Toniolo and Linder, Equation (7)

Percentage Accurate: 32.8% → 82.9%

Time: 14.0s

Alternatives: 4

Speedup: N/A×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, l, t)
use fmin_fmax_functions
    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: 32.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, l, t)
use fmin_fmax_functions
    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: 82.9% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t_3 := \left(x \cdot x\right) \cdot x\\ t_4 := t\_m \cdot {2}^{0.5}\\ t_5 := {\left({2}^{0.5} \cdot t\_m\right)}^{1}\\ t_6 := \sqrt{2} \cdot t\_m\\ t_7 := \frac{t\_6}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(l\_m \cdot l\_m + 2 \cdot \left(t\_m \cdot t\_m\right)\right) - l\_m \cdot l\_m}}\\ t_8 := t\_2 \cdot -1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_7 \leq 0:\\ \;\;\;\;{\left(\frac{x - 1}{1 + x}\right)}^{0.5}\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\frac{t\_6}{\sqrt{\left(\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{\left(t\_m \cdot t\_m\right) \cdot 2}{t\_3}\right) + \mathsf{fma}\left(t\_5, t\_5, \frac{l\_m \cdot l\_m}{t\_3} + \frac{l\_m \cdot l\_m}{x}\right)\right) - \left(\mathsf{fma}\left(\frac{t\_2}{x}, -1, \frac{t\_8}{t\_3}\right) + \frac{t\_2 - t\_8}{x \cdot x} \cdot -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.5, t\_m \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t\_4}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t\_4}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t\_m \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l_m l_m)))
        (t_3 (* (* x x) x))
        (t_4 (* t_m (pow 2.0 0.5)))
        (t_5 (pow (* (pow 2.0 0.5) t_m) 1.0))
        (t_6 (* (sqrt 2.0) t_m))
        (t_7
         (/
          t_6
          (sqrt
           (-
            (* (/ (+ x 1.0) (- x 1.0)) (+ (* l_m l_m) (* 2.0 (* t_m t_m))))
            (* l_m l_m)))))
        (t_8 (* t_2 -1.0)))
   (*
    t_s
    (if (<= t_7 0.0)
      (pow (/ (- x 1.0) (+ 1.0 x)) 0.5)
      (if (<= t_7 INFINITY)
        (/
         t_6
         (sqrt
          (-
           (+
            (fma (/ (* t_m t_m) x) 2.0 (/ (* (* t_m t_m) 2.0) t_3))
            (fma t_5 t_5 (+ (/ (* l_m l_m) t_3) (/ (* l_m l_m) x))))
           (+
            (fma (/ t_2 x) -1.0 (/ t_8 t_3))
            (* (/ (- t_2 t_8) (* x x)) -1.0)))))
        (/
         (*
          x
          (fma
           -0.5
           (* t_m (pow (pow (pow x 3.0) -1.0) 0.5))
           (+
            (*
             -0.0625
             (* (/ t_4 (pow 0.5 0.5)) (pow (pow (pow x 5.0) -1.0) 0.5)))
            (fma
             -0.0078125
             (*
              (/ t_4 (pow (pow 0.5 0.5) 5.0))
              (pow (pow (pow x 7.0) -1.0) 0.5))
             (* t_m (pow (pow x -1.0) 0.5))))))
         l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double t_3 = (x * x) * x;
	double t_4 = t_m * pow(2.0, 0.5);
	double t_5 = pow((pow(2.0, 0.5) * t_m), 1.0);
	double t_6 = sqrt(2.0) * t_m;
	double t_7 = t_6 / sqrt(((((x + 1.0) / (x - 1.0)) * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)));
	double t_8 = t_2 * -1.0;
	double tmp;
	if (t_7 <= 0.0) {
		tmp = pow(((x - 1.0) / (1.0 + x)), 0.5);
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = t_6 / sqrt(((fma(((t_m * t_m) / x), 2.0, (((t_m * t_m) * 2.0) / t_3)) + fma(t_5, t_5, (((l_m * l_m) / t_3) + ((l_m * l_m) / x)))) - (fma((t_2 / x), -1.0, (t_8 / t_3)) + (((t_2 - t_8) / (x * x)) * -1.0))));
	} else {
		tmp = (x * fma(-0.5, (t_m * pow(pow(pow(x, 3.0), -1.0), 0.5)), ((-0.0625 * ((t_4 / pow(0.5, 0.5)) * pow(pow(pow(x, 5.0), -1.0), 0.5))) + fma(-0.0078125, ((t_4 / pow(pow(0.5, 0.5), 5.0)) * pow(pow(pow(x, 7.0), -1.0), 0.5)), (t_m * pow(pow(x, -1.0), 0.5)))))) / l_m;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	t_3 = Float64(Float64(x * x) * x)
	t_4 = Float64(t_m * (2.0 ^ 0.5))
	t_5 = Float64((2.0 ^ 0.5) * t_m) ^ 1.0
	t_6 = Float64(sqrt(2.0) * t_m)
	t_7 = Float64(t_6 / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l_m * l_m) + Float64(2.0 * Float64(t_m * t_m)))) - Float64(l_m * l_m))))
	t_8 = Float64(t_2 * -1.0)
	tmp = 0.0
	if (t_7 <= 0.0)
		tmp = Float64(Float64(x - 1.0) / Float64(1.0 + x)) ^ 0.5;
	elseif (t_7 <= Inf)
		tmp = Float64(t_6 / sqrt(Float64(Float64(fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(Float64(t_m * t_m) * 2.0) / t_3)) + fma(t_5, t_5, Float64(Float64(Float64(l_m * l_m) / t_3) + Float64(Float64(l_m * l_m) / x)))) - Float64(fma(Float64(t_2 / x), -1.0, Float64(t_8 / t_3)) + Float64(Float64(Float64(t_2 - t_8) / Float64(x * x)) * -1.0)))));
	else
		tmp = Float64(Float64(x * fma(-0.5, Float64(t_m * (((x ^ 3.0) ^ -1.0) ^ 0.5)), Float64(Float64(-0.0625 * Float64(Float64(t_4 / (0.5 ^ 0.5)) * (((x ^ 5.0) ^ -1.0) ^ 0.5))) + fma(-0.0078125, Float64(Float64(t_4 / ((0.5 ^ 0.5) ^ 5.0)) * (((x ^ 7.0) ^ -1.0) ^ 0.5)), Float64(t_m * ((x ^ -1.0) ^ 0.5)))))) / l_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[Power[2.0, 0.5], $MachinePrecision] * t$95$m), $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$2 * -1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$7, 0.0], N[Power[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(t$95$6 / N[Sqrt[N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 * t$95$5 + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t$95$2 / x), $MachinePrecision] * -1.0 + N[(t$95$8 / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 - t$95$8), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-0.5 * N[(t$95$m * N[Power[N[Power[N[Power[x, 3.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[(N[(t$95$4 / N[Power[0.5, 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 5.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0078125 * N[(N[(t$95$4 / N[Power[N[Power[0.5, 0.5], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 7.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Power[N[Power[x, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < 0.0

   1. Initial program 40.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{2}^{\frac{1}{2}} \cdot t} \]

      4. lower-pow.f64 19.2

         \[\leadsto \frac{\sqrt{2} \cdot t}{{2}^{0.5} \cdot t} \]

   5. Applied rewrites 19.2%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{2}^{0.5} \cdot t}} \]

   6. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 19.2

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

   8. Applied rewrites 19.2%

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

   if 0.0 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 56.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 88.2%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{32} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{3}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + \frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

   8. Applied rewrites 44.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-0.25, \left(\frac{t}{\ell} \cdot 2\right) \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.03125, \frac{t \cdot {2}^{0.5}}{\ell \cdot \left({0.5}^{0.5} \cdot 0.5\right)} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{\ell \cdot {\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, \frac{t \cdot 1}{\ell} \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)} \]

   9. Taylor expanded in l around 0

      \[\leadsto \frac{x \cdot \left(\frac{-1}{2} \cdot \left(t \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{16} \cdot \left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{{\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + t \cdot \sqrt{\frac{1}{x}}\right)\right)\right)}{\ell} \]

   11. Applied rewrites 52.1%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(-0.5, t \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t \cdot {2}^{0.5}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{\ell} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq 0:\\ \;\;\;\;{\left(\frac{x - 1}{1 + x}\right)}^{0.5}\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\left(t \cdot t\right) \cdot 2}{\left(x \cdot x\right) \cdot x}\right) + \mathsf{fma}\left({\left({2}^{0.5} \cdot t\right)}^{1}, {\left({2}^{0.5} \cdot t\right)}^{1}, \frac{\ell \cdot \ell}{\left(x \cdot x\right) \cdot x} + \frac{\ell \cdot \ell}{x}\right)\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}, -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -1}{\left(x \cdot x\right) \cdot x}\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) - \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -1}{x \cdot x} \cdot -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.5, t \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t \cdot {2}^{0.5}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.7% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {2}^{0.5}\\ t_3 := {\left(\frac{x - 1}{1 + x}\right)}^{0.5}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.05 \cdot 10^{+72}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;l\_m \leq 2.15 \cdot 10^{+162}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{{\left(e^{0.5}\right)}^{\left(\log 2 - \log x\right)} \cdot l\_m}\\ \mathbf{elif}\;l\_m \leq 1.65 \cdot 10^{+173}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.5, t\_m \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t\_2}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t\_2}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t\_m \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (pow 2.0 0.5))) (t_3 (pow (/ (- x 1.0) (+ 1.0 x)) 0.5)))
   (*
    t_s
    (if (<= l_m 1.05e+72)
      t_3
      (if (<= l_m 2.15e+162)
        (/ (* (sqrt 2.0) t_m) (* (pow (exp 0.5) (- (log 2.0) (log x))) l_m))
        (if (<= l_m 1.65e+173)
          t_3
          (/
           (*
            x
            (fma
             -0.5
             (* t_m (pow (pow (pow x 3.0) -1.0) 0.5))
             (+
              (*
               -0.0625
               (* (/ t_2 (pow 0.5 0.5)) (pow (pow (pow x 5.0) -1.0) 0.5)))
              (fma
               -0.0078125
               (*
                (/ t_2 (pow (pow 0.5 0.5) 5.0))
                (pow (pow (pow x 7.0) -1.0) 0.5))
               (* t_m (pow (pow x -1.0) 0.5))))))
           l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * pow(2.0, 0.5);
	double t_3 = pow(((x - 1.0) / (1.0 + x)), 0.5);
	double tmp;
	if (l_m <= 1.05e+72) {
		tmp = t_3;
	} else if (l_m <= 2.15e+162) {
		tmp = (sqrt(2.0) * t_m) / (pow(exp(0.5), (log(2.0) - log(x))) * l_m);
	} else if (l_m <= 1.65e+173) {
		tmp = t_3;
	} else {
		tmp = (x * fma(-0.5, (t_m * pow(pow(pow(x, 3.0), -1.0), 0.5)), ((-0.0625 * ((t_2 / pow(0.5, 0.5)) * pow(pow(pow(x, 5.0), -1.0), 0.5))) + fma(-0.0078125, ((t_2 / pow(pow(0.5, 0.5), 5.0)) * pow(pow(pow(x, 7.0), -1.0), 0.5)), (t_m * pow(pow(x, -1.0), 0.5)))))) / l_m;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * (2.0 ^ 0.5))
	t_3 = Float64(Float64(x - 1.0) / Float64(1.0 + x)) ^ 0.5
	tmp = 0.0
	if (l_m <= 1.05e+72)
		tmp = t_3;
	elseif (l_m <= 2.15e+162)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64((exp(0.5) ^ Float64(log(2.0) - log(x))) * l_m));
	elseif (l_m <= 1.65e+173)
		tmp = t_3;
	else
		tmp = Float64(Float64(x * fma(-0.5, Float64(t_m * (((x ^ 3.0) ^ -1.0) ^ 0.5)), Float64(Float64(-0.0625 * Float64(Float64(t_2 / (0.5 ^ 0.5)) * (((x ^ 5.0) ^ -1.0) ^ 0.5))) + fma(-0.0078125, Float64(Float64(t_2 / ((0.5 ^ 0.5) ^ 5.0)) * (((x ^ 7.0) ^ -1.0) ^ 0.5)), Float64(t_m * ((x ^ -1.0) ^ 0.5)))))) / l_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 1.05e+72], t$95$3, If[LessEqual[l$95$m, 2.15e+162], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Power[N[Exp[0.5], $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 1.65e+173], t$95$3, N[(N[(x * N[(-0.5 * N[(t$95$m * N[Power[N[Power[N[Power[x, 3.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[(N[(t$95$2 / N[Power[0.5, 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 5.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0078125 * N[(N[(t$95$2 / N[Power[N[Power[0.5, 0.5], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 7.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Power[N[Power[x, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.0500000000000001e72 or 2.1500000000000001e162 < l < 1.64999999999999998e173

   1. Initial program 43.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{2}^{\frac{1}{2}} \cdot t} \]

      4. lower-pow.f64 35.1

         \[\leadsto \frac{\sqrt{2} \cdot t}{{2}^{0.5} \cdot t} \]

   5. Applied rewrites 35.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{2}^{0.5} \cdot t}} \]

   6. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 35.6

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

   8. Applied rewrites 35.6%

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

   if 1.0500000000000001e72 < l < 2.1500000000000001e162

   1. Initial program 11.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 21.0%

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

   6. Taylor expanded in x around inf

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

      1. exp-prod N/A

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

      2. lower-pow.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 59.9

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

   8. Applied rewrites 59.9%

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

   if 1.64999999999999998e173 < 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 l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{32} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{3}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + \frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

   8. Applied rewrites 71.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-0.25, \left(\frac{t}{\ell} \cdot 2\right) \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.03125, \frac{t \cdot {2}^{0.5}}{\ell \cdot \left({0.5}^{0.5} \cdot 0.5\right)} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{\ell \cdot {\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, \frac{t \cdot 1}{\ell} \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)} \]

   9. Taylor expanded in l around 0

      \[\leadsto \frac{x \cdot \left(\frac{-1}{2} \cdot \left(t \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{16} \cdot \left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{{\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + t \cdot \sqrt{\frac{1}{x}}\right)\right)\right)}{\ell} \]

   11. Applied rewrites 84.6%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(-0.5, t \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t \cdot {2}^{0.5}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{\ell} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+72}:\\ \;\;\;\;{\left(\frac{x - 1}{1 + x}\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{+162}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{{\left(e^{0.5}\right)}^{\left(\log 2 - \log x\right)} \cdot \ell}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+173}:\\ \;\;\;\;{\left(\frac{x - 1}{1 + x}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.5, t \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t \cdot {2}^{0.5}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {2}^{0.5}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.05 \cdot 10^{+72}:\\ \;\;\;\;{\left(\frac{x - 1}{1 + x}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.5, t\_m \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t\_2}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t\_2}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t\_m \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (pow 2.0 0.5))))
   (*
    t_s
    (if (<= l_m 1.05e+72)
      (pow (/ (- x 1.0) (+ 1.0 x)) 0.5)
      (/
       (*
        x
        (fma
         -0.5
         (* t_m (pow (pow (pow x 3.0) -1.0) 0.5))
         (+
          (*
           -0.0625
           (* (/ t_2 (pow 0.5 0.5)) (pow (pow (pow x 5.0) -1.0) 0.5)))
          (fma
           -0.0078125
           (* (/ t_2 (pow (pow 0.5 0.5) 5.0)) (pow (pow (pow x 7.0) -1.0) 0.5))
           (* t_m (pow (pow x -1.0) 0.5))))))
       l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * pow(2.0, 0.5);
	double tmp;
	if (l_m <= 1.05e+72) {
		tmp = pow(((x - 1.0) / (1.0 + x)), 0.5);
	} else {
		tmp = (x * fma(-0.5, (t_m * pow(pow(pow(x, 3.0), -1.0), 0.5)), ((-0.0625 * ((t_2 / pow(0.5, 0.5)) * pow(pow(pow(x, 5.0), -1.0), 0.5))) + fma(-0.0078125, ((t_2 / pow(pow(0.5, 0.5), 5.0)) * pow(pow(pow(x, 7.0), -1.0), 0.5)), (t_m * pow(pow(x, -1.0), 0.5)))))) / l_m;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * (2.0 ^ 0.5))
	tmp = 0.0
	if (l_m <= 1.05e+72)
		tmp = Float64(Float64(x - 1.0) / Float64(1.0 + x)) ^ 0.5;
	else
		tmp = Float64(Float64(x * fma(-0.5, Float64(t_m * (((x ^ 3.0) ^ -1.0) ^ 0.5)), Float64(Float64(-0.0625 * Float64(Float64(t_2 / (0.5 ^ 0.5)) * (((x ^ 5.0) ^ -1.0) ^ 0.5))) + fma(-0.0078125, Float64(Float64(t_2 / ((0.5 ^ 0.5) ^ 5.0)) * (((x ^ 7.0) ^ -1.0) ^ 0.5)), Float64(t_m * ((x ^ -1.0) ^ 0.5)))))) / l_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 1.05e+72], N[Power[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[(x * N[(-0.5 * N[(t$95$m * N[Power[N[Power[N[Power[x, 3.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[(N[(t$95$2 / N[Power[0.5, 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 5.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0078125 * N[(N[(t$95$2 / N[Power[N[Power[0.5, 0.5], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 7.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Power[N[Power[x, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.0500000000000001e72

   1. Initial program 44.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{2}^{\frac{1}{2}} \cdot t} \]

      4. lower-pow.f64 35.9

         \[\leadsto \frac{\sqrt{2} \cdot t}{{2}^{0.5} \cdot t} \]

   5. Applied rewrites 35.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{2}^{0.5} \cdot t}} \]

   6. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 36.5

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

   8. Applied rewrites 36.5%

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

   if 1.0500000000000001e72 < l

   1. Initial program 5.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{32} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{3}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + \frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

   8. Applied rewrites 48.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-0.25, \left(\frac{t}{\ell} \cdot 2\right) \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.03125, \frac{t \cdot {2}^{0.5}}{\ell \cdot \left({0.5}^{0.5} \cdot 0.5\right)} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{\ell \cdot {\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, \frac{t \cdot 1}{\ell} \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)} \]

   9. Taylor expanded in l around 0

      \[\leadsto \frac{x \cdot \left(\frac{-1}{2} \cdot \left(t \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{16} \cdot \left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{{\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + t \cdot \sqrt{\frac{1}{x}}\right)\right)\right)}{\ell} \]

   11. Applied rewrites 63.6%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(-0.5, t \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t \cdot {2}^{0.5}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+72}:\\ \;\;\;\;{\left(\frac{x - 1}{1 + x}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.5, t \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t \cdot {2}^{0.5}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 22.6% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {2}^{0.5}\\ t\_s \cdot \frac{x \cdot \mathsf{fma}\left(-0.5, t\_m \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t\_2}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t\_2}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t\_m \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{l\_m} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (pow 2.0 0.5))))
   (*
    t_s
    (/
     (*
      x
      (fma
       -0.5
       (* t_m (pow (pow (pow x 3.0) -1.0) 0.5))
       (+
        (* -0.0625 (* (/ t_2 (pow 0.5 0.5)) (pow (pow (pow x 5.0) -1.0) 0.5)))
        (fma
         -0.0078125
         (* (/ t_2 (pow (pow 0.5 0.5) 5.0)) (pow (pow (pow x 7.0) -1.0) 0.5))
         (* t_m (pow (pow x -1.0) 0.5))))))
     l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * pow(2.0, 0.5);
	return t_s * ((x * fma(-0.5, (t_m * pow(pow(pow(x, 3.0), -1.0), 0.5)), ((-0.0625 * ((t_2 / pow(0.5, 0.5)) * pow(pow(pow(x, 5.0), -1.0), 0.5))) + fma(-0.0078125, ((t_2 / pow(pow(0.5, 0.5), 5.0)) * pow(pow(pow(x, 7.0), -1.0), 0.5)), (t_m * pow(pow(x, -1.0), 0.5)))))) / l_m);
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * (2.0 ^ 0.5))
	return Float64(t_s * Float64(Float64(x * fma(-0.5, Float64(t_m * (((x ^ 3.0) ^ -1.0) ^ 0.5)), Float64(Float64(-0.0625 * Float64(Float64(t_2 / (0.5 ^ 0.5)) * (((x ^ 5.0) ^ -1.0) ^ 0.5))) + fma(-0.0078125, Float64(Float64(t_2 / ((0.5 ^ 0.5) ^ 5.0)) * (((x ^ 7.0) ^ -1.0) ^ 0.5)), Float64(t_m * ((x ^ -1.0) ^ 0.5)))))) / l_m))
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(N[(x * N[(-0.5 * N[(t$95$m * N[Power[N[Power[N[Power[x, 3.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[(N[(t$95$2 / N[Power[0.5, 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 5.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0078125 * N[(N[(t$95$2 / N[Power[N[Power[0.5, 0.5], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 7.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Power[N[Power[x, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 37.0%

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

3. Taylor expanded in l around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 9.1%

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

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{32} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{3}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + \frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

8. Applied rewrites 14.5%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-0.25, \left(\frac{t}{\ell} \cdot 2\right) \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.03125, \frac{t \cdot {2}^{0.5}}{\ell \cdot \left({0.5}^{0.5} \cdot 0.5\right)} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{\ell \cdot {\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, \frac{t \cdot 1}{\ell} \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)} \]

9. Taylor expanded in l around 0

   \[\leadsto \frac{x \cdot \left(\frac{-1}{2} \cdot \left(t \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{16} \cdot \left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{{\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + t \cdot \sqrt{\frac{1}{x}}\right)\right)\right)}{\ell} \]

11. Applied rewrites 17.1%

    \[\leadsto \frac{x \cdot \mathsf{fma}\left(-0.5, t \cdot {\left({\left({x}^{3}\right)}^{-1}\right)}^{0.5}, -0.0625 \cdot \left(\frac{t \cdot {2}^{0.5}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}\right) + \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)}{\ell} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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)))))