Toniolo and Linder, Equation (7)

Percentage Accurate: 33.6% → 87.4%

Time: 6.7s

Alternatives: 6

Speedup: 3.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.6% 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: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := 1 - \frac{1}{x}\\ t_2 := \sqrt{2} \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{t\_2}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;\left|t\right| \leq 3.8 \cdot 10^{-216}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|t\right| \leq 1.75 \cdot 10^{+87}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\left(\left(\left|t\right| + \left|t\right|\right) \cdot \left|t\right|\right) \cdot \frac{-1 - x}{1 - x} - \left|\ell\right| \cdot \frac{-1 \cdot \left|\ell\right| - \left|\ell\right|}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ 1.0 x))) (t_2 (* (sqrt 2.0) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 4.8e-233)
      (/ t_2 (* (fabs l) (sqrt (/ 2.0 x))))
      (if (<= (fabs t) 3.8e-216)
        t_1
        (if (<= (fabs t) 1.75e+87)
          (/
           t_2
           (sqrt
            (-
             (* (* (+ (fabs t) (fabs t)) (fabs t)) (/ (- -1.0 x) (- 1.0 x)))
             (* (fabs l) (/ (- (* -1.0 (fabs l)) (fabs l)) x)))))
          t_1))))))

C

double code(double x, double l, double t) {
	double t_1 = 1.0 - (1.0 / x);
	double t_2 = sqrt(2.0) * fabs(t);
	double tmp;
	if (fabs(t) <= 4.8e-233) {
		tmp = t_2 / (fabs(l) * sqrt((2.0 / x)));
	} else if (fabs(t) <= 3.8e-216) {
		tmp = t_1;
	} else if (fabs(t) <= 1.75e+87) {
		tmp = t_2 / sqrt(((((fabs(t) + fabs(t)) * fabs(t)) * ((-1.0 - x) / (1.0 - x))) - (fabs(l) * (((-1.0 * fabs(l)) - fabs(l)) / x))));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double t_1 = 1.0 - (1.0 / x);
	double t_2 = Math.sqrt(2.0) * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 4.8e-233) {
		tmp = t_2 / (Math.abs(l) * Math.sqrt((2.0 / x)));
	} else if (Math.abs(t) <= 3.8e-216) {
		tmp = t_1;
	} else if (Math.abs(t) <= 1.75e+87) {
		tmp = t_2 / Math.sqrt(((((Math.abs(t) + Math.abs(t)) * Math.abs(t)) * ((-1.0 - x) / (1.0 - x))) - (Math.abs(l) * (((-1.0 * Math.abs(l)) - Math.abs(l)) / x))));
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	t_1 = 1.0 - (1.0 / x)
	t_2 = math.sqrt(2.0) * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 4.8e-233:
		tmp = t_2 / (math.fabs(l) * math.sqrt((2.0 / x)))
	elif math.fabs(t) <= 3.8e-216:
		tmp = t_1
	elif math.fabs(t) <= 1.75e+87:
		tmp = t_2 / math.sqrt(((((math.fabs(t) + math.fabs(t)) * math.fabs(t)) * ((-1.0 - x) / (1.0 - x))) - (math.fabs(l) * (((-1.0 * math.fabs(l)) - math.fabs(l)) / x))))
	else:
		tmp = t_1
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	t_1 = Float64(1.0 - Float64(1.0 / x))
	t_2 = Float64(sqrt(2.0) * abs(t))
	tmp = 0.0
	if (abs(t) <= 4.8e-233)
		tmp = Float64(t_2 / Float64(abs(l) * sqrt(Float64(2.0 / x))));
	elseif (abs(t) <= 3.8e-216)
		tmp = t_1;
	elseif (abs(t) <= 1.75e+87)
		tmp = Float64(t_2 / sqrt(Float64(Float64(Float64(Float64(abs(t) + abs(t)) * abs(t)) * Float64(Float64(-1.0 - x) / Float64(1.0 - x))) - Float64(abs(l) * Float64(Float64(Float64(-1.0 * abs(l)) - abs(l)) / x)))));
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = 1.0 - (1.0 / x);
	t_2 = sqrt(2.0) * abs(t);
	tmp = 0.0;
	if (abs(t) <= 4.8e-233)
		tmp = t_2 / (abs(l) * sqrt((2.0 / x)));
	elseif (abs(t) <= 3.8e-216)
		tmp = t_1;
	elseif (abs(t) <= 1.75e+87)
		tmp = t_2 / sqrt(((((abs(t) + abs(t)) * abs(t)) * ((-1.0 - x) / (1.0 - x))) - (abs(l) * (((-1.0 * abs(l)) - abs(l)) / x))));
	else
		tmp = t_1;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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], 4.8e-233], N[(t$95$2 / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-216], t$95$1, If[LessEqual[N[Abs[t], $MachinePrecision], 1.75e+87], N[(t$95$2 / N[Sqrt[N[(N[(N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Abs[l], $MachinePrecision] * N[(N[(N[(-1.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[Abs[l], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.7999999999999998e-233

   1. Initial program 33.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 40.2%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 2.7%

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

   6. Applied rewrites 2.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 14.9%

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

   9. Applied rewrites 14.9%

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

   if 4.7999999999999998e-233 < t < 3.8e-216 or 1.7499999999999999e87 < t

   1. Initial program 33.6%

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

   2. 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 39.5%

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

   4. Applied rewrites 39.5%

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-frac-neg2 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. frac-2neg-rev N/A

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

   6. Applied rewrites 39.5%

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

   7. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 39.2%

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

   9. Applied rewrites 39.2%

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

   if 3.8e-216 < t < 1.7499999999999999e87

   1. Initial program 33.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 40.2%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 57.9%

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

   6. Applied rewrites 57.9%

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

Alternative 2: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_1}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;\left|t\right| \leq 0.012:\\ \;\;\;\;\frac{t\_1}{\sqrt{\left(\left(\left|t\right| + \left|t\right|\right) \cdot \left|t\right|\right) \cdot \frac{-1 - x}{1 - x} - \frac{\left|\ell\right| \cdot \left(-1 \cdot \left|\ell\right| - \left|\ell\right|\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 2.3e-161)
      (/ t_1 (* (fabs l) (sqrt (/ 2.0 x))))
      (if (<= (fabs t) 0.012)
        (/
         t_1
         (sqrt
          (-
           (* (* (+ (fabs t) (fabs t)) (fabs t)) (/ (- -1.0 x) (- 1.0 x)))
           (/ (* (fabs l) (- (* -1.0 (fabs l)) (fabs l))) x))))
        (sqrt (/ (- x 1.0) (- x -1.0))))))))

C

double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) * fabs(t);
	double tmp;
	if (fabs(t) <= 2.3e-161) {
		tmp = t_1 / (fabs(l) * sqrt((2.0 / x)));
	} else if (fabs(t) <= 0.012) {
		tmp = t_1 / sqrt(((((fabs(t) + fabs(t)) * fabs(t)) * ((-1.0 - x) / (1.0 - x))) - ((fabs(l) * ((-1.0 * fabs(l)) - fabs(l))) / x)));
	} else {
		tmp = sqrt(((x - 1.0) / (x - -1.0)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(2.0) * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 2.3e-161) {
		tmp = t_1 / (Math.abs(l) * Math.sqrt((2.0 / x)));
	} else if (Math.abs(t) <= 0.012) {
		tmp = t_1 / Math.sqrt(((((Math.abs(t) + Math.abs(t)) * Math.abs(t)) * ((-1.0 - x) / (1.0 - x))) - ((Math.abs(l) * ((-1.0 * Math.abs(l)) - Math.abs(l))) / x)));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (x - -1.0)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	t_1 = math.sqrt(2.0) * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 2.3e-161:
		tmp = t_1 / (math.fabs(l) * math.sqrt((2.0 / x)))
	elif math.fabs(t) <= 0.012:
		tmp = t_1 / math.sqrt(((((math.fabs(t) + math.fabs(t)) * math.fabs(t)) * ((-1.0 - x) / (1.0 - x))) - ((math.fabs(l) * ((-1.0 * math.fabs(l)) - math.fabs(l))) / x)))
	else:
		tmp = math.sqrt(((x - 1.0) / (x - -1.0)))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	t_1 = Float64(sqrt(2.0) * abs(t))
	tmp = 0.0
	if (abs(t) <= 2.3e-161)
		tmp = Float64(t_1 / Float64(abs(l) * sqrt(Float64(2.0 / x))));
	elseif (abs(t) <= 0.012)
		tmp = Float64(t_1 / sqrt(Float64(Float64(Float64(Float64(abs(t) + abs(t)) * abs(t)) * Float64(Float64(-1.0 - x) / Float64(1.0 - x))) - Float64(Float64(abs(l) * Float64(Float64(-1.0 * abs(l)) - abs(l))) / x))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = sqrt(2.0) * abs(t);
	tmp = 0.0;
	if (abs(t) <= 2.3e-161)
		tmp = t_1 / (abs(l) * sqrt((2.0 / x)));
	elseif (abs(t) <= 0.012)
		tmp = t_1 / sqrt(((((abs(t) + abs(t)) * abs(t)) * ((-1.0 - x) / (1.0 - x))) - ((abs(l) * ((-1.0 * abs(l)) - abs(l))) / x)));
	else
		tmp = sqrt(((x - 1.0) / (x - -1.0)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = 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], 2.3e-161], N[(t$95$1 / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 0.012], N[(t$95$1 / N[Sqrt[N[(N[(N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[l], $MachinePrecision] * N[(N[(-1.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3e-161

   1. Initial program 33.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 40.2%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 2.7%

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

   6. Applied rewrites 2.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 14.9%

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

   9. Applied rewrites 14.9%

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

   if 2.3e-161 < t < 0.012

   1. Initial program 33.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 40.2%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 52.7%

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

   6. Applied rewrites 52.7%

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

   if 0.012 < t

   1. Initial program 33.6%

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

   2. 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 39.5%

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

   4. Applied rewrites 39.5%

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-frac-neg2 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. frac-2neg-rev N/A

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

   6. Applied rewrites 39.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. sub-negate-rev N/A

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

      10. metadata-eval N/A

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

      11. add-flip N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. div-flip-rev N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lift--.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. mul-1-neg N/A

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

      23. frac-2neg-rev N/A

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

      24. lower-/.f64 N/A

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

      25. add-flip N/A

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

   8. Applied rewrites 39.5%

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

Alternative 3: 80.3% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 2.3e-161)
    (/ (* (sqrt 2.0) (fabs t)) (* (fabs l) (sqrt (/ 2.0 x))))
    (sqrt (/ (- x 1.0) (- x -1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (fabs(t) <= 2.3e-161) {
		tmp = (sqrt(2.0) * fabs(t)) / (fabs(l) * sqrt((2.0 / x)));
	} else {
		tmp = sqrt(((x - 1.0) / (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) <= 2.3e-161) {
		tmp = (Math.sqrt(2.0) * Math.abs(t)) / (Math.abs(l) * Math.sqrt((2.0 / x)));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (x - -1.0)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if math.fabs(t) <= 2.3e-161:
		tmp = (math.sqrt(2.0) * math.fabs(t)) / (math.fabs(l) * math.sqrt((2.0 / x)))
	else:
		tmp = math.sqrt(((x - 1.0) / (x - -1.0)))
	return math.copysign(1.0, t) * tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (abs(t) <= 2.3e-161)
		tmp = Float64(Float64(sqrt(2.0) * abs(t)) / Float64(abs(l) * sqrt(Float64(2.0 / x))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / 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) <= 2.3e-161)
		tmp = (sqrt(2.0) * abs(t)) / (abs(l) * sqrt((2.0 / x)));
	else
		tmp = sqrt(((x - 1.0) / (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], 2.3e-161], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.3e-161

   1. Initial program 33.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 40.2%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 2.7%

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

   6. Applied rewrites 2.7%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 14.9%

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

   9. Applied rewrites 14.9%

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

   if 2.3e-161 < t

   1. Initial program 33.6%

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

   2. 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 39.5%

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

   4. Applied rewrites 39.5%

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lift--.f64 N/A

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

      16. distribute-frac-neg2 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. frac-2neg-rev N/A

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

   6. Applied rewrites 39.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. sub-negate-rev N/A

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

      10. metadata-eval N/A

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

      11. add-flip N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. div-flip-rev N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lift--.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. mul-1-neg N/A

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

      23. frac-2neg-rev N/A

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

      24. lower-/.f64 N/A

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

      25. add-flip N/A

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

   8. Applied rewrites 39.5%

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

Alternative 4: 77.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

2. 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 39.5%

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

4. Applied rewrites 39.5%

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

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift--.f64 N/A

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

   14. sub-negate-rev N/A

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

   15. lift--.f64 N/A

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

   16. distribute-frac-neg2 N/A

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

   17. frac-2neg-rev N/A

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

   18. lower-/.f64 N/A

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

   19. frac-2neg-rev N/A

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

6. Applied rewrites 39.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. distribute-neg-frac2 N/A

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

   7. lift--.f64 N/A

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

   8. sub-negate-rev N/A

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

   9. sub-negate-rev N/A

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

   10. metadata-eval N/A

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

   11. add-flip N/A

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

   12. mul-1-neg N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. div-flip-rev N/A

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

   17. lift--.f64 N/A

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

   18. sub-negate-rev N/A

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

   19. lift--.f64 N/A

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

   20. lift-+.f64 N/A

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

   21. +-commutative N/A

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

   22. mul-1-neg N/A

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

   23. frac-2neg-rev N/A

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

   24. lower-/.f64 N/A

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

   25. add-flip N/A

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

8. Applied rewrites 39.5%

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

Alternative 5: 76.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

2. 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 39.5%

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

4. Applied rewrites 39.5%

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

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift--.f64 N/A

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

   14. sub-negate-rev N/A

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

   15. lift--.f64 N/A

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

   16. distribute-frac-neg2 N/A

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

   17. frac-2neg-rev N/A

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

   18. lower-/.f64 N/A

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

   19. frac-2neg-rev N/A

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

6. Applied rewrites 39.5%

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

7. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 39.2%

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

9. Applied rewrites 39.2%

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

Alternative 6: 76.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

2. 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 39.5%

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

4. Applied rewrites 39.5%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 38.9%

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

8. Evaluated real constant 38.9%

   \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025191 
(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)))))