Toniolo and Linder, Equation (7)

Percentage Accurate: 33.0% → 85.3%

Time: 7.9s

Alternatives: 11

Speedup: 85.0×

Specification

Math

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

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: 33.0% accurate, 1.0× speedup

Math

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

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: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)\\ t_3 := x \cdot \sqrt{2}\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{{x}^{-3}} \cdot \sqrt{0.5}\right) \cdot l\_m}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(2, \frac{t\_m}{t\_3}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{l\_m \cdot l\_m}{t\_m \cdot t\_3}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9600000:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, t\_2, \frac{\mathsf{fma}\left(2, \frac{t\_2}{x}, t\_2\right) + t\_2}{-x}\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (fma 2.0 (* t_m t_m) (* l_m l_m)))
        (t_3 (* x (sqrt 2.0)))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.6e-271)
      (/
       t_4
       (*
        (fma (/ 1.0 (sqrt x)) (sqrt 2.0) (* (sqrt (pow x -3.0)) (sqrt 0.5)))
        l_m))
      (if (<= t_m 5.6e-163)
        (/
         t_4
         (fma
          2.0
          (/ t_m t_3)
          (fma t_m (sqrt 2.0) (/ (* l_m l_m) (* t_m t_3)))))
        (if (<= t_m 9600000.0)
          (/
           t_4
           (sqrt
            (fma
             -1.0
             (/ (fma -2.0 t_2 (/ (+ (fma 2.0 (/ t_2 x) t_2) t_2) (- x))) x)
             (* 2.0 (* t_m t_m)))))
          (sqrt (/ (- x 1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = fma(2.0, (t_m * t_m), (l_m * l_m));
	double t_3 = x * sqrt(2.0);
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.6e-271) {
		tmp = t_4 / (fma((1.0 / sqrt(x)), sqrt(2.0), (sqrt(pow(x, -3.0)) * sqrt(0.5))) * l_m);
	} else if (t_m <= 5.6e-163) {
		tmp = t_4 / fma(2.0, (t_m / t_3), fma(t_m, sqrt(2.0), ((l_m * l_m) / (t_m * t_3))));
	} else if (t_m <= 9600000.0) {
		tmp = t_4 / sqrt(fma(-1.0, (fma(-2.0, t_2, ((fma(2.0, (t_2 / x), t_2) + t_2) / -x)) / x), (2.0 * (t_m * t_m))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m))
	t_3 = Float64(x * sqrt(2.0))
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.6e-271)
		tmp = Float64(t_4 / Float64(fma(Float64(1.0 / sqrt(x)), sqrt(2.0), Float64(sqrt((x ^ -3.0)) * sqrt(0.5))) * l_m));
	elseif (t_m <= 5.6e-163)
		tmp = Float64(t_4 / fma(2.0, Float64(t_m / t_3), fma(t_m, sqrt(2.0), Float64(Float64(l_m * l_m) / Float64(t_m * t_3)))));
	elseif (t_m <= 9600000.0)
		tmp = Float64(t_4 / sqrt(fma(-1.0, Float64(fma(-2.0, t_2, Float64(Float64(fma(2.0, Float64(t_2 / x), t_2) + t_2) / Float64(-x))) / x), Float64(2.0 * Float64(t_m * t_m)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-271], N[(t$95$4 / N[(N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[N[Power[x, -3.0], $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e-163], N[(t$95$4 / N[(2.0 * N[(t$95$m / t$95$3), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(t$95$m * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9600000.0], N[(t$95$4 / N[Sqrt[N[(-1.0 * N[(N[(-2.0 * t$95$2 + N[(N[(N[(2.0 * N[(t$95$2 / x), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$2), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999989e-271

   1. Initial program 33.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x around -inf

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

      1. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{-2 \cdot -1} + \sqrt{\frac{1}{{x}^{3}}} \cdot \frac{\sqrt{-1}}{\sqrt{-2}}\right) \cdot \ell} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{2} + \sqrt{\frac{1}{{x}^{3}}} \cdot \frac{\sqrt{-1}}{\sqrt{-2}}\right) \cdot \ell} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \frac{\sqrt{-1}}{\sqrt{-2}}\right) \cdot \ell} \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\sqrt{1}}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \frac{\sqrt{-1}}{\sqrt{-2}}\right) \cdot \ell} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \frac{\sqrt{-1}}{\sqrt{-2}}\right) \cdot \ell} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \frac{\sqrt{-1}}{\sqrt{-2}}\right) \cdot \ell} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \frac{\sqrt{-1}}{\sqrt{-2}}\right) \cdot \ell} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \frac{\sqrt{-1}}{\sqrt{-2}}\right) \cdot \ell} \]

      9. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \sqrt{\frac{-1}{-2}}\right) \cdot \ell} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \ell} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \ell} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{\frac{1}{{x}^{3}}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \ell} \]

      13. pow-flip N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{{x}^{\left(\mathsf{neg}\left(3\right)\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \ell} \]

      14. lower-pow.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{{x}^{\left(\mathsf{neg}\left(3\right)\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \ell} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{{x}^{-3}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \ell} \]

      16. lower-sqrt.f64 16.9

          \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{{x}^{-3}} \cdot \sqrt{0.5}\right) \cdot \ell} \]

   8. Applied rewrites 16.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{{x}^{-3}} \cdot \sqrt{0.5}\right) \cdot \ell} \]

   if 1.59999999999999989e-271 < t < 5.5999999999999999e-163

   1. Initial program 2.6%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 1.3

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if 5.5999999999999999e-163 < t < 9.6e6

   1. Initial program 52.0%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 4.6

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

   5. Applied rewrites 4.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 3.4

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   10. Applied rewrites 97.3%

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

   if 9.6e6 < t

   1. Initial program 28.7%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 95.2

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

   7. Applied rewrites 95.2%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \sqrt{2}, \sqrt{{x}^{-3}} \cdot \sqrt{0.5}\right) \cdot \ell}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{x \cdot \sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\ell \cdot \ell}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 9600000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{-x}\right)}{x}, 2 \cdot \left(t \cdot t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t_4 := x \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\frac{t\_3}{\left(l\_m \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(2, \frac{t\_m}{t\_4}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{l\_m \cdot l\_m}{t\_m \cdot t\_4}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9600000:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, t\_2, \frac{\mathsf{fma}\left(2, \frac{t\_2}{x}, t\_2\right) + t\_2}{-x}\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (fma 2.0 (* t_m t_m) (* l_m l_m)))
        (t_3 (* (sqrt 2.0) t_m))
        (t_4 (* x (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.6e-271)
      (/ t_3 (* (* l_m (sqrt 2.0)) (/ 1.0 (sqrt x))))
      (if (<= t_m 5.6e-163)
        (/
         t_3
         (fma
          2.0
          (/ t_m t_4)
          (fma t_m (sqrt 2.0) (/ (* l_m l_m) (* t_m t_4)))))
        (if (<= t_m 9600000.0)
          (/
           t_3
           (sqrt
            (fma
             -1.0
             (/ (fma -2.0 t_2 (/ (+ (fma 2.0 (/ t_2 x) t_2) t_2) (- x))) x)
             (* 2.0 (* t_m t_m)))))
          (sqrt (/ (- x 1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = fma(2.0, (t_m * t_m), (l_m * l_m));
	double t_3 = sqrt(2.0) * t_m;
	double t_4 = x * sqrt(2.0);
	double tmp;
	if (t_m <= 1.6e-271) {
		tmp = t_3 / ((l_m * sqrt(2.0)) * (1.0 / sqrt(x)));
	} else if (t_m <= 5.6e-163) {
		tmp = t_3 / fma(2.0, (t_m / t_4), fma(t_m, sqrt(2.0), ((l_m * l_m) / (t_m * t_4))));
	} else if (t_m <= 9600000.0) {
		tmp = t_3 / sqrt(fma(-1.0, (fma(-2.0, t_2, ((fma(2.0, (t_2 / x), t_2) + t_2) / -x)) / x), (2.0 * (t_m * t_m))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m))
	t_3 = Float64(sqrt(2.0) * t_m)
	t_4 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.6e-271)
		tmp = Float64(t_3 / Float64(Float64(l_m * sqrt(2.0)) * Float64(1.0 / sqrt(x))));
	elseif (t_m <= 5.6e-163)
		tmp = Float64(t_3 / fma(2.0, Float64(t_m / t_4), fma(t_m, sqrt(2.0), Float64(Float64(l_m * l_m) / Float64(t_m * t_4)))));
	elseif (t_m <= 9600000.0)
		tmp = Float64(t_3 / sqrt(fma(-1.0, Float64(fma(-2.0, t_2, Float64(Float64(fma(2.0, Float64(t_2 / x), t_2) + t_2) / Float64(-x))) / x), Float64(2.0 * Float64(t_m * t_m)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-271], N[(t$95$3 / N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e-163], N[(t$95$3 / N[(2.0 * N[(t$95$m / t$95$4), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(t$95$m * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9600000.0], N[(t$95$3 / N[Sqrt[N[(-1.0 * N[(N[(-2.0 * t$95$2 + N[(N[(N[(2.0 * N[(t$95$2 / x), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$2), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999989e-271

   1. Initial program 33.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 16.8

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

   8. Applied rewrites 16.8%

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

   if 1.59999999999999989e-271 < t < 5.5999999999999999e-163

   1. Initial program 2.6%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 1.3

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if 5.5999999999999999e-163 < t < 9.6e6

   1. Initial program 52.0%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 4.6

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

   5. Applied rewrites 4.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 3.4

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   10. Applied rewrites 97.3%

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

   if 9.6e6 < t

   1. Initial program 28.7%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 95.2

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

   7. Applied rewrites 95.2%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{x \cdot \sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\ell \cdot \ell}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 9600000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{-x}\right)}{x}, 2 \cdot \left(t \cdot t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t_4 := x \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\frac{t\_3}{\left(l\_m \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(2, \frac{t\_m}{t\_4}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{l\_m \cdot l\_m}{t\_m \cdot t\_4}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9600000:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, t\_2, \frac{t\_2 + t\_2}{-x}\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (fma 2.0 (* t_m t_m) (* l_m l_m)))
        (t_3 (* (sqrt 2.0) t_m))
        (t_4 (* x (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.6e-271)
      (/ t_3 (* (* l_m (sqrt 2.0)) (/ 1.0 (sqrt x))))
      (if (<= t_m 5.6e-163)
        (/
         t_3
         (fma
          2.0
          (/ t_m t_4)
          (fma t_m (sqrt 2.0) (/ (* l_m l_m) (* t_m t_4)))))
        (if (<= t_m 9600000.0)
          (/
           t_3
           (sqrt
            (fma
             -1.0
             (/ (fma -2.0 t_2 (/ (+ t_2 t_2) (- x))) x)
             (* 2.0 (* t_m t_m)))))
          (sqrt (/ (- x 1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = fma(2.0, (t_m * t_m), (l_m * l_m));
	double t_3 = sqrt(2.0) * t_m;
	double t_4 = x * sqrt(2.0);
	double tmp;
	if (t_m <= 1.6e-271) {
		tmp = t_3 / ((l_m * sqrt(2.0)) * (1.0 / sqrt(x)));
	} else if (t_m <= 5.6e-163) {
		tmp = t_3 / fma(2.0, (t_m / t_4), fma(t_m, sqrt(2.0), ((l_m * l_m) / (t_m * t_4))));
	} else if (t_m <= 9600000.0) {
		tmp = t_3 / sqrt(fma(-1.0, (fma(-2.0, t_2, ((t_2 + t_2) / -x)) / x), (2.0 * (t_m * t_m))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m))
	t_3 = Float64(sqrt(2.0) * t_m)
	t_4 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.6e-271)
		tmp = Float64(t_3 / Float64(Float64(l_m * sqrt(2.0)) * Float64(1.0 / sqrt(x))));
	elseif (t_m <= 5.6e-163)
		tmp = Float64(t_3 / fma(2.0, Float64(t_m / t_4), fma(t_m, sqrt(2.0), Float64(Float64(l_m * l_m) / Float64(t_m * t_4)))));
	elseif (t_m <= 9600000.0)
		tmp = Float64(t_3 / sqrt(fma(-1.0, Float64(fma(-2.0, t_2, Float64(Float64(t_2 + t_2) / Float64(-x))) / x), Float64(2.0 * Float64(t_m * t_m)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-271], N[(t$95$3 / N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e-163], N[(t$95$3 / N[(2.0 * N[(t$95$m / t$95$4), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(t$95$m * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9600000.0], N[(t$95$3 / N[Sqrt[N[(-1.0 * N[(N[(-2.0 * t$95$2 + N[(N[(t$95$2 + t$95$2), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999989e-271

   1. Initial program 33.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 16.8

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

   8. Applied rewrites 16.8%

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

   if 1.59999999999999989e-271 < t < 5.5999999999999999e-163

   1. Initial program 2.6%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 1.3

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if 5.5999999999999999e-163 < t < 9.6e6

   1. Initial program 52.0%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 4.6

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

   5. Applied rewrites 4.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 3.4

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in x around -inf

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

      1. lower-fma.f64 N/A

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

   10. Applied rewrites 97.0%

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

   if 9.6e6 < t

   1. Initial program 28.7%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 95.2

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

   7. Applied rewrites 95.2%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{x \cdot \sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\ell \cdot \ell}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 9600000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{-x}\right)}{x}, 2 \cdot \left(t \cdot t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := x \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\frac{t\_2}{\left(l\_m \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(2, \frac{t\_m}{t\_3}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{l\_m \cdot l\_m}{t\_m \cdot t\_3}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9600000:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)) (t_3 (* x (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.6e-271)
      (/ t_2 (* (* l_m (sqrt 2.0)) (/ 1.0 (sqrt x))))
      (if (<= t_m 5.6e-163)
        (/
         t_2
         (fma
          2.0
          (/ t_m t_3)
          (fma t_m (sqrt 2.0) (/ (* l_m l_m) (* t_m t_3)))))
        (if (<= t_m 9600000.0)
          (/
           t_2
           (sqrt
            (fma
             2.0
             (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x)
             (* 2.0 (* t_m t_m)))))
          (sqrt (/ (- x 1.0) (+ 1.0 x)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double t_3 = x * sqrt(2.0);
	double tmp;
	if (t_m <= 1.6e-271) {
		tmp = t_2 / ((l_m * sqrt(2.0)) * (1.0 / sqrt(x)));
	} else if (t_m <= 5.6e-163) {
		tmp = t_2 / fma(2.0, (t_m / t_3), fma(t_m, sqrt(2.0), ((l_m * l_m) / (t_m * t_3))));
	} else if (t_m <= 9600000.0) {
		tmp = t_2 / sqrt(fma(2.0, (fma(2.0, (t_m * t_m), (l_m * l_m)) / x), (2.0 * (t_m * t_m))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	t_3 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.6e-271)
		tmp = Float64(t_2 / Float64(Float64(l_m * sqrt(2.0)) * Float64(1.0 / sqrt(x))));
	elseif (t_m <= 5.6e-163)
		tmp = Float64(t_2 / fma(2.0, Float64(t_m / t_3), fma(t_m, sqrt(2.0), Float64(Float64(l_m * l_m) / Float64(t_m * t_3)))));
	elseif (t_m <= 9600000.0)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x), Float64(2.0 * Float64(t_m * t_m)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-271], N[(t$95$2 / N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e-163], N[(t$95$2 / N[(2.0 * N[(t$95$m / t$95$3), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(t$95$m * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9600000.0], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.59999999999999989e-271

   1. Initial program 33.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 16.8

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

   8. Applied rewrites 16.8%

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

   if 1.59999999999999989e-271 < t < 5.5999999999999999e-163

   1. Initial program 2.6%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 1.3

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if 5.5999999999999999e-163 < t < 9.6e6

   1. Initial program 52.0%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 4.6

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

   5. Applied rewrites 4.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 3.4

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 96.2

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

   10. Applied rewrites 96.2%

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

   if 9.6e6 < t

   1. Initial program 28.7%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 95.2

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

   7. Applied rewrites 95.2%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-271}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{x \cdot \sqrt{2}}, \mathsf{fma}\left(t, \sqrt{2}, \frac{\ell \cdot \ell}{t \cdot \left(x \cdot \sqrt{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 9600000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}, 2 \cdot \left(t \cdot t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-192}:\\ \;\;\;\;\frac{t\_2}{\left(l\_m \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{elif}\;t\_m \leq 9600000:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5e-192)
      (/ t_2 (* (* l_m (sqrt 2.0)) (/ 1.0 (sqrt x))))
      (if (<= t_m 9600000.0)
        (/
         t_2
         (sqrt
          (fma
           2.0
           (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x)
           (* 2.0 (* t_m t_m)))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 5e-192) {
		tmp = t_2 / ((l_m * sqrt(2.0)) * (1.0 / sqrt(x)));
	} else if (t_m <= 9600000.0) {
		tmp = t_2 / sqrt(fma(2.0, (fma(2.0, (t_m * t_m), (l_m * l_m)) / x), (2.0 * (t_m * t_m))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 5e-192)
		tmp = Float64(t_2 / Float64(Float64(l_m * sqrt(2.0)) * Float64(1.0 / sqrt(x))));
	elseif (t_m <= 9600000.0)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x), Float64(2.0 * Float64(t_m * t_m)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5e-192], N[(t$95$2 / N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9600000.0], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.0000000000000001e-192

   1. Initial program 31.1%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 18.4

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

   8. Applied rewrites 18.4%

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

   if 5.0000000000000001e-192 < t < 9.6e6

   1. Initial program 48.7%

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

   3. Taylor expanded in l around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 4.4

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

   5. Applied rewrites 4.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 92.1

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

   10. Applied rewrites 92.1%

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

   if 9.6e6 < t

   1. Initial program 28.7%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 95.2

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

   7. Applied rewrites 95.2%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq 9600000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}, 2 \cdot \left(t \cdot t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\left(l\_m \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-145)
    (/ (* (sqrt 2.0) t_m) (* (* l_m (sqrt 2.0)) (/ 1.0 (sqrt x))))
    (sqrt (/ (- x 1.0) (+ 1.0 x))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.2e-145) {
		tmp = (sqrt(2.0) * t_m) / ((l_m * sqrt(2.0)) * (1.0 / sqrt(x)));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
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(t_s, x, l_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 6.2d-145) then
        tmp = (sqrt(2.0d0) * t_m) / ((l_m * sqrt(2.0d0)) * (1.0d0 / sqrt(x)))
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.2e-145) {
		tmp = (Math.sqrt(2.0) * t_m) / ((l_m * Math.sqrt(2.0)) * (1.0 / Math.sqrt(x)));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 6.2e-145:
		tmp = (math.sqrt(2.0) * t_m) / ((l_m * math.sqrt(2.0)) * (1.0 / math.sqrt(x)))
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 6.2e-145)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(Float64(l_m * sqrt(2.0)) * Float64(1.0 / sqrt(x))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 6.2e-145)
		tmp = (sqrt(2.0) * t_m) / ((l_m * sqrt(2.0)) * (1.0 / sqrt(x)));
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-145], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.20000000000000001e-145

   1. Initial program 30.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 18.5

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

   8. Applied rewrites 18.5%

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

   if 6.20000000000000001e-145 < t

   1. Initial program 38.6%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 86.5

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

   7. Applied rewrites 86.5%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{2}{x}} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-145)
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 2.0 x)) l_m))
    (sqrt (/ (- x 1.0) (+ 1.0 x))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.2e-145) {
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
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(t_s, x, l_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 6.2d-145) then
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((2.0d0 / x)) * l_m)
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 6.2e-145) {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((2.0 / x)) * l_m);
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 6.2e-145:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((2.0 / x)) * l_m)
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 6.2e-145)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(2.0 / x)) * l_m));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 6.2e-145)
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l_m);
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-145], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.20000000000000001e-145

   1. Initial program 30.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 18.5

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

   8. Applied rewrites 18.5%

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

   if 6.20000000000000001e-145 < t

   1. Initial program 38.6%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 86.5

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

   7. Applied rewrites 86.5%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \sqrt{\frac{x - 1}{1 + x}} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (sqrt (/ (- x 1.0) (+ 1.0 x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * sqrt(((x - 1.0) / (1.0 + x)));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
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(t_s, x, l_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * sqrt(((x - 1.0d0) / (1.0d0 + x)))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * Math.sqrt(((x - 1.0) / (1.0 + x)));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * math.sqrt(((x - 1.0) / (1.0 + x)))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * sqrt(((x - 1.0) / (1.0 + x)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 39.4

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

5. Applied rewrites 39.4%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-sqrt.f64 39.4

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

7. Applied rewrites 39.4%

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

Alternative 9: 75.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \sqrt{\frac{x - 1}{x}} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (sqrt (/ (- x 1.0) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * sqrt(((x - 1.0) / x));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
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(t_s, x, l_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * sqrt(((x - 1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * Math.sqrt(((x - 1.0) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * math.sqrt(((x - 1.0) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * sqrt(Float64(Float64(x - 1.0) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * sqrt(((x - 1.0) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 39.4

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

5. Applied rewrites 39.4%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-sqrt.f64 39.4

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

7. Applied rewrites 39.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 39.3%

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

Alternative 10: 75.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \sqrt{\frac{x}{1 + x}} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (sqrt (/ x (+ 1.0 x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * sqrt((x / (1.0 + x)));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
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(t_s, x, l_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * sqrt((x / (1.0d0 + x)))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * Math.sqrt((x / (1.0 + x)));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * math.sqrt((x / (1.0 + x)))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * sqrt(Float64(x / Float64(1.0 + x))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * sqrt((x / (1.0 + x)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 39.4

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

5. Applied rewrites 39.4%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-sqrt.f64 39.4

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

7. Applied rewrites 39.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 39.3%

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

Alternative 11: 75.6% accurate, 85.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
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(t_s, x, l_m, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Taylor expanded in x around inf

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

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot 2} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   3. metadata-eval 39.2

      \[\leadsto 1 \]

5. Applied rewrites 39.2%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025059 
(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)))))