Toniolo and Linder, Equation (7)

Percentage Accurate: 33.1% → 82.1%

Time: 7.7s

Alternatives: 6

Speedup: 3.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \left|t\right|\\ t_2 := {\left(\left|t\right|\right)}^{2}\\ t_3 := \mathsf{fma}\left(2, t\_2, {\ell}^{2}\right)\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.6 \cdot 10^{-273}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\mathsf{fma}\left(\left|t\right| + \left|t\right|, -\left(-1 - x\right) \cdot \frac{\left|t\right|}{x - 1}, \frac{{\ell}^{2} - -1 \cdot {\ell}^{2}}{x}\right)}}\\ \mathbf{elif}\;\left|t\right| \leq 1.12 \cdot 10^{-166}:\\ \;\;\;\;\frac{1.4142135623730951}{\sqrt{2}}\\ \mathbf{elif}\;\left|t\right| \leq 1.45 \cdot 10^{+17}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, t\_3 - -1 \cdot t\_3, -1 \cdot \frac{t\_3}{x}\right) - \mathsf{fma}\left(2, \frac{t\_2}{x}, \frac{{\ell}^{2}}{x}\right)}{x}, 2 \cdot t\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (let* ((t_1 (* (sqrt 2.0) (fabs t)))
       (t_2 (pow (fabs t) 2.0))
       (t_3 (fma 2.0 t_2 (pow l 2.0))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 4.6e-273)
     (/
      t_1
      (sqrt
       (fma
        (+ (fabs t) (fabs t))
        (- (* (- -1.0 x) (/ (fabs t) (- x 1.0))))
        (/ (- (pow l 2.0) (* -1.0 (pow l 2.0))) x))))
     (if (<= (fabs t) 1.12e-166)
       (/ 1.4142135623730951 (sqrt 2.0))
       (if (<= (fabs t) 1.45e+17)
         (/
          t_1
          (sqrt
           (fma
            -1.0
            (/
             (-
              (fma -1.0 (- t_3 (* -1.0 t_3)) (* -1.0 (/ t_3 x)))
              (fma 2.0 (/ t_2 x) (/ (pow l 2.0) x)))
             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 = sqrt(2.0) * fabs(t);
	double t_2 = pow(fabs(t), 2.0);
	double t_3 = fma(2.0, t_2, pow(l, 2.0));
	double tmp;
	if (fabs(t) <= 4.6e-273) {
		tmp = t_1 / sqrt(fma((fabs(t) + fabs(t)), -((-1.0 - x) * (fabs(t) / (x - 1.0))), ((pow(l, 2.0) - (-1.0 * pow(l, 2.0))) / x)));
	} else if (fabs(t) <= 1.12e-166) {
		tmp = 1.4142135623730951 / sqrt(2.0);
	} else if (fabs(t) <= 1.45e+17) {
		tmp = t_1 / sqrt(fma(-1.0, ((fma(-1.0, (t_3 - (-1.0 * t_3)), (-1.0 * (t_3 / x))) - fma(2.0, (t_2 / x), (pow(l, 2.0) / x))) / 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 = Float64(sqrt(2.0) * abs(t))
	t_2 = abs(t) ^ 2.0
	t_3 = fma(2.0, t_2, (l ^ 2.0))
	tmp = 0.0
	if (abs(t) <= 4.6e-273)
		tmp = Float64(t_1 / sqrt(fma(Float64(abs(t) + abs(t)), Float64(-Float64(Float64(-1.0 - x) * Float64(abs(t) / Float64(x - 1.0)))), Float64(Float64((l ^ 2.0) - Float64(-1.0 * (l ^ 2.0))) / x))));
	elseif (abs(t) <= 1.12e-166)
		tmp = Float64(1.4142135623730951 / sqrt(2.0));
	elseif (abs(t) <= 1.45e+17)
		tmp = Float64(t_1 / 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_2 / x), Float64((l ^ 2.0) / x))) / 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[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4.6e-273], N[(t$95$1 / N[Sqrt[N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] * (-N[(N[(-1.0 - x), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(N[Power[l, 2.0], $MachinePrecision] - N[(-1.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.12e-166], N[(1.4142135623730951 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.45e+17], N[(t$95$1 / 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$2 / x), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $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 < 4.5999999999999996e-273

   1. Initial program 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 23.5%

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 48.9%

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

   8. Applied rewrites 48.9%

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

   if 4.5999999999999996e-273 < t < 1.1199999999999999e-166

   1. Initial program 33.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.9%

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

   4. Applied rewrites 38.9%

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

   5. Evaluated real constant 38.9%

      \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2 \cdot \frac{1 + x}{x - 1}}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1.4142135623730951}{\sqrt{2}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 38.3%

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

   8. Applied rewrites 38.3%

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

   if 1.1199999999999999e-166 < t < 1.45e17

   1. Initial program 33.1%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{-1 \cdot \frac{\left(-1 \cdot \left(\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)\right) + -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right) - \left(2 \cdot \frac{{t}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)}{x} + 2 \cdot {t}^{2}}}} \]
   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 52.1%

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

   if 1.45e17 < t

   1. Initial program 33.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.9%

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

   4. Applied rewrites 38.9%

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

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

      19. distribute-frac-neg N/A

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

      20. frac-2neg-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

Alternative 2: 82.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|t\right|\right)}^{2}\\ t_2 := \sqrt{2} \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.6 \cdot 10^{-273}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(\left|t\right| + \left|t\right|, -\left(-1 - x\right) \cdot \frac{\left|t\right|}{x - 1}, \frac{{\ell}^{2} - -1 \cdot {\ell}^{2}}{x}\right)}}\\ \mathbf{elif}\;\left|t\right| \leq 1.12 \cdot 10^{-166}:\\ \;\;\;\;\frac{1.4142135623730951}{\sqrt{2}}\\ \mathbf{elif}\;\left|t\right| \leq 1.45 \cdot 10^{+17}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_1}{x}, \mathsf{fma}\left(2, t\_1, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{\mathsf{fma}\left(2, t\_1, {\ell}^{2}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{-1 - x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (let* ((t_1 (pow (fabs t) 2.0)) (t_2 (* (sqrt 2.0) (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 4.6e-273)
     (/
      t_2
      (sqrt
       (fma
        (+ (fabs t) (fabs t))
        (- (* (- -1.0 x) (/ (fabs t) (- x 1.0))))
        (/ (- (pow l 2.0) (* -1.0 (pow l 2.0))) x))))
     (if (<= (fabs t) 1.12e-166)
       (/ 1.4142135623730951 (sqrt 2.0))
       (if (<= (fabs t) 1.45e+17)
         (/
          t_2
          (sqrt
           (-
            (fma 2.0 (/ t_1 x) (fma 2.0 t_1 (/ (pow l 2.0) x)))
            (* -1.0 (/ (fma 2.0 t_1 (pow l 2.0)) x)))))
         (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 = sqrt(2.0) * fabs(t);
	double tmp;
	if (fabs(t) <= 4.6e-273) {
		tmp = t_2 / sqrt(fma((fabs(t) + fabs(t)), -((-1.0 - x) * (fabs(t) / (x - 1.0))), ((pow(l, 2.0) - (-1.0 * pow(l, 2.0))) / x)));
	} else if (fabs(t) <= 1.12e-166) {
		tmp = 1.4142135623730951 / sqrt(2.0);
	} else if (fabs(t) <= 1.45e+17) {
		tmp = t_2 / sqrt((fma(2.0, (t_1 / x), fma(2.0, t_1, (pow(l, 2.0) / x))) - (-1.0 * (fma(2.0, t_1, pow(l, 2.0)) / x))));
	} else {
		tmp = sqrt((-1.0 / ((-1.0 - x) / (x - 1.0))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	t_1 = abs(t) ^ 2.0
	t_2 = Float64(sqrt(2.0) * abs(t))
	tmp = 0.0
	if (abs(t) <= 4.6e-273)
		tmp = Float64(t_2 / sqrt(fma(Float64(abs(t) + abs(t)), Float64(-Float64(Float64(-1.0 - x) * Float64(abs(t) / Float64(x - 1.0)))), Float64(Float64((l ^ 2.0) - Float64(-1.0 * (l ^ 2.0))) / x))));
	elseif (abs(t) <= 1.12e-166)
		tmp = Float64(1.4142135623730951 / sqrt(2.0));
	elseif (abs(t) <= 1.45e+17)
		tmp = Float64(t_2 / sqrt(Float64(fma(2.0, Float64(t_1 / x), fma(2.0, t_1, Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(fma(2.0, t_1, (l ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(-1.0 / Float64(Float64(-1.0 - x) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4.6e-273], N[(t$95$2 / N[Sqrt[N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] * (-N[(N[(-1.0 - x), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(N[Power[l, 2.0], $MachinePrecision] - N[(-1.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.12e-166], N[(1.4142135623730951 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.45e+17], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * N[(t$95$1 / x), $MachinePrecision] + N[(2.0 * t$95$1 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(2.0 * t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[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 < 4.5999999999999996e-273

   1. Initial program 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 23.5%

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 48.9%

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

   8. Applied rewrites 48.9%

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

   if 4.5999999999999996e-273 < t < 1.1199999999999999e-166

   1. Initial program 33.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.9%

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

   4. Applied rewrites 38.9%

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

   5. Evaluated real constant 38.9%

      \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2 \cdot \frac{1 + x}{x - 1}}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1.4142135623730951}{\sqrt{2}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 38.3%

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

   8. Applied rewrites 38.3%

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

   if 1.1199999999999999e-166 < t < 1.45e17

   1. Initial program 33.1%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-pow.f64 51.9%

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

   4. Applied rewrites 51.9%

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

   if 1.45e17 < t

   1. Initial program 33.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.9%

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

   4. Applied rewrites 38.9%

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

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. sub-negate-rev N/A

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

      18. lift--.f64 N/A

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

      19. distribute-frac-neg N/A

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

      20. frac-2neg-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

Alternative 3: 77.5% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.6 \cdot 10^{-273}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left|t\right|}{\sqrt{\mathsf{fma}\left(\left|t\right| + \left|t\right|, -\left(-1 - x\right) \cdot \frac{\left|t\right|}{x - 1}, \frac{{\ell}^{2} - -1 \cdot {\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.4142135623730951}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}}}\\ \end{array} \]

FPCore

(FPCore (x l t)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 4.6e-273)
   (/
    (* (sqrt 2.0) (fabs t))
    (sqrt
     (fma
      (+ (fabs t) (fabs t))
      (- (* (- -1.0 x) (/ (fabs t) (- x 1.0))))
      (/ (- (pow l 2.0) (* -1.0 (pow l 2.0))) x))))
   (/ 1.4142135623730951 (sqrt (/ (fma x 2.0 2.0) (- x 1.0)))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (fabs(t) <= 4.6e-273) {
		tmp = (sqrt(2.0) * fabs(t)) / sqrt(fma((fabs(t) + fabs(t)), -((-1.0 - x) * (fabs(t) / (x - 1.0))), ((pow(l, 2.0) - (-1.0 * pow(l, 2.0))) / x)));
	} else {
		tmp = 1.4142135623730951 / sqrt((fma(x, 2.0, 2.0) / (x - 1.0)));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(x, l, t)
	tmp = 0.0
	if (abs(t) <= 4.6e-273)
		tmp = Float64(Float64(sqrt(2.0) * abs(t)) / sqrt(fma(Float64(abs(t) + abs(t)), Float64(-Float64(Float64(-1.0 - x) * Float64(abs(t) / Float64(x - 1.0)))), Float64(Float64((l ^ 2.0) - Float64(-1.0 * (l ^ 2.0))) / x))));
	else
		tmp = Float64(1.4142135623730951 / sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x - 1.0))));
	end
	return Float64(copysign(1.0, t) * 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], 4.6e-273], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] * (-N[(N[(-1.0 - x), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(N[Power[l, 2.0], $MachinePrecision] - N[(-1.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.4142135623730951 / N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.5999999999999996e-273

   1. Initial program 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 23.5%

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 48.9%

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

   8. Applied rewrites 48.9%

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

   if 4.5999999999999996e-273 < t

   1. Initial program 33.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 38.9%

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

   4. Applied rewrites 38.9%

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

   5. Evaluated real constant 38.9%

      \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2 \cdot \frac{1 + x}{x - 1}}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}} \]

      3. associate-*r/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{\frac{x \cdot 2 + 1 \cdot 2}{x - 1}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{\frac{x \cdot 2 + 2}{x - 1}}} \]

      8. lift-fma.f64 N/A

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

      9. lower-/.f64 38.9%

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

   7. Applied rewrites 38.9%

      \[\leadsto \frac{1.4142135623730951}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.1%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.9%

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

4. Applied rewrites 38.9%

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

5. Evaluated real constant 38.9%

   \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2 \cdot \frac{1 + x}{x - 1}}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}} \]

   3. associate-*r/ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{\frac{x \cdot 2 + 1 \cdot 2}{x - 1}}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{6369051672525773}{4503599627370496}}{\sqrt{\frac{x \cdot 2 + 2}{x - 1}}} \]

   8. lift-fma.f64 N/A

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

   9. lower-/.f64 38.9%

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

7. Applied rewrites 38.9%

   \[\leadsto \frac{1.4142135623730951}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}}} \]
8. Add Preprocessing

Alternative 5: 76.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.1%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.9%

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

4. Applied rewrites 38.9%

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

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

   15. add-flip N/A

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

   16. metadata-eval N/A

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

   17. sub-negate-rev N/A

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

   18. lift--.f64 N/A

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

   19. distribute-frac-neg N/A

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

   20. frac-2neg-rev N/A

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

   21. lower-/.f64 N/A

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

   22. lower-/.f64 38.9%

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

6. Applied rewrites 38.9%

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

Alternative 6: 75.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.1%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 38.9%

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

4. Applied rewrites 38.9%

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

5. Evaluated real constant 38.9%

   \[\leadsto \frac{1.4142135623730951}{\sqrt{\color{blue}{2 \cdot \frac{1 + x}{x - 1}}}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1.4142135623730951}{\sqrt{2}} \]
7. Step-by-step derivation

   1. lower-sqrt.f64 38.3%

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

8. Applied rewrites 38.3%

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

Reproduce

herbie shell --seed 2025210 
(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)))))