Toniolo and Linder, Equation (7)

Percentage Accurate: 34.1% → 81.5%

Time: 6.8s

Alternatives: 9

Speedup: 3.3×

Specification

Math

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

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: 34.1% accurate, 1.0× speedup

Math

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

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: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := -1 \cdot {\ell}^{2}\\ t_2 := \sqrt{2} \cdot \left|t\right|\\ t_3 := {\left(\left|t\right|\right)}^{2}\\ t_4 := \mathsf{fma}\left(2, t\_3, {\ell}^{2}\right)\\ t_5 := -1 \cdot t\_3\\ t_6 := 2 \cdot t\_3\\ t_7 := -1 \cdot t\_4\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.25 \cdot 10^{-285}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, {\ell}^{2} - t\_1, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_1 - {\ell}^{2}, 2 \cdot \left(t\_3 - t\_5\right)\right)}{x}, 2 \cdot \left(t\_5 - t\_3\right)\right)\right)}{x}, t\_6\right)}}\\ \mathbf{elif}\;\left|t\right| \leq 3.25 \cdot 10^{-224}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{elif}\;\left|t\right| \leq 1.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_4 - t\_7, -1 \cdot \frac{\mathsf{fma}\left(-1, t\_7 - t\_4, \mathsf{fma}\left(2, \frac{t\_3}{x}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{t\_4}{x}}{x}\right)}{x}, t\_6\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* -1.0 (pow l 2.0)))
        (t_2 (* (sqrt 2.0) (fabs t)))
        (t_3 (pow (fabs t) 2.0))
        (t_4 (fma 2.0 t_3 (pow l 2.0)))
        (t_5 (* -1.0 t_3))
        (t_6 (* 2.0 t_3))
        (t_7 (* -1.0 t_4)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.25e-285)
      (/
       t_2
       (sqrt
        (fma
         -1.0
         (/
          (fma
           -1.0
           (- (pow l 2.0) t_1)
           (fma
            -1.0
            (/ (fma -1.0 (- t_1 (pow l 2.0)) (* 2.0 (- t_3 t_5))) x)
            (* 2.0 (- t_5 t_3))))
          x)
         t_6)))
      (if (<= (fabs t) 3.25e-224)
        (- 1.0 (/ 1.0 x))
        (if (<= (fabs t) 1.4e+29)
          (/
           t_2
           (sqrt
            (fma
             -1.0
             (/
              (fma
               -1.0
               (- t_4 t_7)
               (*
                -1.0
                (/
                 (-
                  (fma -1.0 (- t_7 t_4) (fma 2.0 (/ t_3 x) (/ (pow l 2.0) x)))
                  (* -1.0 (/ t_4 x)))
                 x)))
              x)
             t_6)))
          (sqrt (/ -1.0 (/ (- -1.0 x) (- x 1.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = -1.0 * pow(l, 2.0);
	double t_2 = sqrt(2.0) * fabs(t);
	double t_3 = pow(fabs(t), 2.0);
	double t_4 = fma(2.0, t_3, pow(l, 2.0));
	double t_5 = -1.0 * t_3;
	double t_6 = 2.0 * t_3;
	double t_7 = -1.0 * t_4;
	double tmp;
	if (fabs(t) <= 1.25e-285) {
		tmp = t_2 / sqrt(fma(-1.0, (fma(-1.0, (pow(l, 2.0) - t_1), fma(-1.0, (fma(-1.0, (t_1 - pow(l, 2.0)), (2.0 * (t_3 - t_5))) / x), (2.0 * (t_5 - t_3)))) / x), t_6));
	} else if (fabs(t) <= 3.25e-224) {
		tmp = 1.0 - (1.0 / x);
	} else if (fabs(t) <= 1.4e+29) {
		tmp = t_2 / sqrt(fma(-1.0, (fma(-1.0, (t_4 - t_7), (-1.0 * ((fma(-1.0, (t_7 - t_4), fma(2.0, (t_3 / x), (pow(l, 2.0) / x))) - (-1.0 * (t_4 / x))) / x))) / x), t_6));
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(-1.0 * (l ^ 2.0))
	t_2 = Float64(sqrt(2.0) * abs(t))
	t_3 = abs(t) ^ 2.0
	t_4 = fma(2.0, t_3, (l ^ 2.0))
	t_5 = Float64(-1.0 * t_3)
	t_6 = Float64(2.0 * t_3)
	t_7 = Float64(-1.0 * t_4)
	tmp = 0.0
	if (abs(t) <= 1.25e-285)
		tmp = Float64(t_2 / sqrt(fma(-1.0, Float64(fma(-1.0, Float64((l ^ 2.0) - t_1), fma(-1.0, Float64(fma(-1.0, Float64(t_1 - (l ^ 2.0)), Float64(2.0 * Float64(t_3 - t_5))) / x), Float64(2.0 * Float64(t_5 - t_3)))) / x), t_6)));
	elseif (abs(t) <= 3.25e-224)
		tmp = Float64(1.0 - Float64(1.0 / x));
	elseif (abs(t) <= 1.4e+29)
		tmp = Float64(t_2 / sqrt(fma(-1.0, Float64(fma(-1.0, Float64(t_4 - t_7), Float64(-1.0 * Float64(Float64(fma(-1.0, Float64(t_7 - t_4), fma(2.0, Float64(t_3 / x), Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(t_4 / x))) / x))) / x), t_6)));
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(-1.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * t$95$3 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 * t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * t$95$3), $MachinePrecision]}, Block[{t$95$7 = N[(-1.0 * t$95$4), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.25e-285], N[(t$95$2 / N[Sqrt[N[(-1.0 * N[(N[(-1.0 * N[(N[Power[l, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] + N[(-1.0 * N[(N[(-1.0 * N[(t$95$1 - N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$3 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$5 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3.25e-224], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.4e+29], N[(t$95$2 / N[Sqrt[N[(-1.0 * N[(N[(-1.0 * N[(t$95$4 - t$95$7), $MachinePrecision] + N[(-1.0 * N[(N[(N[(-1.0 * N[(t$95$7 - t$95$4), $MachinePrecision] + N[(2.0 * N[(t$95$3 / x), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.25000000000000005e-285

   1. Initial program 34.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. lift-*.f64 N/A

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

   3. Applied rewrites 33.2%

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

   4. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   6. Applied rewrites 53.1%

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

   if 1.25000000000000005e-285 < t < 3.25e-224

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.5

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

   9. Applied rewrites 38.5%

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

   if 3.25e-224 < t < 1.4e29

   1. Initial program 34.1%

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

   2. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 53.2%

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

   if 1.4e29 < t

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := -1 \cdot {\ell}^{2}\\ t_2 := {\left(\left|t\right|\right)}^{2}\\ t_3 := 2 \cdot t\_2\\ t_4 := \mathsf{fma}\left(2, t\_2, {\ell}^{2}\right)\\ t_5 := -1 \cdot t\_2\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.25 \cdot 10^{-285}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, {\ell}^{2} - t\_1, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_1 - {\ell}^{2}, 2 \cdot \left(t\_2 - t\_5\right)\right)}{x}, 2 \cdot \left(t\_5 - t\_2\right)\right)\right)}{x}, t\_3\right)}}\\ \mathbf{elif}\;\left|t\right| \leq 3.25 \cdot 10^{-224}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{elif}\;\left|t\right| \leq 1.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{1.4142135623730951 \cdot \left|t\right|}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_4 - -1 \cdot t\_4, -1 \cdot \frac{t\_4}{x}\right) - \mathsf{fma}\left(2, \frac{t\_2}{x}, \frac{{\ell}^{2}}{x}\right)}{x}, t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* -1.0 (pow l 2.0)))
        (t_2 (pow (fabs t) 2.0))
        (t_3 (* 2.0 t_2))
        (t_4 (fma 2.0 t_2 (pow l 2.0)))
        (t_5 (* -1.0 t_2)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.25e-285)
      (/
       (* (sqrt 2.0) (fabs t))
       (sqrt
        (fma
         -1.0
         (/
          (fma
           -1.0
           (- (pow l 2.0) t_1)
           (fma
            -1.0
            (/ (fma -1.0 (- t_1 (pow l 2.0)) (* 2.0 (- t_2 t_5))) x)
            (* 2.0 (- t_5 t_2))))
          x)
         t_3)))
      (if (<= (fabs t) 3.25e-224)
        (- 1.0 (/ 1.0 x))
        (if (<= (fabs t) 1.4e+29)
          (/
           (* 1.4142135623730951 (fabs t))
           (sqrt
            (fma
             -1.0
             (/
              (-
               (fma -1.0 (- t_4 (* -1.0 t_4)) (* -1.0 (/ t_4 x)))
               (fma 2.0 (/ t_2 x) (/ (pow l 2.0) x)))
              x)
             t_3)))
          (sqrt (/ -1.0 (/ (- -1.0 x) (- x 1.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = -1.0 * pow(l, 2.0);
	double t_2 = pow(fabs(t), 2.0);
	double t_3 = 2.0 * t_2;
	double t_4 = fma(2.0, t_2, pow(l, 2.0));
	double t_5 = -1.0 * t_2;
	double tmp;
	if (fabs(t) <= 1.25e-285) {
		tmp = (sqrt(2.0) * fabs(t)) / sqrt(fma(-1.0, (fma(-1.0, (pow(l, 2.0) - t_1), fma(-1.0, (fma(-1.0, (t_1 - pow(l, 2.0)), (2.0 * (t_2 - t_5))) / x), (2.0 * (t_5 - t_2)))) / x), t_3));
	} else if (fabs(t) <= 3.25e-224) {
		tmp = 1.0 - (1.0 / x);
	} else if (fabs(t) <= 1.4e+29) {
		tmp = (1.4142135623730951 * fabs(t)) / sqrt(fma(-1.0, ((fma(-1.0, (t_4 - (-1.0 * t_4)), (-1.0 * (t_4 / x))) - fma(2.0, (t_2 / x), (pow(l, 2.0) / x))) / x), t_3));
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(-1.0 * (l ^ 2.0))
	t_2 = abs(t) ^ 2.0
	t_3 = Float64(2.0 * t_2)
	t_4 = fma(2.0, t_2, (l ^ 2.0))
	t_5 = Float64(-1.0 * t_2)
	tmp = 0.0
	if (abs(t) <= 1.25e-285)
		tmp = Float64(Float64(sqrt(2.0) * abs(t)) / sqrt(fma(-1.0, Float64(fma(-1.0, Float64((l ^ 2.0) - t_1), fma(-1.0, Float64(fma(-1.0, Float64(t_1 - (l ^ 2.0)), Float64(2.0 * Float64(t_2 - t_5))) / x), Float64(2.0 * Float64(t_5 - t_2)))) / x), t_3)));
	elseif (abs(t) <= 3.25e-224)
		tmp = Float64(1.0 - Float64(1.0 / x));
	elseif (abs(t) <= 1.4e+29)
		tmp = Float64(Float64(1.4142135623730951 * abs(t)) / sqrt(fma(-1.0, Float64(Float64(fma(-1.0, Float64(t_4 - Float64(-1.0 * t_4)), Float64(-1.0 * Float64(t_4 / x))) - fma(2.0, Float64(t_2 / x), Float64((l ^ 2.0) / x))) / x), t_3)));
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(-1.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 * t$95$2), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.25e-285], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-1.0 * N[(N[(-1.0 * N[(N[Power[l, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] + N[(-1.0 * N[(N[(-1.0 * N[(t$95$1 - N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$2 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3.25e-224], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.4e+29], N[(N[(1.4142135623730951 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-1.0 * N[(N[(N[(-1.0 * N[(t$95$4 - N[(-1.0 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(t$95$2 / x), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.25000000000000005e-285

   1. Initial program 34.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. lift-*.f64 N/A

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

   3. Applied rewrites 33.2%

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

   4. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   6. Applied rewrites 53.1%

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

   if 1.25000000000000005e-285 < t < 3.25e-224

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.5

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

   9. Applied rewrites 38.5%

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

   if 3.25e-224 < t < 1.4e29

   1. Initial program 34.1%

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

   2. Evaluated real constant 34.1%

      \[\leadsto \frac{\color{blue}{\frac{6369051672525773}{4503599627370496}} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Taylor expanded in x around -inf

      \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\color{blue}{-1 \cdot \frac{\left(-1 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right) - \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)}{x} + 2 \cdot {t}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \color{blue}{\frac{\left(-1 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right) - \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)}{x}}, 2 \cdot {t}^{2}\right)}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \frac{1.4142135623730951 \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right) - -1 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right), -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}\right) - \mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \frac{{\ell}^{2}}{x}\right)}{x}, 2 \cdot {t}^{2}\right)}}} \]

   if 1.4e29 < t

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ t_2 := \mathsf{fma}\left(2, t\_1, {\ell}^{2}\right)\\ t_3 := \frac{1.4142135623730951 \cdot \left|t\right|}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_2 - -1 \cdot t\_2, -1 \cdot \frac{t\_2}{x}\right) - \mathsf{fma}\left(2, \frac{t\_1}{x}, \frac{{\ell}^{2}}{x}\right)}{x}, 2 \cdot t\_1\right)}}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.25 \cdot 10^{-285}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\left|t\right| \leq 3.25 \cdot 10^{-224}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{elif}\;\left|t\right| \leq 1.4 \cdot 10^{+29}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (pow (fabs t) 2.0))
        (t_2 (fma 2.0 t_1 (pow l 2.0)))
        (t_3
         (/
          (* 1.4142135623730951 (fabs t))
          (sqrt
           (fma
            -1.0
            (/
             (-
              (fma -1.0 (- t_2 (* -1.0 t_2)) (* -1.0 (/ t_2 x)))
              (fma 2.0 (/ t_1 x) (/ (pow l 2.0) x)))
             x)
            (* 2.0 t_1))))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.25e-285)
      t_3
      (if (<= (fabs t) 3.25e-224)
        (- 1.0 (/ 1.0 x))
        (if (<= (fabs t) 1.4e+29)
          t_3
          (sqrt (/ -1.0 (/ (- -1.0 x) (- x 1.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(t), 2.0);
	double t_2 = fma(2.0, t_1, pow(l, 2.0));
	double t_3 = (1.4142135623730951 * fabs(t)) / sqrt(fma(-1.0, ((fma(-1.0, (t_2 - (-1.0 * t_2)), (-1.0 * (t_2 / x))) - fma(2.0, (t_1 / x), (pow(l, 2.0) / x))) / x), (2.0 * t_1)));
	double tmp;
	if (fabs(t) <= 1.25e-285) {
		tmp = t_3;
	} else if (fabs(t) <= 3.25e-224) {
		tmp = 1.0 - (1.0 / x);
	} else if (fabs(t) <= 1.4e+29) {
		tmp = t_3;
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	t_2 = fma(2.0, t_1, (l ^ 2.0))
	t_3 = Float64(Float64(1.4142135623730951 * abs(t)) / sqrt(fma(-1.0, Float64(Float64(fma(-1.0, Float64(t_2 - Float64(-1.0 * t_2)), Float64(-1.0 * Float64(t_2 / x))) - fma(2.0, Float64(t_1 / x), Float64((l ^ 2.0) / x))) / x), Float64(2.0 * t_1))))
	tmp = 0.0
	if (abs(t) <= 1.25e-285)
		tmp = t_3;
	elseif (abs(t) <= 3.25e-224)
		tmp = Float64(1.0 - Float64(1.0 / x));
	elseif (abs(t) <= 1.4e+29)
		tmp = t_3;
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.4142135623730951 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-1.0 * N[(N[(N[(-1.0 * N[(t$95$2 - N[(-1.0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(t$95$1 / x), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.25e-285], t$95$3, If[LessEqual[N[Abs[t], $MachinePrecision], 3.25e-224], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.4e+29], t$95$3, N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.25000000000000005e-285 or 3.25e-224 < t < 1.4e29

   1. Initial program 34.1%

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

   2. Evaluated real constant 34.1%

      \[\leadsto \frac{\color{blue}{\frac{6369051672525773}{4503599627370496}} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Taylor expanded in x around -inf

      \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\color{blue}{-1 \cdot \frac{\left(-1 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right) - \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)}{x} + 2 \cdot {t}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \color{blue}{\frac{\left(-1 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right) - \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)}{x}}, 2 \cdot {t}^{2}\right)}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \frac{1.4142135623730951 \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right) - -1 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right), -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}\right) - \mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \frac{{\ell}^{2}}{x}\right)}{x}, 2 \cdot {t}^{2}\right)}}} \]

   if 1.25000000000000005e-285 < t < 3.25e-224

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.5

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

   9. Applied rewrites 38.5%

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

   if 1.4e29 < t

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

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

Alternative 4: 80.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.25 \cdot 10^{-285}:\\ \;\;\;\;\frac{1.4142135623730951 \cdot \left|t\right|}{\sqrt{\mathsf{fma}\left(2, \frac{t\_1}{x}, \mathsf{fma}\left(2, t\_1, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, t\_1, {\ell}^{2}\right)}{x}}}\\ \mathbf{elif}\;\left|t\right| \leq 3.25 \cdot 10^{-224}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{elif}\;\left|t\right| \leq 2.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, {\ell}^{2} - -1 \cdot {\ell}^{2}, 2 \cdot \left(-1 \cdot t\_1 - t\_1\right)\right)}{x}, 2 \cdot t\_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (pow (fabs t) 2.0)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.25e-285)
      (/
       (* 1.4142135623730951 (fabs t))
       (sqrt
        (-
         (fma 2.0 (/ t_1 x) (fma 2.0 t_1 (/ (pow l 2.0) x)))
         (* -1.0 (/ (fma 2.0 t_1 (pow l 2.0)) x)))))
      (if (<= (fabs t) 3.25e-224)
        (- 1.0 (/ 1.0 x))
        (if (<= (fabs t) 2.6e-43)
          (/
           (* (sqrt 2.0) (fabs t))
           (sqrt
            (fma
             -1.0
             (/
              (fma
               -1.0
               (- (pow l 2.0) (* -1.0 (pow l 2.0)))
               (* 2.0 (- (* -1.0 t_1) t_1)))
              x)
             (* 2.0 t_1))))
          (sqrt (/ -1.0 (/ (- -1.0 x) (- x 1.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(t), 2.0);
	double tmp;
	if (fabs(t) <= 1.25e-285) {
		tmp = (1.4142135623730951 * fabs(t)) / sqrt((fma(2.0, (t_1 / x), fma(2.0, t_1, (pow(l, 2.0) / x))) - (-1.0 * (fma(2.0, t_1, pow(l, 2.0)) / x))));
	} else if (fabs(t) <= 3.25e-224) {
		tmp = 1.0 - (1.0 / x);
	} else if (fabs(t) <= 2.6e-43) {
		tmp = (sqrt(2.0) * fabs(t)) / sqrt(fma(-1.0, (fma(-1.0, (pow(l, 2.0) - (-1.0 * pow(l, 2.0))), (2.0 * ((-1.0 * t_1) - t_1))) / x), (2.0 * t_1)));
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	tmp = 0.0
	if (abs(t) <= 1.25e-285)
		tmp = Float64(Float64(1.4142135623730951 * abs(t)) / sqrt(Float64(fma(2.0, Float64(t_1 / x), fma(2.0, t_1, Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(fma(2.0, t_1, (l ^ 2.0)) / x)))));
	elseif (abs(t) <= 3.25e-224)
		tmp = Float64(1.0 - Float64(1.0 / x));
	elseif (abs(t) <= 2.6e-43)
		tmp = Float64(Float64(sqrt(2.0) * abs(t)) / sqrt(fma(-1.0, Float64(fma(-1.0, Float64((l ^ 2.0) - Float64(-1.0 * (l ^ 2.0))), Float64(2.0 * Float64(Float64(-1.0 * t_1) - t_1))) / x), Float64(2.0 * t_1))));
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.25e-285], N[(N[(1.4142135623730951 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(t$95$1 / x), $MachinePrecision] + N[(2.0 * t$95$1 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(2.0 * t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3.25e-224], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.6e-43], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-1.0 * N[(N[(-1.0 * N[(N[Power[l, 2.0], $MachinePrecision] - N[(-1.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(-1.0 * t$95$1), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.25000000000000005e-285

   1. Initial program 34.1%

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

   2. Evaluated real constant 34.1%

      \[\leadsto \frac{\color{blue}{\frac{6369051672525773}{4503599627370496}} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - \color{blue}{-1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) - \color{blue}{-1} \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \color{blue}{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{\color{blue}{x}}}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}} \]

      13. lower-pow.f64 52.9

          \[\leadsto \frac{1.4142135623730951 \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}} \]

   5. Applied rewrites 52.9%

      \[\leadsto \frac{1.4142135623730951 \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}} \]

   if 1.25000000000000005e-285 < t < 3.25e-224

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.5

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

   9. Applied rewrites 38.5%

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

   if 3.25e-224 < t < 2.6e-43

   1. Initial program 34.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. lift-*.f64 N/A

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

   3. Applied rewrites 33.2%

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

   4. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   6. Applied rewrites 52.9%

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

   if 2.6e-43 < t

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ t_2 := \frac{1.4142135623730951 \cdot \left|t\right|}{\sqrt{\mathsf{fma}\left(2, \frac{t\_1}{x}, \mathsf{fma}\left(2, t\_1, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, t\_1, {\ell}^{2}\right)}{x}}}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.25 \cdot 10^{-285}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\left|t\right| \leq 3.25 \cdot 10^{-224}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{elif}\;\left|t\right| \leq 2.6 \cdot 10^{-43}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (pow (fabs t) 2.0))
        (t_2
         (/
          (* 1.4142135623730951 (fabs t))
          (sqrt
           (-
            (fma 2.0 (/ t_1 x) (fma 2.0 t_1 (/ (pow l 2.0) x)))
            (* -1.0 (/ (fma 2.0 t_1 (pow l 2.0)) x)))))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.25e-285)
      t_2
      (if (<= (fabs t) 3.25e-224)
        (- 1.0 (/ 1.0 x))
        (if (<= (fabs t) 2.6e-43)
          t_2
          (sqrt (/ -1.0 (/ (- -1.0 x) (- x 1.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(t), 2.0);
	double t_2 = (1.4142135623730951 * fabs(t)) / sqrt((fma(2.0, (t_1 / x), fma(2.0, t_1, (pow(l, 2.0) / x))) - (-1.0 * (fma(2.0, t_1, pow(l, 2.0)) / x))));
	double tmp;
	if (fabs(t) <= 1.25e-285) {
		tmp = t_2;
	} else if (fabs(t) <= 3.25e-224) {
		tmp = 1.0 - (1.0 / x);
	} else if (fabs(t) <= 2.6e-43) {
		tmp = t_2;
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	t_2 = Float64(Float64(1.4142135623730951 * abs(t)) / sqrt(Float64(fma(2.0, Float64(t_1 / x), fma(2.0, t_1, Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(fma(2.0, t_1, (l ^ 2.0)) / x)))))
	tmp = 0.0
	if (abs(t) <= 1.25e-285)
		tmp = t_2;
	elseif (abs(t) <= 3.25e-224)
		tmp = Float64(1.0 - Float64(1.0 / x));
	elseif (abs(t) <= 2.6e-43)
		tmp = t_2;
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.4142135623730951 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(t$95$1 / x), $MachinePrecision] + N[(2.0 * t$95$1 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(2.0 * t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.25e-285], t$95$2, If[LessEqual[N[Abs[t], $MachinePrecision], 3.25e-224], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.6e-43], t$95$2, N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.25000000000000005e-285 or 3.25e-224 < t < 2.6e-43

   1. Initial program 34.1%

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

   2. Evaluated real constant 34.1%

      \[\leadsto \frac{\color{blue}{\frac{6369051672525773}{4503599627370496}} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - \color{blue}{-1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) - \color{blue}{-1} \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \color{blue}{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{\color{blue}{x}}}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\frac{6369051672525773}{4503599627370496} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}} \]

      13. lower-pow.f64 52.9

          \[\leadsto \frac{1.4142135623730951 \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}} \]

   5. Applied rewrites 52.9%

      \[\leadsto \frac{1.4142135623730951 \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}} \]

   if 1.25000000000000005e-285 < t < 3.25e-224

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.5

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

   9. Applied rewrites 38.5%

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

   if 2.6e-43 < t

   1. Initial program 34.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.8

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

   4. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      18. lift--.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

      19. distribute-frac-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. Applied rewrites 38.8%

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

Alternative 6: 76.8% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.1%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.8

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

4. Applied rewrites 38.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lift-/.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. add-flip N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

   16. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

   17. sub-negate-rev N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

   18. lift--.f64 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

   19. distribute-frac-neg N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

6. Applied rewrites 38.8%

   \[\leadsto \color{blue}{\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}} \]
7. Add Preprocessing

Alternative 7: 76.8% accurate, 2.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \sqrt{\frac{x - 1}{x - -1}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.1%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.8

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

4. Applied rewrites 38.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lift-/.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. add-flip N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

   16. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

   17. sub-negate-rev N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

   18. lift--.f64 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

   19. distribute-frac-neg N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

6. Applied rewrites 38.8%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   4. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   5. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

   6. sub-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-out N/A

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

   9. lift-+.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. lift--.f64 N/A

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

   13. frac-2neg N/A

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

   14. lift-/.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lift-/.f64 N/A

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

   17. associate-*r/ N/A

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

   18. div-flip-rev N/A

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

   19. lift--.f64 N/A

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

   20. sub-negate-rev N/A

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

   21. lift--.f64 N/A

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

   22. lift-+.f64 N/A

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

   23. +-commutative N/A

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

   24. mul-1-neg N/A

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

8. Applied rewrites 38.8%

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

Alternative 8: 76.3% accurate, 3.1× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \left(1 - \frac{1}{x}\right) \]

FPCore

(FPCore (x l t) :precision binary64 (* (copysign 1.0 t) (- 1.0 (/ 1.0 x))))

C

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

Java

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

Python

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

Julia

function code(x, l, t)
	return Float64(copysign(1.0, t) * Float64(1.0 - Float64(1.0 / x)))
end

MATLAB

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

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.1%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.8

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

4. Applied rewrites 38.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lift-/.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. add-flip N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x - 1}}} \]

   16. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{x - -1}{x - 1}}} \]

   17. sub-negate-rev N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

   18. lift--.f64 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\frac{\mathsf{neg}\left(\left(-1 - x\right)\right)}{x - 1}}} \]

   19. distribute-frac-neg N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{-1 - x}{x - 1}\right)}} \]

6. Applied rewrites 38.8%

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

7. Taylor expanded in x around inf

   \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
8. Step-by-step derivation

   1. lower--.f64 N/A

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

   2. lower-/.f64 38.5

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

9. Applied rewrites 38.5%

   \[\leadsto 1 - \color{blue}{\frac{1}{x}} \]
10. Add Preprocessing

Alternative 9: 75.6% accurate, 3.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \frac{1.4142135623730951}{\sqrt{2}} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (* (copysign 1.0 t) (/ 1.4142135623730951 (sqrt 2.0))))

C

double code(double x, double l, double t) {
	return copysign(1.0, t) * (1.4142135623730951 / sqrt(2.0));
}

Java

public static double code(double x, double l, double t) {
	return Math.copySign(1.0, t) * (1.4142135623730951 / Math.sqrt(2.0));
}

Python

def code(x, l, t):
	return math.copysign(1.0, t) * (1.4142135623730951 / math.sqrt(2.0))

Julia

function code(x, l, t)
	return Float64(copysign(1.0, t) * Float64(1.4142135623730951 / sqrt(2.0)))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sign(t) * abs(1.0)) * (1.4142135623730951 / sqrt(2.0));
end

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(1.4142135623730951 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.1%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.8

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

4. Applied rewrites 38.8%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\sqrt{2}}{\sqrt{2}} \]
6. Step-by-step derivation

7. Applied rewrites 38.2%

   \[\leadsto \frac{\sqrt{2}}{\sqrt{2}} \]

8. Evaluated real constant 38.2%

   \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{\color{blue}{2}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025178 
(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)))))