Toniolo and Linder, Equation (7)

Percentage Accurate: 32.8% → 85.2%

Time: 6.0s

Alternatives: 8

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: 32.8% 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: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ t_2 := {\left(\left|\ell\right|\right)}^{2}\\ t_3 := \mathsf{fma}\left(2, t\_1, t\_2\right)\\ t_4 := -1 \cdot t\_3\\ t_5 := \sqrt{2} \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{t\_5}{\left|\ell\right| \cdot \sqrt{\frac{2 + 2 \cdot \frac{1}{x}}{x}}}\\ \mathbf{elif}\;\left|t\right| \leq 5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{1.4142135623730951}{\sqrt{2}}\\ \mathbf{elif}\;\left|t\right| \leq 2.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{t\_5}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_3 - t\_4, -1 \cdot \frac{\mathsf{fma}\left(-1, t\_4 - t\_3, \mathsf{fma}\left(2, \frac{t\_1}{x}, \frac{t\_2}{x}\right)\right) - -1 \cdot \frac{t\_3}{x}}{x}\right)}{x}, 2 \cdot t\_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (pow (fabs t) 2.0))
        (t_2 (pow (fabs l) 2.0))
        (t_3 (fma 2.0 t_1 t_2))
        (t_4 (* -1.0 t_3))
        (t_5 (* (sqrt 2.0) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 7.5e-181)
      (/ t_5 (* (fabs l) (sqrt (/ (+ 2.0 (* 2.0 (/ 1.0 x))) x))))
      (if (<= (fabs t) 5.2e-159)
        (/ 1.4142135623730951 (sqrt 2.0))
        (if (<= (fabs t) 2.3e+64)
          (/
           t_5
           (sqrt
            (fma
             -1.0
             (/
              (fma
               -1.0
               (- t_3 t_4)
               (*
                -1.0
                (/
                 (-
                  (fma -1.0 (- t_4 t_3) (fma 2.0 (/ t_1 x) (/ t_2 x)))
                  (* -1.0 (/ t_3 x)))
                 x)))
              x)
             (* 2.0 t_1))))
          (sqrt (/ -1.0 (/ (- -1.0 x) (- x 1.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(t), 2.0);
	double t_2 = pow(fabs(l), 2.0);
	double t_3 = fma(2.0, t_1, t_2);
	double t_4 = -1.0 * t_3;
	double t_5 = sqrt(2.0) * fabs(t);
	double tmp;
	if (fabs(t) <= 7.5e-181) {
		tmp = t_5 / (fabs(l) * sqrt(((2.0 + (2.0 * (1.0 / x))) / x)));
	} else if (fabs(t) <= 5.2e-159) {
		tmp = 1.4142135623730951 / sqrt(2.0);
	} else if (fabs(t) <= 2.3e+64) {
		tmp = t_5 / sqrt(fma(-1.0, (fma(-1.0, (t_3 - t_4), (-1.0 * ((fma(-1.0, (t_4 - t_3), fma(2.0, (t_1 / x), (t_2 / x))) - (-1.0 * (t_3 / x))) / x))) / x), (2.0 * t_1)));
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	t_2 = abs(l) ^ 2.0
	t_3 = fma(2.0, t_1, t_2)
	t_4 = Float64(-1.0 * t_3)
	t_5 = Float64(sqrt(2.0) * abs(t))
	tmp = 0.0
	if (abs(t) <= 7.5e-181)
		tmp = Float64(t_5 / Float64(abs(l) * sqrt(Float64(Float64(2.0 + Float64(2.0 * Float64(1.0 / x))) / x))));
	elseif (abs(t) <= 5.2e-159)
		tmp = Float64(1.4142135623730951 / sqrt(2.0));
	elseif (abs(t) <= 2.3e+64)
		tmp = Float64(t_5 / sqrt(fma(-1.0, Float64(fma(-1.0, Float64(t_3 - t_4), Float64(-1.0 * Float64(Float64(fma(-1.0, Float64(t_4 - t_3), fma(2.0, Float64(t_1 / x), Float64(t_2 / x))) - Float64(-1.0 * Float64(t_3 / x))) / x))) / x), Float64(2.0 * t_1))));
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$1 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 * 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], 7.5e-181], N[(t$95$5 / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(2.0 + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5.2e-159], N[(1.4142135623730951 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.3e+64], N[(t$95$5 / N[Sqrt[N[(-1.0 * N[(N[(-1.0 * N[(t$95$3 - t$95$4), $MachinePrecision] + N[(-1.0 * N[(N[(N[(-1.0 * N[(t$95$4 - t$95$3), $MachinePrecision] + N[(2.0 * N[(t$95$1 / x), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 7.5000000000000002e-181

   1. Initial program 32.8%

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

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

      5. associate-*l/ N/A

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

      6. sub-to-fraction N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 23.4%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 8.4%

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

   6. Applied rewrites 8.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 15.5%

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

   9. Applied rewrites 15.5%

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

   if 7.5000000000000002e-181 < t < 5.1999999999999997e-159

   1. Initial program 32.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 38.0%

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

   8. Evaluated real constant 38.0%

      \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2}}} \]

   if 5.1999999999999997e-159 < t < 2.3e64

   1. Initial program 32.8%

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

   2. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 52.6%

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

   if 2.3e64 < t

   1. Initial program 32.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lift--.f64 N/A

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

   6. Applied rewrites 38.6%

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

Alternative 2: 85.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ t_2 := {\left(\left|\ell\right|\right)}^{2}\\ t_3 := \mathsf{fma}\left(2, t\_1, t\_2\right)\\ t_4 := \sqrt{2} \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{t\_4}{\left|\ell\right| \cdot \sqrt{\frac{2 + 2 \cdot \frac{1}{x}}{x}}}\\ \mathbf{elif}\;\left|t\right| \leq 5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{1.4142135623730951}{\sqrt{2}}\\ \mathbf{elif}\;\left|t\right| \leq 2.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_3 - -1 \cdot t\_3, -1 \cdot \frac{t\_3}{x}\right) - \mathsf{fma}\left(2, \frac{t\_1}{x}, \frac{t\_2}{x}\right)}{x}, 2 \cdot t\_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (pow (fabs t) 2.0))
        (t_2 (pow (fabs l) 2.0))
        (t_3 (fma 2.0 t_1 t_2))
        (t_4 (* (sqrt 2.0) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 7.5e-181)
      (/ t_4 (* (fabs l) (sqrt (/ (+ 2.0 (* 2.0 (/ 1.0 x))) x))))
      (if (<= (fabs t) 5.2e-159)
        (/ 1.4142135623730951 (sqrt 2.0))
        (if (<= (fabs t) 2.3e+64)
          (/
           t_4
           (sqrt
            (fma
             -1.0
             (/
              (-
               (fma -1.0 (- t_3 (* -1.0 t_3)) (* -1.0 (/ t_3 x)))
               (fma 2.0 (/ t_1 x) (/ t_2 x)))
              x)
             (* 2.0 t_1))))
          (sqrt (/ -1.0 (/ (- -1.0 x) (- x 1.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(t), 2.0);
	double t_2 = pow(fabs(l), 2.0);
	double t_3 = fma(2.0, t_1, t_2);
	double t_4 = sqrt(2.0) * fabs(t);
	double tmp;
	if (fabs(t) <= 7.5e-181) {
		tmp = t_4 / (fabs(l) * sqrt(((2.0 + (2.0 * (1.0 / x))) / x)));
	} else if (fabs(t) <= 5.2e-159) {
		tmp = 1.4142135623730951 / sqrt(2.0);
	} else if (fabs(t) <= 2.3e+64) {
		tmp = t_4 / sqrt(fma(-1.0, ((fma(-1.0, (t_3 - (-1.0 * t_3)), (-1.0 * (t_3 / x))) - fma(2.0, (t_1 / x), (t_2 / x))) / x), (2.0 * t_1)));
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	t_2 = abs(l) ^ 2.0
	t_3 = fma(2.0, t_1, t_2)
	t_4 = Float64(sqrt(2.0) * abs(t))
	tmp = 0.0
	if (abs(t) <= 7.5e-181)
		tmp = Float64(t_4 / Float64(abs(l) * sqrt(Float64(Float64(2.0 + Float64(2.0 * Float64(1.0 / x))) / x))));
	elseif (abs(t) <= 5.2e-159)
		tmp = Float64(1.4142135623730951 / sqrt(2.0));
	elseif (abs(t) <= 2.3e+64)
		tmp = Float64(t_4 / sqrt(fma(-1.0, Float64(Float64(fma(-1.0, Float64(t_3 - Float64(-1.0 * t_3)), Float64(-1.0 * Float64(t_3 / x))) - fma(2.0, Float64(t_1 / x), Float64(t_2 / x))) / x), Float64(2.0 * t_1))));
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$1 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = 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], 7.5e-181], N[(t$95$4 / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(2.0 + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5.2e-159], N[(1.4142135623730951 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.3e+64], N[(t$95$4 / N[Sqrt[N[(-1.0 * N[(N[(N[(-1.0 * N[(t$95$3 - N[(-1.0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(t$95$1 / x), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 7.5000000000000002e-181

   1. Initial program 32.8%

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

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

      5. associate-*l/ N/A

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

      6. sub-to-fraction N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 23.4%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 8.4%

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

   6. Applied rewrites 8.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 15.5%

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

   9. Applied rewrites 15.5%

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

   if 7.5000000000000002e-181 < t < 5.1999999999999997e-159

   1. Initial program 32.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 38.0%

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

   8. Evaluated real constant 38.0%

      \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2}}} \]

   if 5.1999999999999997e-159 < t < 2.3e64

   1. Initial program 32.8%

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

   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-fma.f64 N/A

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

   4. Applied rewrites 52.5%

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

   if 2.3e64 < t

   1. Initial program 32.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lift--.f64 N/A

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

   6. Applied rewrites 38.6%

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

Alternative 3: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|\ell\right|\right)}^{2}\\ t_2 := {\left(\left|t\right|\right)}^{2}\\ t_3 := \sqrt{2} \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{t\_3}{\left|\ell\right| \cdot \sqrt{\frac{2 + 2 \cdot \frac{1}{x}}{x}}}\\ \mathbf{elif}\;\left|t\right| \leq 5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{1.4142135623730951}{\sqrt{2}}\\ \mathbf{elif}\;\left|t\right| \leq 2.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_1, 2 \cdot \left(-1 \cdot t\_2 - t\_2\right)\right) - t\_1}{x}, 2 \cdot t\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (pow (fabs l) 2.0))
        (t_2 (pow (fabs t) 2.0))
        (t_3 (* (sqrt 2.0) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 7.5e-181)
      (/ t_3 (* (fabs l) (sqrt (/ (+ 2.0 (* 2.0 (/ 1.0 x))) x))))
      (if (<= (fabs t) 5.2e-159)
        (/ 1.4142135623730951 (sqrt 2.0))
        (if (<= (fabs t) 2.3e+64)
          (/
           t_3
           (sqrt
            (fma
             -1.0
             (/ (- (fma -1.0 t_1 (* 2.0 (- (* -1.0 t_2) t_2))) t_1) x)
             (* 2.0 t_2))))
          (sqrt (/ -1.0 (/ (- -1.0 x) (- x 1.0))))))))))

C

double code(double x, double l, double t) {
	double t_1 = pow(fabs(l), 2.0);
	double t_2 = pow(fabs(t), 2.0);
	double t_3 = sqrt(2.0) * fabs(t);
	double tmp;
	if (fabs(t) <= 7.5e-181) {
		tmp = t_3 / (fabs(l) * sqrt(((2.0 + (2.0 * (1.0 / x))) / x)));
	} else if (fabs(t) <= 5.2e-159) {
		tmp = 1.4142135623730951 / sqrt(2.0);
	} else if (fabs(t) <= 2.3e+64) {
		tmp = t_3 / sqrt(fma(-1.0, ((fma(-1.0, t_1, (2.0 * ((-1.0 * t_2) - t_2))) - t_1) / x), (2.0 * t_2)));
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = abs(l) ^ 2.0
	t_2 = abs(t) ^ 2.0
	t_3 = Float64(sqrt(2.0) * abs(t))
	tmp = 0.0
	if (abs(t) <= 7.5e-181)
		tmp = Float64(t_3 / Float64(abs(l) * sqrt(Float64(Float64(2.0 + Float64(2.0 * Float64(1.0 / x))) / x))));
	elseif (abs(t) <= 5.2e-159)
		tmp = Float64(1.4142135623730951 / sqrt(2.0));
	elseif (abs(t) <= 2.3e+64)
		tmp = Float64(t_3 / sqrt(fma(-1.0, Float64(Float64(fma(-1.0, t_1, Float64(2.0 * Float64(Float64(-1.0 * t_2) - t_2))) - t_1) / x), Float64(2.0 * t_2))));
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = 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], 7.5e-181], N[(t$95$3 / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(2.0 + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5.2e-159], N[(1.4142135623730951 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.3e+64], N[(t$95$3 / N[Sqrt[N[(-1.0 * N[(N[(N[(-1.0 * t$95$1 + N[(2.0 * N[(N[(-1.0 * t$95$2), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 / N[(N[(-1.0 - x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 7.5000000000000002e-181

   1. Initial program 32.8%

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

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

      5. associate-*l/ N/A

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

      6. sub-to-fraction N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 23.4%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 8.4%

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

   6. Applied rewrites 8.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 15.5%

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

   9. Applied rewrites 15.5%

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

   if 7.5000000000000002e-181 < t < 5.1999999999999997e-159

   1. Initial program 32.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 38.0%

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

   8. Evaluated real constant 38.0%

      \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2}}} \]

   if 5.1999999999999997e-159 < t < 2.3e64

   1. Initial program 32.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*r* N/A

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

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

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

      12. lift-*.f64 N/A

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

   3. Applied rewrites 33.3%

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

   4. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   6. Applied rewrites 52.3%

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

   if 2.3e64 < t

   1. Initial program 32.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lift--.f64 N/A

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

   6. Applied rewrites 38.6%

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

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{2 + 2 \cdot \frac{1}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 7.5e-181)
    (/
     (* (sqrt 2.0) (fabs t))
     (* (fabs l) (sqrt (/ (+ 2.0 (* 2.0 (/ 1.0 x))) x))))
    (sqrt (/ (- x 1.0) (- x -1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (fabs(t) <= 7.5e-181) {
		tmp = (sqrt(2.0) * fabs(t)) / (fabs(l) * sqrt(((2.0 + (2.0 * (1.0 / x))) / 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) <= 7.5e-181) {
		tmp = (Math.sqrt(2.0) * Math.abs(t)) / (Math.abs(l) * Math.sqrt(((2.0 + (2.0 * (1.0 / x))) / 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) <= 7.5e-181:
		tmp = (math.sqrt(2.0) * math.fabs(t)) / (math.fabs(l) * math.sqrt(((2.0 + (2.0 * (1.0 / x))) / 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) <= 7.5e-181)
		tmp = Float64(Float64(sqrt(2.0) * abs(t)) / Float64(abs(l) * sqrt(Float64(Float64(2.0 + Float64(2.0 * Float64(1.0 / x))) / 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) <= 7.5e-181)
		tmp = (sqrt(2.0) * abs(t)) / (abs(l) * sqrt(((2.0 + (2.0 * (1.0 / x))) / 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], 7.5e-181], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(2.0 + N[(2.0 * N[(1.0 / x), $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 2 regimes

2. if t < 7.5000000000000002e-181

   1. Initial program 32.8%

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

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

      5. associate-*l/ N/A

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

      6. sub-to-fraction N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 23.4%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 8.4%

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

   6. Applied rewrites 8.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 15.5%

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

   9. Applied rewrites 15.5%

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

   if 7.5000000000000002e-181 < t

   1. Initial program 32.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lift--.f64 N/A

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

   6. Applied rewrites 38.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. distribute-frac-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. div-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lower-/.f64 38.6%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. metadata-eval N/A

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

      22. lower--.f64 38.6%

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

   8. Applied rewrites 38.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 38.6%

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

   10. Applied rewrites 38.6%

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

Alternative 5: 80.8% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\left|\ell\right| \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 7.5e-181)
    (/ (* (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) <= 7.5e-181) {
		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) <= 7.5e-181) {
		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) <= 7.5e-181:
		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) <= 7.5e-181)
		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) <= 7.5e-181)
		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], 7.5e-181], 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 < 7.5000000000000002e-181

   1. Initial program 32.8%

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

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

      5. associate-*l/ N/A

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

      6. sub-to-fraction N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 23.4%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 8.4%

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

   6. Applied rewrites 8.4%

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

   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 15.4%

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

   9. Applied rewrites 15.4%

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

   if 7.5000000000000002e-181 < t

   1. Initial program 32.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. sub-negate-rev N/A

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

      17. lift--.f64 N/A

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

      18. lift--.f64 N/A

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

   6. Applied rewrites 38.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. distribute-frac-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. div-flip-rev N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. metadata-eval N/A

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

      14. add-flip N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lower-/.f64 38.6%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. metadata-eval N/A

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

      22. lower--.f64 38.6%

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

   8. Applied rewrites 38.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 38.6%

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

   10. Applied rewrites 38.6%

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

Alternative 6: 76.4% 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 32.8%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.6%

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

4. Applied rewrites 38.6%

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

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. add-flip N/A

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

   15. metadata-eval N/A

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

   16. sub-negate-rev N/A

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

   17. lift--.f64 N/A

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

   18. lift--.f64 N/A

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

6. Applied rewrites 38.6%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. sub-negate-rev N/A

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

   9. lift--.f64 N/A

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

   10. div-flip-rev N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. metadata-eval N/A

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

   14. add-flip N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. lower-/.f64 38.6%

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. add-flip N/A

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

   21. metadata-eval N/A

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

   22. lower--.f64 38.6%

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

8. Applied rewrites 38.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 38.6%

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

10. Applied rewrites 38.6%

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

Alternative 7: 75.8% 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 32.8%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.6%

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

4. Applied rewrites 38.6%

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

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. add-flip N/A

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

   15. metadata-eval N/A

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

   16. sub-negate-rev N/A

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

   17. lift--.f64 N/A

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

   18. lift--.f64 N/A

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

6. Applied rewrites 38.6%

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

7. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 38.3%

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

9. Applied rewrites 38.3%

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

Alternative 8: 75.1% 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 32.8%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.6%

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

4. Applied rewrites 38.6%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 38.0%

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

8. Evaluated real constant 38.0%

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

Reproduce

herbie shell --seed 2025187 
(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)))))