Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 84.0%

Time: 9.2s

Alternatives: 13

Speedup: 0.7×

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: 33.3% 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: 84.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ t_2 := 2 \cdot t\_1\\ t_3 := {\left(\left|\ell\right|\right)}^{2}\\ t_4 := t\_2 + t\_3\\ t_5 := \sqrt{2} \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{t\_5}{\left|\ell\right| \cdot \sqrt{\frac{4}{\left(x - 1\right) \cdot 2}}}\\ \mathbf{elif}\;\left|t\right| \leq 4.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{t\_5}{\sqrt{-1 \cdot \frac{\left(-1 \cdot \left(t\_4 - -1 \cdot t\_4\right) + -1 \cdot \frac{t\_4}{x}\right) - \left(2 \cdot \frac{t\_1}{x} + \frac{t\_3}{x}\right)}{x} + t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (let* ((t_1 (pow (fabs t) 2.0))
       (t_2 (* 2.0 t_1))
       (t_3 (pow (fabs l) 2.0))
       (t_4 (+ t_2 t_3))
       (t_5 (* (sqrt 2.0) (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 5.4e-142)
     (/ t_5 (* (fabs l) (sqrt (/ 4.0 (* (- x 1.0) 2.0)))))
     (if (<= (fabs t) 4.5e+32)
       (/
        t_5
        (sqrt
         (+
          (*
           -1.0
           (/
            (-
             (+ (* -1.0 (- t_4 (* -1.0 t_4))) (* -1.0 (/ t_4 x)))
             (+ (* 2.0 (/ t_1 x)) (/ t_3 x)))
            x))
          t_2)))
       (/ 1.0 (sqrt (/ (- -1.0 x) (- 1.0 x)))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(t), 2.0);
	double t_2 = 2.0 * t_1;
	double t_3 = pow(fabs(l), 2.0);
	double t_4 = t_2 + t_3;
	double t_5 = sqrt(2.0) * fabs(t);
	double tmp;
	if (fabs(t) <= 5.4e-142) {
		tmp = t_5 / (fabs(l) * sqrt((4.0 / ((x - 1.0) * 2.0))));
	} else if (fabs(t) <= 4.5e+32) {
		tmp = t_5 / sqrt(((-1.0 * ((((-1.0 * (t_4 - (-1.0 * t_4))) + (-1.0 * (t_4 / x))) - ((2.0 * (t_1 / x)) + (t_3 / x))) / x)) + t_2));
	} else {
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double t_1 = Math.pow(Math.abs(t), 2.0);
	double t_2 = 2.0 * t_1;
	double t_3 = Math.pow(Math.abs(l), 2.0);
	double t_4 = t_2 + t_3;
	double t_5 = Math.sqrt(2.0) * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 5.4e-142) {
		tmp = t_5 / (Math.abs(l) * Math.sqrt((4.0 / ((x - 1.0) * 2.0))));
	} else if (Math.abs(t) <= 4.5e+32) {
		tmp = t_5 / Math.sqrt(((-1.0 * ((((-1.0 * (t_4 - (-1.0 * t_4))) + (-1.0 * (t_4 / x))) - ((2.0 * (t_1 / x)) + (t_3 / x))) / x)) + t_2));
	} else {
		tmp = 1.0 / Math.sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	t_1 = math.pow(math.fabs(t), 2.0)
	t_2 = 2.0 * t_1
	t_3 = math.pow(math.fabs(l), 2.0)
	t_4 = t_2 + t_3
	t_5 = math.sqrt(2.0) * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 5.4e-142:
		tmp = t_5 / (math.fabs(l) * math.sqrt((4.0 / ((x - 1.0) * 2.0))))
	elif math.fabs(t) <= 4.5e+32:
		tmp = t_5 / math.sqrt(((-1.0 * ((((-1.0 * (t_4 - (-1.0 * t_4))) + (-1.0 * (t_4 / x))) - ((2.0 * (t_1 / x)) + (t_3 / x))) / x)) + t_2))
	else:
		tmp = 1.0 / math.sqrt(((-1.0 - x) / (1.0 - x)))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	t_2 = Float64(2.0 * t_1)
	t_3 = abs(l) ^ 2.0
	t_4 = Float64(t_2 + t_3)
	t_5 = Float64(sqrt(2.0) * abs(t))
	tmp = 0.0
	if (abs(t) <= 5.4e-142)
		tmp = Float64(t_5 / Float64(abs(l) * sqrt(Float64(4.0 / Float64(Float64(x - 1.0) * 2.0)))));
	elseif (abs(t) <= 4.5e+32)
		tmp = Float64(t_5 / sqrt(Float64(Float64(-1.0 * Float64(Float64(Float64(Float64(-1.0 * Float64(t_4 - Float64(-1.0 * t_4))) + Float64(-1.0 * Float64(t_4 / x))) - Float64(Float64(2.0 * Float64(t_1 / x)) + Float64(t_3 / x))) / x)) + t_2)));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = abs(t) ^ 2.0;
	t_2 = 2.0 * t_1;
	t_3 = abs(l) ^ 2.0;
	t_4 = t_2 + t_3;
	t_5 = sqrt(2.0) * abs(t);
	tmp = 0.0;
	if (abs(t) <= 5.4e-142)
		tmp = t_5 / (abs(l) * sqrt((4.0 / ((x - 1.0) * 2.0))));
	elseif (abs(t) <= 4.5e+32)
		tmp = t_5 / sqrt(((-1.0 * ((((-1.0 * (t_4 - (-1.0 * t_4))) + (-1.0 * (t_4 / x))) - ((2.0 * (t_1 / x)) + (t_3 / x))) / x)) + t_2));
	else
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * 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), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $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], 5.4e-142], N[(t$95$5 / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(4.0 / N[(N[(x - 1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 4.5e+32], N[(t$95$5 / N[Sqrt[N[(N[(-1.0 * N[(N[(N[(N[(-1.0 * N[(t$95$4 - N[(-1.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.3999999999999996e-142

   1. Initial program 33.3%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 2.7%

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

   4. Applied rewrites 2.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-add N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 2.9%

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

   6. Applied rewrites 2.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 15.0%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{4}{\left(x - 1\right) \cdot 2}}} \]

   if 5.3999999999999996e-142 < t < 4.5000000000000003e32

   1. Initial program 33.3%

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

   2. Taylor expanded in x around -inf

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 51.2%

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

   if 4.5000000000000003e32 < t

   1. Initial program 33.3%

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

   2. 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 40.1%

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

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
   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. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 40.1%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. sub-negate-rev N/A

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

      20. lower--.f64 N/A

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

      21. lift--.f64 N/A

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

      22. sub-negate-rev N/A

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

      23. lower--.f64 40.1%

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

   6. Applied rewrites 40.1%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval 40.1%

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

   8. Applied rewrites 40.1%

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

Alternative 2: 83.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ t_2 := 2 \cdot t\_1\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{4}{\left(x - 1\right) \cdot 2}}}\\ \mathbf{elif}\;\left|t\right| \leq 4.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-1 \cdot \frac{-2 \cdot t\_1 - \left(2 \cdot {\left(\left|\ell\right|\right)}^{2} + t\_2\right)}{x} + t\_2}}{\left|t\right| \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (let* ((t_1 (pow (fabs t) 2.0)) (t_2 (* 2.0 t_1)))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 5.4e-142)
     (/
      (* (sqrt 2.0) (fabs t))
      (* (fabs l) (sqrt (/ 4.0 (* (- x 1.0) 2.0)))))
     (if (<= (fabs t) 4.5e+32)
       (/
        1.0
        (/
         (sqrt
          (+
           (*
            -1.0
            (/ (- (* -2.0 t_1) (+ (* 2.0 (pow (fabs l) 2.0)) t_2)) x))
           t_2))
         (* (fabs t) (sqrt 2.0))))
       (/ 1.0 (sqrt (/ (- -1.0 x) (- 1.0 x)))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(t), 2.0);
	double t_2 = 2.0 * t_1;
	double tmp;
	if (fabs(t) <= 5.4e-142) {
		tmp = (sqrt(2.0) * fabs(t)) / (fabs(l) * sqrt((4.0 / ((x - 1.0) * 2.0))));
	} else if (fabs(t) <= 4.5e+32) {
		tmp = 1.0 / (sqrt(((-1.0 * (((-2.0 * t_1) - ((2.0 * pow(fabs(l), 2.0)) + t_2)) / x)) + t_2)) / (fabs(t) * sqrt(2.0)));
	} else {
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double t_1 = Math.pow(Math.abs(t), 2.0);
	double t_2 = 2.0 * t_1;
	double tmp;
	if (Math.abs(t) <= 5.4e-142) {
		tmp = (Math.sqrt(2.0) * Math.abs(t)) / (Math.abs(l) * Math.sqrt((4.0 / ((x - 1.0) * 2.0))));
	} else if (Math.abs(t) <= 4.5e+32) {
		tmp = 1.0 / (Math.sqrt(((-1.0 * (((-2.0 * t_1) - ((2.0 * Math.pow(Math.abs(l), 2.0)) + t_2)) / x)) + t_2)) / (Math.abs(t) * Math.sqrt(2.0)));
	} else {
		tmp = 1.0 / Math.sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	t_1 = math.pow(math.fabs(t), 2.0)
	t_2 = 2.0 * t_1
	tmp = 0
	if math.fabs(t) <= 5.4e-142:
		tmp = (math.sqrt(2.0) * math.fabs(t)) / (math.fabs(l) * math.sqrt((4.0 / ((x - 1.0) * 2.0))))
	elif math.fabs(t) <= 4.5e+32:
		tmp = 1.0 / (math.sqrt(((-1.0 * (((-2.0 * t_1) - ((2.0 * math.pow(math.fabs(l), 2.0)) + t_2)) / x)) + t_2)) / (math.fabs(t) * math.sqrt(2.0)))
	else:
		tmp = 1.0 / math.sqrt(((-1.0 - x) / (1.0 - x)))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	t_2 = Float64(2.0 * t_1)
	tmp = 0.0
	if (abs(t) <= 5.4e-142)
		tmp = Float64(Float64(sqrt(2.0) * abs(t)) / Float64(abs(l) * sqrt(Float64(4.0 / Float64(Float64(x - 1.0) * 2.0)))));
	elseif (abs(t) <= 4.5e+32)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(-1.0 * Float64(Float64(Float64(-2.0 * t_1) - Float64(Float64(2.0 * (abs(l) ^ 2.0)) + t_2)) / x)) + t_2)) / Float64(abs(t) * sqrt(2.0))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = abs(t) ^ 2.0;
	t_2 = 2.0 * t_1;
	tmp = 0.0;
	if (abs(t) <= 5.4e-142)
		tmp = (sqrt(2.0) * abs(t)) / (abs(l) * sqrt((4.0 / ((x - 1.0) * 2.0))));
	elseif (abs(t) <= 4.5e+32)
		tmp = 1.0 / (sqrt(((-1.0 * (((-2.0 * t_1) - ((2.0 * (abs(l) ^ 2.0)) + t_2)) / x)) + t_2)) / (abs(t) * sqrt(2.0)));
	else
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * 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), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.4e-142], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(4.0 / N[(N[(x - 1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 4.5e+32], N[(1.0 / N[(N[Sqrt[N[(N[(-1.0 * N[(N[(N[(-2.0 * t$95$1), $MachinePrecision] - N[(N[(2.0 * N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.3999999999999996e-142

   1. Initial program 33.3%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 2.7%

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

   4. Applied rewrites 2.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-add N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 2.9%

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

   6. Applied rewrites 2.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 15.0%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{4}{\left(x - 1\right) \cdot 2}}} \]

   if 5.3999999999999996e-142 < t < 4.5000000000000003e32

   1. Initial program 33.3%

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

   3. Applied rewrites 28.1%

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in x around -inf

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

      1. lower-+.f64 N/A

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

   8. Applied rewrites 50.9%

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

   if 4.5000000000000003e32 < t

   1. Initial program 33.3%

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

   2. 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 40.1%

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

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
   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. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 40.1%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. sub-negate-rev N/A

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

      20. lower--.f64 N/A

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

      21. lift--.f64 N/A

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

      22. sub-negate-rev N/A

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

      23. lower--.f64 40.1%

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

   6. Applied rewrites 40.1%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval 40.1%

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

   8. Applied rewrites 40.1%

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

Alternative 3: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{4}{\left(x - 1\right) \cdot 2}}}\\ \mathbf{elif}\;\left|t\right| \leq 4.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\left(-1 \cdot \frac{-2 \cdot t\_1 - {\left(\left|\ell\right|\right)}^{2}}{x} + 2 \cdot t\_1\right) - \frac{\left|\ell\right| \cdot \left|\ell\right| + \left(\left|t\right| + \left|t\right|\right) \cdot \left|t\right|}{1 - x}}}{\left|t\right| \cdot \sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (let* ((t_1 (pow (fabs t) 2.0)))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 5.4e-142)
     (/
      (* (sqrt 2.0) (fabs t))
      (* (fabs l) (sqrt (/ 4.0 (* (- x 1.0) 2.0)))))
     (if (<= (fabs t) 4.5e+32)
       (/
        1.0
        (/
         (sqrt
          (-
           (+
            (* -1.0 (/ (- (* -2.0 t_1) (pow (fabs l) 2.0)) x))
            (* 2.0 t_1))
           (/
            (+
             (* (fabs l) (fabs l))
             (* (+ (fabs t) (fabs t)) (fabs t)))
            (- 1.0 x))))
         (* (fabs t) (sqrt 2.0))))
       (/ 1.0 (sqrt (/ (- -1.0 x) (- 1.0 x)))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(t), 2.0);
	double tmp;
	if (fabs(t) <= 5.4e-142) {
		tmp = (sqrt(2.0) * fabs(t)) / (fabs(l) * sqrt((4.0 / ((x - 1.0) * 2.0))));
	} else if (fabs(t) <= 4.5e+32) {
		tmp = 1.0 / (sqrt((((-1.0 * (((-2.0 * t_1) - pow(fabs(l), 2.0)) / x)) + (2.0 * t_1)) - (((fabs(l) * fabs(l)) + ((fabs(t) + fabs(t)) * fabs(t))) / (1.0 - x)))) / (fabs(t) * sqrt(2.0)));
	} else {
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double t_1 = Math.pow(Math.abs(t), 2.0);
	double tmp;
	if (Math.abs(t) <= 5.4e-142) {
		tmp = (Math.sqrt(2.0) * Math.abs(t)) / (Math.abs(l) * Math.sqrt((4.0 / ((x - 1.0) * 2.0))));
	} else if (Math.abs(t) <= 4.5e+32) {
		tmp = 1.0 / (Math.sqrt((((-1.0 * (((-2.0 * t_1) - Math.pow(Math.abs(l), 2.0)) / x)) + (2.0 * t_1)) - (((Math.abs(l) * Math.abs(l)) + ((Math.abs(t) + Math.abs(t)) * Math.abs(t))) / (1.0 - x)))) / (Math.abs(t) * Math.sqrt(2.0)));
	} else {
		tmp = 1.0 / Math.sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	t_1 = math.pow(math.fabs(t), 2.0)
	tmp = 0
	if math.fabs(t) <= 5.4e-142:
		tmp = (math.sqrt(2.0) * math.fabs(t)) / (math.fabs(l) * math.sqrt((4.0 / ((x - 1.0) * 2.0))))
	elif math.fabs(t) <= 4.5e+32:
		tmp = 1.0 / (math.sqrt((((-1.0 * (((-2.0 * t_1) - math.pow(math.fabs(l), 2.0)) / x)) + (2.0 * t_1)) - (((math.fabs(l) * math.fabs(l)) + ((math.fabs(t) + math.fabs(t)) * math.fabs(t))) / (1.0 - x)))) / (math.fabs(t) * math.sqrt(2.0)))
	else:
		tmp = 1.0 / math.sqrt(((-1.0 - x) / (1.0 - x)))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	tmp = 0.0
	if (abs(t) <= 5.4e-142)
		tmp = Float64(Float64(sqrt(2.0) * abs(t)) / Float64(abs(l) * sqrt(Float64(4.0 / Float64(Float64(x - 1.0) * 2.0)))));
	elseif (abs(t) <= 4.5e+32)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(-2.0 * t_1) - (abs(l) ^ 2.0)) / x)) + Float64(2.0 * t_1)) - Float64(Float64(Float64(abs(l) * abs(l)) + Float64(Float64(abs(t) + abs(t)) * abs(t))) / Float64(1.0 - x)))) / Float64(abs(t) * sqrt(2.0))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = abs(t) ^ 2.0;
	tmp = 0.0;
	if (abs(t) <= 5.4e-142)
		tmp = (sqrt(2.0) * abs(t)) / (abs(l) * sqrt((4.0 / ((x - 1.0) * 2.0))));
	elseif (abs(t) <= 4.5e+32)
		tmp = 1.0 / (sqrt((((-1.0 * (((-2.0 * t_1) - (abs(l) ^ 2.0)) / x)) + (2.0 * t_1)) - (((abs(l) * abs(l)) + ((abs(t) + abs(t)) * abs(t))) / (1.0 - x)))) / (abs(t) * sqrt(2.0)));
	else
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * 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], 5.4e-142], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(4.0 / N[(N[(x - 1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 4.5e+32], N[(1.0 / N[(N[Sqrt[N[(N[(N[(-1.0 * N[(N[(N[(-2.0 * t$95$1), $MachinePrecision] - N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.3999999999999996e-142

   1. Initial program 33.3%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 2.7%

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

   4. Applied rewrites 2.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-add N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 2.9%

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

   6. Applied rewrites 2.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 15.0%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{4}{\left(x - 1\right) \cdot 2}}} \]

   if 5.3999999999999996e-142 < t < 4.5000000000000003e32

   1. Initial program 33.3%

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

   3. Applied rewrites 28.1%

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in x around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 51.0%

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

   8. Applied rewrites 51.0%

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

   if 4.5000000000000003e32 < t

   1. Initial program 33.3%

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

   2. 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 40.1%

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

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
   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. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 40.1%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. sub-negate-rev N/A

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

      20. lower--.f64 N/A

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

      21. lift--.f64 N/A

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

      22. sub-negate-rev N/A

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

      23. lower--.f64 40.1%

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

   6. Applied rewrites 40.1%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval 40.1%

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

   8. Applied rewrites 40.1%

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

Alternative 4: 80.2% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{4}{\left(x - 1\right) \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}\\ \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 5.4e-137)
   (/
    (* (sqrt 2.0) (fabs t))
    (* (fabs l) (sqrt (/ 4.0 (* (- x 1.0) 2.0)))))
   (/ (sqrt 2.0) (sqrt (* 2.0 (/ (+ 1.0 x) (- x 1.0))))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (fabs(t) <= 5.4e-137) {
		tmp = (sqrt(2.0) * fabs(t)) / (fabs(l) * sqrt((4.0 / ((x - 1.0) * 2.0))));
	} else {
		tmp = sqrt(2.0) / sqrt((2.0 * ((1.0 + x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (Math.abs(t) <= 5.4e-137) {
		tmp = (Math.sqrt(2.0) * Math.abs(t)) / (Math.abs(l) * Math.sqrt((4.0 / ((x - 1.0) * 2.0))));
	} else {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 * ((1.0 + x) / (x - 1.0))));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if math.fabs(t) <= 5.4e-137:
		tmp = (math.sqrt(2.0) * math.fabs(t)) / (math.fabs(l) * math.sqrt((4.0 / ((x - 1.0) * 2.0))))
	else:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 * ((1.0 + x) / (x - 1.0))))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (abs(t) <= 5.4e-137)
		tmp = Float64(Float64(sqrt(2.0) * abs(t)) / Float64(abs(l) * sqrt(Float64(4.0 / Float64(Float64(x - 1.0) * 2.0)))));
	else
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(1.0 + x) / Float64(x - 1.0)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (abs(t) <= 5.4e-137)
		tmp = (sqrt(2.0) * abs(t)) / (abs(l) * sqrt((4.0 / ((x - 1.0) * 2.0))));
	else
		tmp = sqrt(2.0) / sqrt((2.0 * ((1.0 + x) / (x - 1.0))));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.4e-137], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(4.0 / N[(N[(x - 1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.3999999999999999e-137

   1. Initial program 33.3%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 2.7%

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

   4. Applied rewrites 2.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-add N/A

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

      7. metadata-eval N/A

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

      8. frac-sub N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 2.9%

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

   6. Applied rewrites 2.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 15.0%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{4}{\left(x - 1\right) \cdot 2}}} \]

   if 5.3999999999999999e-137 < t

   1. Initial program 33.3%

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

   2. 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 40.1%

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

   4. Applied rewrites 40.1%

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

Alternative 5: 80.0% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(1.189207115002721 \cdot 1.189207115002721\right) \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}\\ \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 5.4e-137)
   (/
    (* (* 1.189207115002721 1.189207115002721) (fabs t))
    (* (fabs l) (sqrt (/ 2.0 x))))
   (/ (sqrt 2.0) (sqrt (* 2.0 (/ (+ 1.0 x) (- x 1.0))))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (fabs(t) <= 5.4e-137) {
		tmp = ((1.189207115002721 * 1.189207115002721) * fabs(t)) / (fabs(l) * sqrt((2.0 / x)));
	} else {
		tmp = sqrt(2.0) / sqrt((2.0 * ((1.0 + x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (Math.abs(t) <= 5.4e-137) {
		tmp = ((1.189207115002721 * 1.189207115002721) * Math.abs(t)) / (Math.abs(l) * Math.sqrt((2.0 / x)));
	} else {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 * ((1.0 + x) / (x - 1.0))));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if math.fabs(t) <= 5.4e-137:
		tmp = ((1.189207115002721 * 1.189207115002721) * math.fabs(t)) / (math.fabs(l) * math.sqrt((2.0 / x)))
	else:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 * ((1.0 + x) / (x - 1.0))))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (abs(t) <= 5.4e-137)
		tmp = Float64(Float64(Float64(1.189207115002721 * 1.189207115002721) * abs(t)) / Float64(abs(l) * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(1.0 + x) / Float64(x - 1.0)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (abs(t) <= 5.4e-137)
		tmp = ((1.189207115002721 * 1.189207115002721) * abs(t)) / (abs(l) * sqrt((2.0 / x)));
	else
		tmp = sqrt(2.0) / sqrt((2.0 * ((1.0 + x) / (x - 1.0))));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.4e-137], N[(N[(N[(1.189207115002721 * 1.189207115002721), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.3999999999999999e-137

   1. Initial program 33.3%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 2.7%

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

   4. Applied rewrites 2.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 14.8%

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

   7. Applied rewrites 14.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lower-unsound-sqrt.f64 14.8%

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

   9. Applied rewrites 14.8%

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

   10. Evaluated real constant 14.8%

       \[\leadsto \frac{\left(\color{blue}{1.189207115002721} \cdot \sqrt{\sqrt{2}}\right) \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]

   11. Evaluated real constant 14.8%

       \[\leadsto \frac{\left(1.189207115002721 \cdot \color{blue}{1.189207115002721}\right) \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]

   if 5.3999999999999999e-137 < t

   1. Initial program 33.3%

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

   2. 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 40.1%

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

   4. Applied rewrites 40.1%

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

Alternative 6: 80.0% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(1.189207115002721 \cdot 1.189207115002721\right) \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 5.4e-137)
   (/
    (* (* 1.189207115002721 1.189207115002721) (fabs t))
    (* (fabs l) (sqrt (/ 2.0 x))))
   (/ 1.0 (sqrt (/ (- -1.0 x) (- 1.0 x)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (fabs(t) <= 5.4e-137) {
		tmp = ((1.189207115002721 * 1.189207115002721) * fabs(t)) / (fabs(l) * sqrt((2.0 / x)));
	} else {
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (Math.abs(t) <= 5.4e-137) {
		tmp = ((1.189207115002721 * 1.189207115002721) * Math.abs(t)) / (Math.abs(l) * Math.sqrt((2.0 / x)));
	} else {
		tmp = 1.0 / Math.sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if math.fabs(t) <= 5.4e-137:
		tmp = ((1.189207115002721 * 1.189207115002721) * math.fabs(t)) / (math.fabs(l) * math.sqrt((2.0 / x)))
	else:
		tmp = 1.0 / math.sqrt(((-1.0 - x) / (1.0 - x)))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (abs(t) <= 5.4e-137)
		tmp = Float64(Float64(Float64(1.189207115002721 * 1.189207115002721) * abs(t)) / Float64(abs(l) * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (abs(t) <= 5.4e-137)
		tmp = ((1.189207115002721 * 1.189207115002721) * abs(t)) / (abs(l) * sqrt((2.0 / x)));
	else
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.4e-137], N[(N[(N[(1.189207115002721 * 1.189207115002721), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.3999999999999999e-137

   1. Initial program 33.3%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 2.7%

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

   4. Applied rewrites 2.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 14.8%

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

   7. Applied rewrites 14.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lower-unsound-sqrt.f64 14.8%

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

   9. Applied rewrites 14.8%

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

   10. Evaluated real constant 14.8%

       \[\leadsto \frac{\left(\color{blue}{1.189207115002721} \cdot \sqrt{\sqrt{2}}\right) \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]

   11. Evaluated real constant 14.8%

       \[\leadsto \frac{\left(1.189207115002721 \cdot \color{blue}{1.189207115002721}\right) \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]

   if 5.3999999999999999e-137 < t

   1. Initial program 33.3%

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

   2. 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 40.1%

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

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
   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. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 40.1%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. sub-negate-rev N/A

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

      20. lower--.f64 N/A

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

      21. lift--.f64 N/A

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

      22. sub-negate-rev N/A

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

      23. lower--.f64 40.1%

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

   6. Applied rewrites 40.1%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval 40.1%

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

   8. Applied rewrites 40.1%

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

Alternative 7: 80.0% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{1.4142135623730951 \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 5.4e-137)
   (/ (* 1.4142135623730951 (fabs t)) (* (fabs l) (sqrt (/ 2.0 x))))
   (/ 1.0 (sqrt (/ (- -1.0 x) (- 1.0 x)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (fabs(t) <= 5.4e-137) {
		tmp = (1.4142135623730951 * fabs(t)) / (fabs(l) * sqrt((2.0 / x)));
	} else {
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (Math.abs(t) <= 5.4e-137) {
		tmp = (1.4142135623730951 * Math.abs(t)) / (Math.abs(l) * Math.sqrt((2.0 / x)));
	} else {
		tmp = 1.0 / Math.sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if math.fabs(t) <= 5.4e-137:
		tmp = (1.4142135623730951 * math.fabs(t)) / (math.fabs(l) * math.sqrt((2.0 / x)))
	else:
		tmp = 1.0 / math.sqrt(((-1.0 - x) / (1.0 - x)))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (abs(t) <= 5.4e-137)
		tmp = Float64(Float64(1.4142135623730951 * abs(t)) / Float64(abs(l) * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (abs(t) <= 5.4e-137)
		tmp = (1.4142135623730951 * abs(t)) / (abs(l) * sqrt((2.0 / x)));
	else
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.4e-137], N[(N[(1.4142135623730951 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.3999999999999999e-137

   1. Initial program 33.3%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 2.7%

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

   4. Applied rewrites 2.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 14.8%

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

   7. Applied rewrites 14.8%

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

   8. Evaluated real constant 14.8%

      \[\leadsto \frac{\color{blue}{1.4142135623730951} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]

   if 5.3999999999999999e-137 < t

   1. Initial program 33.3%

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

   2. 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 40.1%

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

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
   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. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 40.1%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. sub-negate-rev N/A

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

      20. lower--.f64 N/A

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

      21. lift--.f64 N/A

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

      22. sub-negate-rev N/A

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

      23. lower--.f64 40.1%

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

   6. Applied rewrites 40.1%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval 40.1%

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

   8. Applied rewrites 40.1%

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

Alternative 8: 80.0% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left|t\right|}{\sqrt{\frac{2}{x}} \cdot \left|\ell\right|} \cdot 1.4142135623730951\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{-1 - x}{1 - x}}}\\ \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 5.4e-137)
   (* (/ (fabs t) (* (sqrt (/ 2.0 x)) (fabs l))) 1.4142135623730951)
   (/ 1.0 (sqrt (/ (- -1.0 x) (- 1.0 x)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (fabs(t) <= 5.4e-137) {
		tmp = (fabs(t) / (sqrt((2.0 / x)) * fabs(l))) * 1.4142135623730951;
	} else {
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (Math.abs(t) <= 5.4e-137) {
		tmp = (Math.abs(t) / (Math.sqrt((2.0 / x)) * Math.abs(l))) * 1.4142135623730951;
	} else {
		tmp = 1.0 / Math.sqrt(((-1.0 - x) / (1.0 - x)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if math.fabs(t) <= 5.4e-137:
		tmp = (math.fabs(t) / (math.sqrt((2.0 / x)) * math.fabs(l))) * 1.4142135623730951
	else:
		tmp = 1.0 / math.sqrt(((-1.0 - x) / (1.0 - x)))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (abs(t) <= 5.4e-137)
		tmp = Float64(Float64(abs(t) / Float64(sqrt(Float64(2.0 / x)) * abs(l))) * 1.4142135623730951);
	else
		tmp = Float64(1.0 / sqrt(Float64(Float64(-1.0 - x) / Float64(1.0 - x))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (abs(t) <= 5.4e-137)
		tmp = (abs(t) / (sqrt((2.0 / x)) * abs(l))) * 1.4142135623730951;
	else
		tmp = 1.0 / sqrt(((-1.0 - x) / (1.0 - x)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[x_, l_, t_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.4e-137], N[(N[(N[Abs[t], $MachinePrecision] / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.4142135623730951), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.3999999999999999e-137

   1. Initial program 33.3%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 2.7%

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

   4. Applied rewrites 2.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 14.8%

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

   7. Applied rewrites 14.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 14.8%

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

   10. Evaluated real constant 14.8%

       \[\leadsto \frac{t}{\sqrt{\frac{2}{x}} \cdot \ell} \cdot \color{blue}{1.4142135623730951} \]

   if 5.3999999999999999e-137 < t

   1. Initial program 33.3%

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

   2. 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 40.1%

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

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
   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. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 40.1%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. sub-negate-rev N/A

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

      20. lower--.f64 N/A

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

      21. lift--.f64 N/A

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

      22. sub-negate-rev N/A

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

      23. lower--.f64 40.1%

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

   6. Applied rewrites 40.1%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval 40.1%

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

   8. Applied rewrites 40.1%

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

Alternative 9: 77.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, l, t)
	tmp = (sign(t) * abs(1.0)) * (1.0 / sqrt(((-1.0 - x) / (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[Sqrt[N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

2. 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 40.1%

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

4. Applied rewrites 40.1%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
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. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. sqrt-div N/A

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

   9. lower-unsound-/.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. lower-unsound-sqrt.f64 40.1%

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-/.f64 N/A

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

   15. frac-2neg N/A

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

   16. lower-/.f64 N/A

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

   17. add-flip N/A

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

   18. metadata-eval N/A

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

   19. sub-negate-rev N/A

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

   20. lower--.f64 N/A

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

   21. lift--.f64 N/A

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

   22. sub-negate-rev N/A

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

   23. lower--.f64 40.1%

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

6. Applied rewrites 40.1%

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

   1. lift-sqrt.f64 N/A

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

   2. metadata-eval 40.1%

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

8. Applied rewrites 40.1%

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

Alternative 10: 76.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x, l, t)
	tmp = (sign(t) * abs(1.0)) * sqrt(((-2.0 / (2.0 * (-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[(N[(-2.0 / N[(2.0 * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

2. 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 40.1%

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

4. Applied rewrites 40.1%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
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. associate-*r/ N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

6. Applied rewrites 40.0%

   \[\leadsto \sqrt{\frac{-2}{2 \cdot \left(-1 - x\right)} \cdot \left(x - 1\right)} \]
7. Add Preprocessing

Alternative 11: 76.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, l, t)
	tmp = (sign(t) * abs(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[(1.0 / N[(x / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

2. 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 40.1%

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

4. Applied rewrites 40.1%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
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. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. sqrt-div N/A

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

   9. lower-unsound-/.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. lower-unsound-sqrt.f64 40.1%

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-/.f64 N/A

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

   15. frac-2neg N/A

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

   16. lower-/.f64 N/A

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

   17. add-flip N/A

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

   18. metadata-eval N/A

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

   19. sub-negate-rev N/A

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

   20. lower--.f64 N/A

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

   21. lift--.f64 N/A

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

   22. sub-negate-rev N/A

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

   23. lower--.f64 40.1%

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

6. Applied rewrites 40.1%

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

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 39.8%

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

9. Applied rewrites 39.8%

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

    1. lift--.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. sub-to-fraction N/A

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

    4. div-flip N/A

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

    5. sub-flip N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

    8. distribute-lft-in N/A

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

    9. metadata-eval N/A

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

    10. sub-flip N/A

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

    11. *-lft-identity N/A

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

    12. lower-unsound-/.f64 N/A

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

    13. lift--.f64 N/A

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

    14. lower-unsound-/.f64 39.8%

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

11. Applied rewrites 39.8%

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

Alternative 12: 76.5% accurate, 0.7× 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 33.3%

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

2. 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 40.1%

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

4. Applied rewrites 40.1%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
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. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. sqrt-div N/A

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

   9. lower-unsound-/.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. lower-unsound-sqrt.f64 40.1%

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-/.f64 N/A

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

   15. frac-2neg N/A

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

   16. lower-/.f64 N/A

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

   17. add-flip N/A

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

   18. metadata-eval N/A

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

   19. sub-negate-rev N/A

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

   20. lower--.f64 N/A

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

   21. lift--.f64 N/A

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

   22. sub-negate-rev N/A

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

   23. lower--.f64 40.1%

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

6. Applied rewrites 40.1%

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

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 39.8%

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

9. Applied rewrites 39.8%

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

Alternative 13: 4.0% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

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 = (-1.0d0) / x
end function

Java

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

Python

def code(x, l, t):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

code[x_, l_, t_] := N[(-1.0 / x), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

2. 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 40.1%

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

4. Applied rewrites 40.1%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
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. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. sqrt-div N/A

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

   9. lower-unsound-/.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. lower-unsound-sqrt.f64 40.1%

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-/.f64 N/A

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

   15. frac-2neg N/A

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

   16. lower-/.f64 N/A

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

   17. add-flip N/A

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

   18. metadata-eval N/A

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

   19. sub-negate-rev N/A

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

   20. lower--.f64 N/A

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

   21. lift--.f64 N/A

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

   22. sub-negate-rev N/A

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

   23. lower--.f64 40.1%

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

6. Applied rewrites 40.1%

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

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 39.8%

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

9. Applied rewrites 39.8%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{x} \]
11. Step-by-step derivation

    1. lower-/.f64 4.0%

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

12. Applied rewrites 4.0%

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

Reproduce

herbie shell --seed 2025258 
(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)))))