
           (+
            t_4
            (/
             (+
              (+ t_4 (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l 2.0) x)))
              (/ t_3 x))
             x))
           x)))))
      (sqrt (/ (+ -1.0 x) (* x (+ 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) {
	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 <= 2.65e+23) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((t_4 + (((t_4 + ((2.0 * (pow(t_m, 2.0) / x)) + (pow(l, 2.0) / x))) + (t_3 / x)) / x)) / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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) :: 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 <= 2.65d+23) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((t_2 + ((t_4 + (((t_4 + ((2.0d0 * ((t_m ** 2.0d0) / x)) + ((l ** 2.0d0) / x))) + (t_3 / x)) / x)) / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x * (1.0d0 + (1.0d0 / 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 t_3 = t_2 + Math.pow(l, 2.0);
	double t_4 = t_3 + t_3;
	double tmp;
	if (t_m <= 2.65e+23) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((t_2 + ((t_4 + (((t_4 + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (Math.pow(l, 2.0) / x))) + (t_3 / x)) / x)) / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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)
	t_3 = t_2 + math.pow(l, 2.0)
	t_4 = t_3 + t_3
	tmp = 0
	if t_m <= 2.65e+23:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((t_2 + ((t_4 + (((t_4 + ((2.0 * (math.pow(t_m, 2.0) / x)) + (math.pow(l, 2.0) / x))) + (t_3 / x)) / x)) / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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))
	t_3 = Float64(t_2 + (l ^ 2.0))
	t_4 = Float64(t_3 + t_3)
	tmp = 0.0
	if (t_m <= 2.65e+23)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_2 + Float64(Float64(t_4 + Float64(Float64(Float64(t_4 + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l ^ 2.0) / x))) + Float64(t_3 / x)) / x)) / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x * Float64(1.0 + Float64(1.0 / 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);
	t_3 = t_2 + (l ^ 2.0);
	t_4 = t_3 + t_3;
	tmp = 0.0;
	if (t_m <= 2.65e+23)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((t_4 + (((t_4 + ((2.0 * ((t_m ^ 2.0) / x)) + ((l ^ 2.0) / x))) + (t_3 / x)) / x)) / x))));
	else
		tmp = sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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]}, 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, 2.65e+23], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$2 + N[(N[(t$95$4 + N[(N[(N[(t$95$4 + 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]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.6500000000000001e23

   1. Initial program 30.2%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in x around -inf 55.3%

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

   if 2.6500000000000001e23 < t

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in t around inf 95.8%

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

   6. Taylor expanded in t around 0 95.9%

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

   7. Taylor expanded in x around inf 95.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.65 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \frac{\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{\left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x \cdot \left(1 + \frac{1}{x}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_2 + \frac{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(t\_3 + t\_3\right) + \frac{t\_3}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x \cdot \left(1 + \frac{1}{x}\right)}}\\ \end{array} \end{array} \end{array} \]

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_3 (+ t_2 (pow l 2.0))))
   (*
    t_s
    (if (<= t_m 2e+23)
      (*
       (sqrt 2.0)
       (/
        t_m
        (sqrt
         (+
          t_2
          (/
           (+
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l 2.0) x))
            (+ (+ t_3 t_3) (/ t_3 x)))
           x)))))
      (sqrt (/ (+ -1.0 x) (* x (+ 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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double tmp;
	if (t_m <= 2e+23) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * (pow(t_m, 2.0) / x)) + (pow(l, 2.0) / x)) + ((t_3 + t_3) + (t_3 / x))) / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    if (t_m <= 2d+23) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((t_2 + ((((2.0d0 * ((t_m ** 2.0d0) / x)) + ((l ** 2.0d0) / x)) + ((t_3 + t_3) + (t_3 / x))) / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x * (1.0d0 + (1.0d0 / 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 t_3 = t_2 + Math.pow(l, 2.0);
	double tmp;
	if (t_m <= 2e+23) {
		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_3 + t_3) + (t_3 / x))) / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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)
	t_3 = t_2 + math.pow(l, 2.0)
	tmp = 0
	if t_m <= 2e+23:
		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_3 + t_3) + (t_3 / x))) / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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))
	t_3 = Float64(t_2 + (l ^ 2.0))
	tmp = 0.0
	if (t_m <= 2e+23)
		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(Float64(t_3 + t_3) + Float64(t_3 / x))) / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x * Float64(1.0 + Float64(1.0 / 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);
	t_3 = t_2 + (l ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2e+23)
		tmp = sqrt(2.0) * (t_m / sqrt((t_2 + ((((2.0 * ((t_m ^ 2.0) / x)) + ((l ^ 2.0) / x)) + ((t_3 + t_3) + (t_3 / x))) / x))));
	else
		tmp = sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2e+23], 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[(N[(t$95$3 + t$95$3), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.9999999999999998e23

   1. Initial program 30.2%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in x around -inf 55.2%

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

   if 1.9999999999999998e23 < t

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in t around inf 95.8%

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

   6. Taylor expanded in t around 0 95.9%

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

   7. Taylor expanded in x around inf 95.9%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x \cdot \left(1 + \frac{1}{x}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{t\_2 + 2 \cdot \frac{t\_2 + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x \cdot \left(1 + \frac{1}{x}\right)}}\\ \end{array} \end{array} \end{array} \]

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 2.6e+22)
      (/ (* t_m (sqrt 2.0)) (sqrt (+ t_2 (* 2.0 (/ (+ t_2 (pow l 2.0)) x)))))
      (sqrt (/ (+ -1.0 x) (* x (+ 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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.6e+22) {
		tmp = (t_m * sqrt(2.0)) / sqrt((t_2 + (2.0 * ((t_2 + pow(l, 2.0)) / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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 <= 2.6d+22) then
        tmp = (t_m * sqrt(2.0d0)) / sqrt((t_2 + (2.0d0 * ((t_2 + (l ** 2.0d0)) / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x * (1.0d0 + (1.0d0 / 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 <= 2.6e+22) {
		tmp = (t_m * Math.sqrt(2.0)) / Math.sqrt((t_2 + (2.0 * ((t_2 + Math.pow(l, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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 <= 2.6e+22:
		tmp = (t_m * math.sqrt(2.0)) / math.sqrt((t_2 + (2.0 * ((t_2 + math.pow(l, 2.0)) / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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 <= 2.6e+22)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(t_2 + Float64(2.0 * Float64(Float64(t_2 + (l ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x * Float64(1.0 + Float64(1.0 / 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 <= 2.6e+22)
		tmp = (t_m * sqrt(2.0)) / sqrt((t_2 + (2.0 * ((t_2 + (l ^ 2.0)) / x))));
	else
		tmp = sqrt(((-1.0 + x) / (x * (1.0 + (1.0 / 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, 2.6e+22], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(2.0 * N[(N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.6e22

   1. Initial program 30.2%

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

      1. flip-- 16.9%

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

      2. associate-/r/ 16.8%

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

      3. metadata-eval 16.8%

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

      4. fma-neg 16.8%

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

      5. metadata-eval 16.8%

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

   4. Applied egg-rr 16.8%

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

   5. Taylor expanded in x around inf 55.0%

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

   if 2.6e22 < t

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   5. Taylor expanded in t around inf 95.8%

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

   6. Taylor expanded in t around 0 95.9%

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

   7. Taylor expanded in x around inf 95.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x \cdot \left(1 + \frac{1}{x}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {\ell}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{t\_2 + \frac{t\_2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 l 2.0))))
   (*
    t_s
    (if (<= t_m 2.7e-215)
      (* (sqrt 2.0) (/ t_m (sqrt (/ (+ t_2 (/ t_2 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(l, 2.0);
	double tmp;
	if (t_m <= 2.7e-215) {
		tmp = sqrt(2.0) * (t_m / sqrt(((t_2 + (t_2 / 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) :: tmp
    t_2 = 2.0d0 * (l ** 2.0d0)
    if (t_m <= 2.7d-215) then
        tmp = sqrt(2.0d0) * (t_m / sqrt(((t_2 + (t_2 / 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(l, 2.0);
	double tmp;
	if (t_m <= 2.7e-215) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt(((t_2 + (t_2 / 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(l, 2.0)
	tmp = 0
	if t_m <= 2.7e-215:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt(((t_2 + (t_2 / 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 * (l ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.7e-215)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(t_2 + Float64(t_2 / 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 * (l ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.7e-215)
		tmp = sqrt(2.0) * (t_m / sqrt(((t_2 + (t_2 / 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[l, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.7e-215], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(t$95$2 + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision] / x), $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 2 regimes

2. if t < 2.70000000000000018e-215

   1. Initial program 26.9%

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

   3. Simplified 23.5%

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

   5. Taylor expanded in t around 0 3.1%

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

   6. Taylor expanded in x around inf 21.9%

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

   8. Simplified 21.9%

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

   if 2.70000000000000018e-215 < t

   1. Initial program 33.7%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in t around inf 85.7%

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

   6. Taylor expanded in t around 0 85.8%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {\ell}^{2} + \frac{2 \cdot {\ell}^{2}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{2 \cdot {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

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 1.3e-215)
    (* (sqrt 2.0) (/ t_m (sqrt (/ (* 2.0 (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 tmp;
	if (t_m <= 1.3e-215) {
		tmp = sqrt(2.0) * (t_m / sqrt(((2.0 * 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) :: tmp
    if (t_m <= 1.3d-215) then
        tmp = sqrt(2.0d0) * (t_m / sqrt(((2.0d0 * (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 tmp;
	if (t_m <= 1.3e-215) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt(((2.0 * 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):
	tmp = 0
	if t_m <= 1.3e-215:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt(((2.0 * 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)
	tmp = 0.0
	if (t_m <= 1.3e-215)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(2.0 * (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)
	tmp = 0.0;
	if (t_m <= 1.3e-215)
		tmp = sqrt(2.0) * (t_m / sqrt(((2.0 * (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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.3e-215], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $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 2 regimes

2. if t < 1.3e-215

   1. Initial program 26.9%

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

   3. Simplified 23.5%

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

   5. Taylor expanded in t around 0 3.1%

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

   6. Taylor expanded in x around inf 21.5%

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

      1. div-sub 21.5%

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

      2. remove-double-neg 21.5%

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

      3. mul-1-neg 21.5%

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

      4. rem-square-sqrt 0.0%

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

      5. unpow2 0.0%

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

      6. *-commutative 0.0%

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

      7. mul-1-neg 0.0%

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

      8. rem-square-sqrt 0.0%

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

      9. unpow2 0.0%

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

      10. *-commutative 0.0%

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

      11. div-sub 0.0%

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

   8. Simplified 21.5%

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

   if 1.3e-215 < t

   1. Initial program 33.7%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in t around inf 85.7%

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

   6. Taylor expanded in t around 0 85.8%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_m \cdot \sqrt{\frac{x}{2 \cdot {\ell}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

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 3.7e-216)
    (* (sqrt 2.0) (* t_m (sqrt (/ x (* 2.0 (pow l 2.0))))))
    (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 tmp;
	if (t_m <= 3.7e-216) {
		tmp = sqrt(2.0) * (t_m * sqrt((x / (2.0 * pow(l, 2.0)))));
	} 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) :: tmp
    if (t_m <= 3.7d-216) then
        tmp = sqrt(2.0d0) * (t_m * sqrt((x / (2.0d0 * (l ** 2.0d0)))))
    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 tmp;
	if (t_m <= 3.7e-216) {
		tmp = Math.sqrt(2.0) * (t_m * Math.sqrt((x / (2.0 * Math.pow(l, 2.0)))));
	} 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):
	tmp = 0
	if t_m <= 3.7e-216:
		tmp = math.sqrt(2.0) * (t_m * math.sqrt((x / (2.0 * math.pow(l, 2.0)))))
	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)
	tmp = 0.0
	if (t_m <= 3.7e-216)
		tmp = Float64(sqrt(2.0) * Float64(t_m * sqrt(Float64(x / Float64(2.0 * (l ^ 2.0))))));
	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)
	tmp = 0.0;
	if (t_m <= 3.7e-216)
		tmp = sqrt(2.0) * (t_m * sqrt((x / (2.0 * (l ^ 2.0)))));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.7e-216], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * N[Sqrt[N[(x / N[(2.0 * N[Power[l, 2.0], $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 2 regimes

2. if t < 3.69999999999999996e-216

   1. Initial program 26.9%

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

   3. Simplified 23.5%

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

   5. Taylor expanded in t around 0 3.1%

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

   6. Taylor expanded in x around inf 20.9%

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

      1. cancel-sign-sub-inv 20.9%

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

      2. metadata-eval 20.9%

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

      3. distribute-rgt1-in 20.9%

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

      4. metadata-eval 20.9%

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

   8. Simplified 20.9%

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

   if 3.69999999999999996e-216 < t

   1. Initial program 33.7%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in t around inf 85.7%

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

   6. Taylor expanded in t around 0 85.8%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x}{2 \cdot {\ell}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-217}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{x}{2 \cdot {\ell}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

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 5.2e-217)
    (* t_m (* (sqrt 2.0) (sqrt (/ x (* 2.0 (pow l 2.0))))))
    (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 tmp;
	if (t_m <= 5.2e-217) {
		tmp = t_m * (sqrt(2.0) * sqrt((x / (2.0 * pow(l, 2.0)))));
	} 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) :: tmp
    if (t_m <= 5.2d-217) then
        tmp = t_m * (sqrt(2.0d0) * sqrt((x / (2.0d0 * (l ** 2.0d0)))))
    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 tmp;
	if (t_m <= 5.2e-217) {
		tmp = t_m * (Math.sqrt(2.0) * Math.sqrt((x / (2.0 * Math.pow(l, 2.0)))));
	} 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):
	tmp = 0
	if t_m <= 5.2e-217:
		tmp = t_m * (math.sqrt(2.0) * math.sqrt((x / (2.0 * math.pow(l, 2.0)))))
	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)
	tmp = 0.0
	if (t_m <= 5.2e-217)
		tmp = Float64(t_m * Float64(sqrt(2.0) * sqrt(Float64(x / Float64(2.0 * (l ^ 2.0))))));
	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)
	tmp = 0.0;
	if (t_m <= 5.2e-217)
		tmp = t_m * (sqrt(2.0) * sqrt((x / (2.0 * (l ^ 2.0)))));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-217], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(x / N[(2.0 * N[Power[l, 2.0], $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 2 regimes

2. if t < 5.19999999999999986e-217

   1. Initial program 26.9%

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

   3. Simplified 23.5%

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

   5. Taylor expanded in t around 0 3.1%

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

   6. Taylor expanded in x around inf 20.9%

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

      1. associate-*l* 20.9%

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

      2. cancel-sign-sub-inv 20.9%

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

      3. metadata-eval 20.9%

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

      4. distribute-rgt1-in 20.9%

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

      5. metadata-eval 20.9%

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

   8. Simplified 20.9%

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

   if 5.19999999999999986e-217 < t

   1. Initial program 33.7%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in t around inf 85.7%

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

   6. Taylor expanded in t around 0 85.8%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot \sqrt{\frac{x}{2 \cdot {\ell}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% 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 29.8%

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

3. Simplified 26.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1}, -\ell \cdot \ell\right)}}} \]
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 38.5%

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

7. Final simplification 38.5%

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

Alternative 10: 77.0% accurate, 17.3× speedup

Math

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

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 (/ -0.5 x)) 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 + (-0.5 / x)) / 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 + ((-0.5d0) / x)) / 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 + (-0.5 / x)) / 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 + (-0.5 / x)) / 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(Float64(0.5 + Float64(-0.5 / x)) / 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 + (-0.5 / x)) / 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[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.8%

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

3. Simplified 26.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1}, -\ell \cdot \ell\right)}}} \]
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 38.5%

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

7. Taylor expanded in x around inf 38.5%

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

8. Taylor expanded in x around -inf 38.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}{x}} \]
9. Step-by-step derivation

   1. mul-1-neg 38.2%

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

   2. unsub-neg 38.2%

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

   3. mul-1-neg 38.2%

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

   4. unsub-neg 38.2%

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

   5. sub-neg 38.2%

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

   6. associate-*r/ 38.2%

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

   7. metadata-eval 38.2%

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

   8. distribute-neg-frac 38.2%

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

   9. metadata-eval 38.2%

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

10. Simplified 38.2%

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

11. Final simplification 38.2%

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

Alternative 11: 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 29.8%

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

3. Simplified 26.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1}, -\ell \cdot \ell\right)}}} \]
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 38.5%

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

7. Taylor expanded in x around inf 38.5%

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

8. Taylor expanded in x around -inf 38.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 - 0.5 \cdot \frac{1}{x}}{x}} \]
9. Step-by-step derivation

   1. mul-1-neg 38.2%

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

   2. unsub-neg 38.2%

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

   3. sub-neg 38.2%

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

   4. associate-*r/ 38.2%

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

   5. metadata-eval 38.2%

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

   6. distribute-neg-frac 38.2%

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

   7. metadata-eval 38.2%

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

10. Simplified 38.2%

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

11. Final simplification 38.2%

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

Alternative 12: 76.6% 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 29.8%

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

3. Simplified 26.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1}, -\ell \cdot \ell\right)}}} \]
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 38.1%

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

7. Final simplification 38.1%

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

Alternative 13: 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 29.8%

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

3. Simplified 26.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1}, -\ell \cdot \ell\right)}}} \]
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 37.8%

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

Reproduce

herbie shell --seed 2024135 
(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)))))