Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 85.7%

Time: 17.1s

Alternatives: 12

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 85.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_3 + t\_3\\ t_5 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_5}{0.5 \cdot \frac{t\_4}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_5}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\frac{t\_5}{\sqrt{t\_2 + \frac{t\_4 + \frac{\left(t\_4 + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{t\_3}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_5 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 8e-160)
      (/ t_5 (+ (* 0.5 (/ t_4 (* t_m (* (sqrt 2.0) x)))) t_5))
      (if (<= t_m 2e+53)
        (/
         t_5
         (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))))))))

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 t_5 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 8e-160) {
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + t_5);
	} else if (t_m <= 2e+53) {
		tmp = t_5 / 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)));
	}
	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 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 + t_3
    t_5 = t_m * sqrt(2.0d0)
    if (t_m <= 8d-160) then
        tmp = t_5 / ((0.5d0 * (t_4 / (t_m * (sqrt(2.0d0) * x)))) + t_5)
    else if (t_m <= 2d+53) then
        tmp = t_5 / 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)))
    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 t_5 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 8e-160) {
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (Math.sqrt(2.0) * x)))) + t_5);
	} else if (t_m <= 2e+53) {
		tmp = t_5 / 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)));
	}
	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
	t_5 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 8e-160:
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (math.sqrt(2.0) * x)))) + t_5)
	elif t_m <= 2e+53:
		tmp = t_5 / 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)))
	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)
	t_5 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 8e-160)
		tmp = Float64(t_5 / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) * x)))) + t_5));
	elseif (t_m <= 2e+53)
		tmp = Float64(t_5 / 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 + 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;
	t_5 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 8e-160)
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + t_5);
	elseif (t_m <= 2e+53)
		tmp = t_5 / 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)));
	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]}, Block[{t$95$5 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8e-160], N[(t$95$5 / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+53], N[(t$95$5 / 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], 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 < 7.9999999999999999e-160

   1. Initial program 30.0%

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

   3. Taylor expanded in x around inf 19.3%

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

   if 7.9999999999999999e-160 < t < 2e53

   1. Initial program 50.4%

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

   3. Taylor expanded in x around -inf 81.9%

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

   if 2e53 < t

   1. Initial program 28.4%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in t around inf 98.2%

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

   6. Taylor expanded in t around 0 98.5%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot {t}^{2} + \frac{\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{\left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_3 + t\_3\\ t_5 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_5}{0.5 \cdot \frac{t\_4}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_5}\\ \mathbf{elif}\;t\_m \leq 6.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{t\_5}{\sqrt{t\_2 + \frac{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(t\_4 + \frac{t\_3}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_5 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 8.2e-160)
      (/ t_5 (+ (* 0.5 (/ t_4 (* t_m (* (sqrt 2.0) x)))) t_5))
      (if (<= t_m 6.1e+53)
        (/
         t_5
         (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 t_5 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 8.2e-160) {
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + t_5);
	} else if (t_m <= 6.1e+53) {
		tmp = t_5 / 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) :: t_5
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 + t_3
    t_5 = t_m * sqrt(2.0d0)
    if (t_m <= 8.2d-160) then
        tmp = t_5 / ((0.5d0 * (t_4 / (t_m * (sqrt(2.0d0) * x)))) + t_5)
    else if (t_m <= 6.1d+53) then
        tmp = t_5 / 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 t_5 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 8.2e-160) {
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (Math.sqrt(2.0) * x)))) + t_5);
	} else if (t_m <= 6.1e+53) {
		tmp = t_5 / 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
	t_5 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 8.2e-160:
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (math.sqrt(2.0) * x)))) + t_5)
	elif t_m <= 6.1e+53:
		tmp = t_5 / 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)
	t_5 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 8.2e-160)
		tmp = Float64(t_5 / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) * x)))) + t_5));
	elseif (t_m <= 6.1e+53)
		tmp = Float64(t_5 / 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;
	t_5 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 8.2e-160)
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (sqrt(2.0) * x)))) + t_5);
	elseif (t_m <= 6.1e+53)
		tmp = t_5 / 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]}, Block[{t$95$5 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.2e-160], N[(t$95$5 / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.1e+53], N[(t$95$5 / 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], 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 < 8.20000000000000003e-160

   1. Initial program 30.0%

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

   3. Taylor expanded in x around inf 19.3%

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

   if 8.20000000000000003e-160 < t < 6.1000000000000002e53

   1. Initial program 50.4%

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

   3. Taylor expanded in x around -inf 81.9%

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

   if 6.1000000000000002e53 < t

   1. Initial program 28.4%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in t around inf 98.2%

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

   6. Taylor expanded in t around 0 98.5%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot {t}^{2} + \frac{\left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right) + \left(\left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_4}{0.5 \cdot \frac{t\_3 + t\_3}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_4}\\ \mathbf{elif}\;t\_m \leq 10^{+54}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\frac{t\_3}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_4 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 7.5e-160)
      (/ t_4 (+ (* 0.5 (/ (+ t_3 t_3) (* t_m (* (sqrt 2.0) x)))) t_4))
      (if (<= t_m 1e+54)
        (/
         t_4
         (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 t_4 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 7.5e-160) {
		tmp = t_4 / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + t_4);
	} else if (t_m <= 1e+54) {
		tmp = t_4 / 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) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_m * sqrt(2.0d0)
    if (t_m <= 7.5d-160) then
        tmp = t_4 / ((0.5d0 * ((t_3 + t_3) / (t_m * (sqrt(2.0d0) * x)))) + t_4)
    else if (t_m <= 1d+54) then
        tmp = t_4 / 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 t_4 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 7.5e-160) {
		tmp = t_4 / ((0.5 * ((t_3 + t_3) / (t_m * (Math.sqrt(2.0) * x)))) + t_4);
	} else if (t_m <= 1e+54) {
		tmp = t_4 / 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)
	t_4 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 7.5e-160:
		tmp = t_4 / ((0.5 * ((t_3 + t_3) / (t_m * (math.sqrt(2.0) * x)))) + t_4)
	elif t_m <= 1e+54:
		tmp = t_4 / 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))
	t_4 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 7.5e-160)
		tmp = Float64(t_4 / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(sqrt(2.0) * x)))) + t_4));
	elseif (t_m <= 1e+54)
		tmp = Float64(t_4 / 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);
	t_4 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 7.5e-160)
		tmp = t_4 / ((0.5 * ((t_3 + t_3) / (t_m * (sqrt(2.0) * x)))) + t_4);
	elseif (t_m <= 1e+54)
		tmp = t_4 / 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]}, Block[{t$95$4 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.5e-160], N[(t$95$4 / 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] + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+54], N[(t$95$4 / 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], 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 < 7.50000000000000023e-160

   1. Initial program 30.0%

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

   3. Taylor expanded in x around inf 19.3%

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

   if 7.50000000000000023e-160 < t < 1.0000000000000001e54

   1. Initial program 50.4%

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

   3. Taylor expanded in x around inf 81.9%

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

   if 1.0000000000000001e54 < t

   1. Initial program 28.4%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in t around inf 98.2%

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

   6. Taylor expanded in t around 0 98.5%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 10^{+54}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_2}{t\_2 + \ell \cdot \frac{\ell}{x \cdot t\_2}}\\ \mathbf{elif}\;t\_m \leq 7 \cdot 10^{+52}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{t\_3 + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_3 + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* t_m (sqrt 2.0))) (t_3 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 2.5e-162)
      (/ t_2 (+ t_2 (* l (/ l (* x t_2)))))
      (if (<= t_m 7e+52)
        (/
         t_2
         (sqrt
          (+
           (/ (+ t_3 (pow l 2.0)) x)
           (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_3 (/ (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 = t_m * sqrt(2.0);
	double t_3 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.5e-162) {
		tmp = t_2 / (t_2 + (l * (l / (x * t_2))));
	} else if (t_m <= 7e+52) {
		tmp = t_2 / sqrt((((t_3 + pow(l, 2.0)) / x) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_3 + (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 = t_m * sqrt(2.0d0)
    t_3 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 2.5d-162) then
        tmp = t_2 / (t_2 + (l * (l / (x * t_2))))
    else if (t_m <= 7d+52) then
        tmp = t_2 / sqrt((((t_3 + (l ** 2.0d0)) / x) + ((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_3 + ((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 = t_m * Math.sqrt(2.0);
	double t_3 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 2.5e-162) {
		tmp = t_2 / (t_2 + (l * (l / (x * t_2))));
	} else if (t_m <= 7e+52) {
		tmp = t_2 / Math.sqrt((((t_3 + Math.pow(l, 2.0)) / x) + ((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_3 + (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 = t_m * math.sqrt(2.0)
	t_3 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 2.5e-162:
		tmp = t_2 / (t_2 + (l * (l / (x * t_2))))
	elif t_m <= 7e+52:
		tmp = t_2 / math.sqrt((((t_3 + math.pow(l, 2.0)) / x) + ((2.0 * (math.pow(t_m, 2.0) / x)) + (t_3 + (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(t_m * sqrt(2.0))
	t_3 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.5e-162)
		tmp = Float64(t_2 / Float64(t_2 + Float64(l * Float64(l / Float64(x * t_2)))));
	elseif (t_m <= 7e+52)
		tmp = Float64(t_2 / sqrt(Float64(Float64(Float64(t_3 + (l ^ 2.0)) / x) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_3 + 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 = t_m * sqrt(2.0);
	t_3 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.5e-162)
		tmp = t_2 / (t_2 + (l * (l / (x * t_2))));
	elseif (t_m <= 7e+52)
		tmp = t_2 / sqrt((((t_3 + (l ^ 2.0)) / x) + ((2.0 * ((t_m ^ 2.0) / x)) + (t_3 + ((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[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.5e-162], N[(t$95$2 / N[(t$95$2 + N[(l * N[(l / N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7e+52], N[(t$95$2 / N[Sqrt[N[(N[(N[(t$95$3 + 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$3 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.50000000000000007e-162

   1. Initial program 29.9%

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

   3. Taylor expanded in x around inf 19.1%

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

   4. Taylor expanded in l around inf 19.1%

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

      1. *-commutative 19.1%

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

   6. Simplified 19.1%

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

      1. unpow2 19.1%

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

      2. associate-/l* 19.2%

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

      3. associate-*l* 19.2%

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

      4. *-commutative 19.2%

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

   8. Applied egg-rr 19.2%

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

   if 2.50000000000000007e-162 < t < 7e52

   1. Initial program 50.2%

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

   3. Taylor expanded in x around inf 81.1%

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

   if 7e52 < t

   1. Initial program 28.4%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in t around inf 98.2%

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

   6. Taylor expanded in t around 0 98.5%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \ell \cdot \frac{\ell}{x \cdot \left(t \cdot \sqrt{2}\right)}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+52}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \frac{t\_2}{t\_2 + \frac{\ell}{\left(\sqrt{2} \cdot x\right) \cdot \frac{t\_m}{\ell}}} \end{array} \end{array} \]

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 (* t_m (sqrt 2.0))))
   (* t_s (/ t_2 (+ t_2 (/ l (* (* (sqrt 2.0) x) (/ t_m l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	return t_s * (t_2 / (t_2 + (l / ((sqrt(2.0) * x) * (t_m / l)))));
}

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
    t_2 = t_m * sqrt(2.0d0)
    code = t_s * (t_2 / (t_2 + (l / ((sqrt(2.0d0) * x) * (t_m / l)))))
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 = t_m * Math.sqrt(2.0);
	return t_s * (t_2 / (t_2 + (l / ((Math.sqrt(2.0) * x) * (t_m / l)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = t_m * math.sqrt(2.0)
	return t_s * (t_2 / (t_2 + (l / ((math.sqrt(2.0) * x) * (t_m / l)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	return Float64(t_s * Float64(t_2 / Float64(t_2 + Float64(l / Float64(Float64(sqrt(2.0) * x) * Float64(t_m / l))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	t_2 = t_m * sqrt(2.0);
	tmp = t_s * (t_2 / (t_2 + (l / ((sqrt(2.0) * x) * (t_m / l)))));
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[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(t$95$2 / N[(t$95$2 + N[(l / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.4%

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

3. Taylor expanded in x around inf 31.0%

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

4. Taylor expanded in l around inf 43.6%

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

   1. *-commutative 43.6%

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

6. Simplified 43.6%

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

   1. unpow2 43.6%

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

   2. times-frac 49.0%

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

8. Applied egg-rr 49.0%

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

   1. *-commutative 49.0%

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

   2. clear-num 49.0%

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

   3. rem-exp-log 26.3%

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

   4. frac-times 26.4%

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

   5. *-un-lft-identity 26.4%

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

   6. rem-exp-log 49.1%

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

10. Applied egg-rr 49.1%

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

11. Final simplification 49.1%

    \[\leadsto \frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{\ell}{\left(\sqrt{2} \cdot x\right) \cdot \frac{t}{\ell}}} \]
12. Add Preprocessing

Alternative 6: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \frac{t\_2}{t\_2 + \frac{\ell}{\sqrt{2} \cdot x} \cdot \frac{\ell}{t\_m}} \end{array} \end{array} \]

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 (* t_m (sqrt 2.0))))
   (* t_s (/ t_2 (+ t_2 (* (/ l (* (sqrt 2.0) x)) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	return t_s * (t_2 / (t_2 + ((l / (sqrt(2.0) * x)) * (l / t_m))));
}

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
    t_2 = t_m * sqrt(2.0d0)
    code = t_s * (t_2 / (t_2 + ((l / (sqrt(2.0d0) * x)) * (l / t_m))))
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 = t_m * Math.sqrt(2.0);
	return t_s * (t_2 / (t_2 + ((l / (Math.sqrt(2.0) * x)) * (l / t_m))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = t_m * math.sqrt(2.0)
	return t_s * (t_2 / (t_2 + ((l / (math.sqrt(2.0) * x)) * (l / t_m))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	return Float64(t_s * Float64(t_2 / Float64(t_2 + Float64(Float64(l / Float64(sqrt(2.0) * x)) * Float64(l / t_m)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	t_2 = t_m * sqrt(2.0);
	tmp = t_s * (t_2 / (t_2 + ((l / (sqrt(2.0) * x)) * (l / t_m))));
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[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(t$95$2 / N[(t$95$2 + N[(N[(l / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.4%

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

3. Taylor expanded in x around inf 31.0%

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

4. Taylor expanded in l around inf 43.6%

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

   1. *-commutative 43.6%

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

6. Simplified 43.6%

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

   1. unpow2 43.6%

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

   2. times-frac 49.0%

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

8. Applied egg-rr 49.0%

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

9. Final simplification 49.0%

   \[\leadsto \frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \frac{\ell}{\sqrt{2} \cdot x} \cdot \frac{\ell}{t}} \]
10. Add Preprocessing

Alternative 7: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \frac{t\_2}{t\_2 + \ell \cdot \frac{\ell}{x \cdot t\_2}} \end{array} \end{array} \]

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 (* t_m (sqrt 2.0))))
   (* t_s (/ t_2 (+ t_2 (* l (/ l (* x t_2))))))))

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 = t_m * sqrt(2.0);
	return t_s * (t_2 / (t_2 + (l * (l / (x * t_2)))));
}

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
    t_2 = t_m * sqrt(2.0d0)
    code = t_s * (t_2 / (t_2 + (l * (l / (x * t_2)))))
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 = t_m * Math.sqrt(2.0);
	return t_s * (t_2 / (t_2 + (l * (l / (x * t_2)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = t_m * math.sqrt(2.0)
	return t_s * (t_2 / (t_2 + (l * (l / (x * t_2)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	return Float64(t_s * Float64(t_2 / Float64(t_2 + Float64(l * Float64(l / Float64(x * t_2))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	t_2 = t_m * sqrt(2.0);
	tmp = t_s * (t_2 / (t_2 + (l * (l / (x * t_2)))));
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[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(t$95$2 / N[(t$95$2 + N[(l * N[(l / N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.4%

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

3. Taylor expanded in x around inf 31.0%

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

4. Taylor expanded in l around inf 43.6%

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

   1. *-commutative 43.6%

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

6. Simplified 43.6%

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

   1. unpow2 43.6%

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

   2. associate-/l* 49.1%

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

   3. associate-*l* 49.1%

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

   4. *-commutative 49.1%

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

8. Applied egg-rr 49.1%

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

9. Final simplification 49.1%

   \[\leadsto \frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \ell \cdot \frac{\ell}{x \cdot \left(t \cdot \sqrt{2}\right)}} \]
10. Add Preprocessing

Alternative 8: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.3 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{\ell}\\ \end{array} \end{array} \]

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 5.3e+250) (sqrt (/ (+ -1.0 x) (+ x 1.0))) (/ (* t_m (sqrt x)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 5.3e+250) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = (t_m * sqrt(x)) / l;
	}
	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 <= 5.3d+250) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else
        tmp = (t_m * sqrt(x)) / l
    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 <= 5.3e+250) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = (t_m * Math.sqrt(x)) / l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 5.3e+250:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	else:
		tmp = (t_m * math.sqrt(x)) / l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 5.3e+250)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(t_m * sqrt(x)) / l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 5.3e+250)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	else
		tmp = (t_m * sqrt(x)) / l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 5.3e+250], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.2999999999999999e250

   1. Initial program 34.6%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in t around inf 46.5%

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

   6. Taylor expanded in t around 0 46.6%

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

   if 5.2999999999999999e250 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. metadata-eval 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in l around inf 67.0%

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

      1. associate-*l/ 67.6%

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

      2. sqrt-unprod 67.6%

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

      3. metadata-eval 67.6%

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

      4. metadata-eval 67.6%

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

      5. *-rgt-identity 67.6%

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

   8. Applied egg-rr 67.6%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.3 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+249}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{\ell}\\ \end{array} \end{array} \]

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 1.2e+249) (+ 1.0 (/ -1.0 x)) (/ (* t_m (sqrt x)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.2e+249) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = (t_m * sqrt(x)) / l;
	}
	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 <= 1.2d+249) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = (t_m * sqrt(x)) / l
    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 <= 1.2e+249) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = (t_m * Math.sqrt(x)) / l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 1.2e+249:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = (t_m * math.sqrt(x)) / l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 1.2e+249)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(t_m * sqrt(x)) / l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 1.2e+249)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = (t_m * sqrt(x)) / l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 1.2e+249], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.2e249

   1. Initial program 34.6%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in t around inf 46.5%

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

   6. Taylor expanded in x around inf 46.1%

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

   if 1.2e249 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. metadata-eval 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in l around inf 67.0%

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

      1. associate-*l/ 67.6%

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

      2. sqrt-unprod 67.6%

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

      3. metadata-eval 67.6%

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

      4. metadata-eval 67.6%

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

      5. *-rgt-identity 67.6%

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

   8. Applied egg-rr 67.6%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+249}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{+250}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \end{array} \]

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 8.2e+250) (+ 1.0 (/ -1.0 x)) (* t_m (/ (sqrt x) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 8.2e+250) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (sqrt(x) / l);
	}
	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 <= 8.2d+250) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_m * (sqrt(x) / l)
    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 <= 8.2e+250) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (Math.sqrt(x) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 8.2e+250:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_m * (math.sqrt(x) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 8.2e+250)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 8.2e+250)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_m * (sqrt(x) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 8.2e+250], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.19999999999999999e250

   1. Initial program 34.6%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in t around inf 46.5%

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

   6. Taylor expanded in x around inf 46.1%

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

   if 8.19999999999999999e250 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. metadata-eval 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in l around inf 67.0%

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

      1. associate-*l/ 67.6%

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

      2. clear-num 67.0%

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

      3. sqrt-unprod 67.0%

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

      4. metadata-eval 67.0%

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

      5. metadata-eval 67.0%

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

      6. *-rgt-identity 67.0%

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

   8. Applied egg-rr 67.0%

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

   9. Taylor expanded in l around 0 67.2%

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

       1. associate-*l/ 67.6%

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

       2. associate-*r/ 67.6%

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

   11. Simplified 67.6%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{+250}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 33.4%

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

3. Simplified 27.4%

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

5. Taylor expanded in t around inf 45.4%

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

6. Taylor expanded in x around inf 45.0%

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

7. Final simplification 45.0%

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

Alternative 12: 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 33.4%

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

3. Simplified 27.4%

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

5. Taylor expanded in t around inf 45.4%

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

6. Taylor expanded in x around inf 44.6%

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

Reproduce

herbie shell --seed 2024172 
(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)))))