Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 76.7%

Time: 5.8s

Alternatives: 7

Speedup: 1.5×

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: 33.7% 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: 76.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-248}:\\ \;\;\;\;\frac{t\_3}{{\left(\mathsf{fma}\left(-\ell, \ell, t\_2 \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, t\_2 \cdot \left(\frac{x}{x - 1} - \frac{-1}{x - 1}\right)\right)\right)}^{0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\left(t\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l l))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.9e-248)
      (/
       t_3
       (*
        (pow (fma (- l) l (* t_2 (/ (- x -1.0) (- x 1.0)))) 0.25)
        (pow
         (fma (- l) l (* t_2 (- (/ x (- x 1.0)) (/ -1.0 (- x 1.0)))))
         0.25)))
      (/ t_3 (* (* t_m (sqrt 2.0)) (sqrt (/ (+ 1.0 x) (- x 1.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.9e-248) {
		tmp = t_3 / (pow(fma(-l, l, (t_2 * ((x - -1.0) / (x - 1.0)))), 0.25) * pow(fma(-l, l, (t_2 * ((x / (x - 1.0)) - (-1.0 / (x - 1.0))))), 0.25));
	} else {
		tmp = t_3 / ((t_m * sqrt(2.0)) * sqrt(((1.0 + x) / (x - 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 2.9e-248)
		tmp = Float64(t_3 / Float64((fma(Float64(-l), l, Float64(t_2 * Float64(Float64(x - -1.0) / Float64(x - 1.0)))) ^ 0.25) * (fma(Float64(-l), l, Float64(t_2 * Float64(Float64(x / Float64(x - 1.0)) - Float64(-1.0 / Float64(x - 1.0))))) ^ 0.25)));
	else
		tmp = Float64(t_3 / Float64(Float64(t_m * sqrt(2.0)) * sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-248], N[(t$95$3 / N[(N[Power[N[((-l) * l + N[(t$95$2 * N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[((-l) * l + N[(t$95$2 * N[(N[(x / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.9000000000000001e-248

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 28.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \color{blue}{\frac{x - -1}{x - 1}}\right)\right)}^{\frac{1}{4}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{\color{blue}{x - -1}}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. div-sub N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{x}{x - 1} - \frac{-1}{x - 1}\right)}\right)\right)}^{\frac{1}{4}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{x}{x - 1} - \frac{-1}{x - 1}\right)}\right)\right)}^{\frac{1}{4}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{x}{x - 1}} - \frac{-1}{x - 1}\right)\right)\right)}^{\frac{1}{4}}} \]

      6. lower-/.f64 28.7

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \left(\frac{x}{x - 1} - \color{blue}{\frac{-1}{x - 1}}\right)\right)\right)}^{0.25}} \]

   5. Applied rewrites 28.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{x}{x - 1} - \frac{-1}{x - 1}\right)}\right)\right)}^{0.25}} \]

   if 2.9000000000000001e-248 < t

   1. Initial program 36.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 39.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{2 \cdot \left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{\left(2 \cdot t\right) \cdot t} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lower-*.f64 39.3

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   5. Applied rewrites 39.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\left(-\ell\right) \cdot \ell + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(-\ell\right) \cdot \ell} + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} + \left(-\ell\right) \cdot \ell\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1}} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{\left(x - -1\right) \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(x - -1\right) \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      10. lower-/.f64 30.2

          \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(x - -1, \color{blue}{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}}, \left(-\ell\right) \cdot \ell\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   7. Applied rewrites 30.2%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 80.4

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

   10. Applied rewrites 80.4%

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

Alternative 2: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - -1}{x - 1}\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-248}:\\ \;\;\;\;\frac{t\_3}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right) \cdot t\_2\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t\_m + t\_m, t\_m, \ell \cdot \ell\right) \cdot t\_2\right)\right)}^{0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\left(t\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (/ (- x -1.0) (- x 1.0))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.9e-248)
      (/
       t_3
       (*
        (pow (fma (- l) l (* (fma (* t_m t_m) 2.0 (* l l)) t_2)) 0.25)
        (pow (fma (- l) l (* (fma (+ t_m t_m) t_m (* l l)) t_2)) 0.25)))
      (/ t_3 (* (* t_m (sqrt 2.0)) (sqrt (/ (+ 1.0 x) (- x 1.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = (x - -1.0) / (x - 1.0);
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.9e-248) {
		tmp = t_3 / (pow(fma(-l, l, (fma((t_m * t_m), 2.0, (l * l)) * t_2)), 0.25) * pow(fma(-l, l, (fma((t_m + t_m), t_m, (l * l)) * t_2)), 0.25));
	} else {
		tmp = t_3 / ((t_m * sqrt(2.0)) * sqrt(((1.0 + x) / (x - 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(Float64(x - -1.0) / Float64(x - 1.0))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 2.9e-248)
		tmp = Float64(t_3 / Float64((fma(Float64(-l), l, Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) * t_2)) ^ 0.25) * (fma(Float64(-l), l, Float64(fma(Float64(t_m + t_m), t_m, Float64(l * l)) * t_2)) ^ 0.25)));
	else
		tmp = Float64(t_3 / Float64(Float64(t_m * sqrt(2.0)) * sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-248], N[(t$95$3 / N[(N[Power[N[((-l) * l + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[((-l) * l + N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] * t$95$m + N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.9000000000000001e-248

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 28.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{2 \cdot \left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{\left(2 \cdot t\right) \cdot t} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lower-*.f64 28.7

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   5. Applied rewrites 28.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. count-2-rev N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{t + t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. lower-+.f64 28.7

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{t + t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   7. Applied rewrites 28.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{t + t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   if 2.9000000000000001e-248 < t

   1. Initial program 36.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 39.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{2 \cdot \left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{\left(2 \cdot t\right) \cdot t} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lower-*.f64 39.3

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   5. Applied rewrites 39.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\left(-\ell\right) \cdot \ell + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(-\ell\right) \cdot \ell} + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} + \left(-\ell\right) \cdot \ell\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1}} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{\left(x - -1\right) \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(x - -1\right) \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      10. lower-/.f64 30.2

          \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(x - -1, \color{blue}{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}}, \left(-\ell\right) \cdot \ell\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   7. Applied rewrites 30.2%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 80.4

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

   10. Applied rewrites 80.4%

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

Alternative 3: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-248}:\\ \;\;\;\;\frac{t\_2}{e^{\log \left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\left(t\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.9e-248)
      (/
       t_2
       (exp
        (*
         (log
          (fma
           (- l)
           l
           (* (fma (* t_m t_m) 2.0 (* l l)) (/ (- x -1.0) (- x 1.0)))))
         0.5)))
      (/ t_2 (* (* t_m (sqrt 2.0)) (sqrt (/ (+ 1.0 x) (- x 1.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.9e-248) {
		tmp = t_2 / exp((log(fma(-l, l, (fma((t_m * t_m), 2.0, (l * l)) * ((x - -1.0) / (x - 1.0))))) * 0.5));
	} else {
		tmp = t_2 / ((t_m * sqrt(2.0)) * sqrt(((1.0 + x) / (x - 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 2.9e-248)
		tmp = Float64(t_2 / exp(Float64(log(fma(Float64(-l), l, Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) * Float64(Float64(x - -1.0) / Float64(x - 1.0))))) * 0.5)));
	else
		tmp = Float64(t_2 / Float64(Float64(t_m * sqrt(2.0)) * sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, 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, 2.9e-248], N[(t$95$2 / N[Exp[N[(N[Log[N[((-l) * l + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.9000000000000001e-248

   1. Initial program 3.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 28.7%

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

   if 2.9000000000000001e-248 < t

   1. Initial program 36.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 39.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{2 \cdot \left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{\left(2 \cdot t\right) \cdot t} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lower-*.f64 39.3

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   5. Applied rewrites 39.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\left(-\ell\right) \cdot \ell + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(-\ell\right) \cdot \ell} + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} + \left(-\ell\right) \cdot \ell\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1}} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{\left(x - -1\right) \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(x - -1\right) \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      10. lower-/.f64 30.2

          \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(x - -1, \color{blue}{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}}, \left(-\ell\right) \cdot \ell\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   7. Applied rewrites 30.2%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 80.4

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

   10. Applied rewrites 80.4%

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

Alternative 4: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-248}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t\_m, t\_m, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\left(t\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.9e-248)
    (*
     t_m
     (/
      (sqrt 2.0)
      (sqrt
       (fma
        (- l)
        l
        (* (fma (* 2.0 t_m) t_m (* l l)) (/ (- x -1.0) (- x 1.0)))))))
    (/
     (* (sqrt 2.0) t_m)
     (* (* t_m (sqrt 2.0)) (sqrt (/ (+ 1.0 x) (- x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 2.9e-248) {
		tmp = t_m * (sqrt(2.0) / sqrt(fma(-l, l, (fma((2.0 * t_m), t_m, (l * l)) * ((x - -1.0) / (x - 1.0))))));
	} else {
		tmp = (sqrt(2.0) * t_m) / ((t_m * sqrt(2.0)) * sqrt(((1.0 + x) / (x - 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 2.9e-248)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(fma(Float64(-l), l, Float64(fma(Float64(2.0 * t_m), t_m, Float64(l * l)) * Float64(Float64(x - -1.0) / Float64(x - 1.0)))))));
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(Float64(t_m * sqrt(2.0)) * sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.9e-248], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[((-l) * l + N[(N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.9000000000000001e-248

   1. Initial program 3.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 3.5

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

      7. lift--.f64 N/A

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

      8. sub-negate1 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 27.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 27.8

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

   5. Applied rewrites 27.8%

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

   if 2.9000000000000001e-248 < t

   1. Initial program 36.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 39.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{2 \cdot \left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{\left(2 \cdot t\right) \cdot t} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lower-*.f64 39.3

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   5. Applied rewrites 39.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\left(-\ell\right) \cdot \ell + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(-\ell\right) \cdot \ell} + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} + \left(-\ell\right) \cdot \ell\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1}} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{\left(x - -1\right) \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(x - -1\right) \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      10. lower-/.f64 30.2

          \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(x - -1, \color{blue}{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}}, \left(-\ell\right) \cdot \ell\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   7. Applied rewrites 30.2%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 80.4

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

   10. Applied rewrites 80.4%

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

Alternative 5: 77.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(2, t\_m, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\left(t\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7e-265)
    (*
     (sqrt
      (/ -2.0 (fma l l (/ (* (- -1.0 x) (fma 2.0 t_m (* l l))) (- x 1.0)))))
     t_m)
    (/
     (* (sqrt 2.0) t_m)
     (* (* t_m (sqrt 2.0)) (sqrt (/ (+ 1.0 x) (- x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 7e-265) {
		tmp = sqrt((-2.0 / fma(l, l, (((-1.0 - x) * fma(2.0, t_m, (l * l))) / (x - 1.0))))) * t_m;
	} else {
		tmp = (sqrt(2.0) * t_m) / ((t_m * sqrt(2.0)) * sqrt(((1.0 + x) / (x - 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 7e-265)
		tmp = Float64(sqrt(Float64(-2.0 / fma(l, l, Float64(Float64(Float64(-1.0 - x) * fma(2.0, t_m, Float64(l * l))) / Float64(x - 1.0))))) * t_m);
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(Float64(t_m * sqrt(2.0)) * sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 7e-265], N[(N[Sqrt[N[(-2.0 / N[(l * l + N[(N[(N[(-1.0 - x), $MachinePrecision] * N[(2.0 * t$95$m + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.00000000000000031e-265

   1. Initial program 4.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 29.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot t}}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \sqrt{2}}}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{\sqrt{2}}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto t \cdot \frac{\sqrt{2}}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}}} \]

   5. Applied rewrites 4.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. fp-cancel-sub-sign N/A

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

      9. lift-*.f64 N/A

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

      10. sub-negate2 N/A

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

      11. *-commutative N/A

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

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

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 30.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 30.4

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. flip-+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-inverses N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{0}}{t - t}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      13. +-inverses N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{0}{\color{blue}{0}}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{1 - 1}}{0}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{1 \cdot 1} - 1}{0}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{1 \cdot 1 - \color{blue}{1 \cdot 1}}{0}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      17. metadata-eval N/A

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

      18. flip-+ N/A

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

      19. metadata-eval 32.4

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

   9. Applied rewrites 32.4%

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

   if 7.00000000000000031e-265 < t

   1. Initial program 35.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 39.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{2 \cdot \left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{\left(2 \cdot t\right) \cdot t} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lower-*.f64 39.1

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\left(-\ell\right) \cdot \ell + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(-\ell\right) \cdot \ell} + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} + \left(-\ell\right) \cdot \ell\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1}} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{\left(x - -1\right) \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(x - -1\right) \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      10. lower-/.f64 29.7

          \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(x - -1, \color{blue}{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}}, \left(-\ell\right) \cdot \ell\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   7. Applied rewrites 29.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 79.7

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

   10. Applied rewrites 79.7%

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

Alternative 6: 76.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(2, t\_m, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7e-265)
    (*
     (sqrt
      (/ -2.0 (fma l l (/ (* (- -1.0 x) (fma 2.0 t_m (* l l))) (- x 1.0)))))
     t_m)
    (* (* (sqrt 0.5) (sqrt 2.0)) (sqrt (/ (- x 1.0) (+ 1.0 x)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 7e-265) {
		tmp = sqrt((-2.0 / fma(l, l, (((-1.0 - x) * fma(2.0, t_m, (l * l))) / (x - 1.0))))) * t_m;
	} else {
		tmp = (sqrt(0.5) * sqrt(2.0)) * sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 7e-265)
		tmp = Float64(sqrt(Float64(-2.0 / fma(l, l, Float64(Float64(Float64(-1.0 - x) * fma(2.0, t_m, Float64(l * l))) / Float64(x - 1.0))))) * t_m);
	else
		tmp = Float64(Float64(sqrt(0.5) * sqrt(2.0)) * sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 7e-265], N[(N[Sqrt[N[(-2.0 / N[(l * l + N[(N[(N[(-1.0 - x), $MachinePrecision] * N[(2.0 * t$95$m + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.00000000000000031e-265

   1. Initial program 4.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 29.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot t}}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \sqrt{2}}}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{\sqrt{2}}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto t \cdot \frac{\sqrt{2}}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}}} \]

   5. Applied rewrites 4.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. fp-cancel-sub-sign N/A

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

      9. lift-*.f64 N/A

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

      10. sub-negate2 N/A

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

      11. *-commutative N/A

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

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

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 30.4%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 30.4

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. flip-+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-inverses N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{0}}{t - t}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      13. +-inverses N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{0}{\color{blue}{0}}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{1 - 1}}{0}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{1 \cdot 1} - 1}{0}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{-2}{\mathsf{fma}\left(\ell, \ell, \frac{\left(-1 - x\right) \cdot \mathsf{fma}\left(\frac{1 \cdot 1 - \color{blue}{1 \cdot 1}}{0}, t, \ell \cdot \ell\right)}{x - 1}\right)}} \cdot t \]

      17. metadata-eval N/A

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

      18. flip-+ N/A

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

      19. metadata-eval 32.4

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

   9. Applied rewrites 32.4%

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

   if 7.00000000000000031e-265 < t

   1. Initial program 35.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 39.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{2 \cdot \left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{\left(2 \cdot t\right) \cdot t} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lower-*.f64 39.1

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\left(-\ell\right) \cdot \ell + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(-\ell\right) \cdot \ell} + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} + \left(-\ell\right) \cdot \ell\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1}} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{\left(x - -1\right) \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(x - -1\right) \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

      10. lower-/.f64 29.7

          \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(x - -1, \color{blue}{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}}, \left(-\ell\right) \cdot \ell\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   7. Applied rewrites 29.7%

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 78.5

         \[\leadsto \left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}} \]

   10. Applied rewrites 78.5%

       \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (* (* (sqrt 0.5) (sqrt 2.0)) (sqrt (/ (- x 1.0) (+ 1.0 x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * ((sqrt(0.5) * sqrt(2.0)) * sqrt(((x - 1.0) / (1.0 + x))));
}

Fortran

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, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * ((sqrt(0.5d0) * sqrt(2.0d0)) * sqrt(((x - 1.0d0) / (1.0d0 + x))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * ((Math.sqrt(0.5) * Math.sqrt(2.0)) * Math.sqrt(((x - 1.0) / (1.0 + x))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * ((math.sqrt(0.5) * math.sqrt(2.0)) * math.sqrt(((x - 1.0) / (1.0 + x))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(sqrt(0.5) * sqrt(2.0)) * sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * ((sqrt(0.5) * sqrt(2.0)) * sqrt(((x - 1.0) / (1.0 + x))));
end

Wolfram

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_, t$95$m_] := N[(t$95$s * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.7%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

3. Applied rewrites 38.6%

   \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}}} \]
4. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{2 \cdot \left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \left(\color{blue}{\left(2 \cdot t\right) \cdot t} + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   6. lower-*.f64 38.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(\color{blue}{2 \cdot t}, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

5. Applied rewrites 38.6%

   \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)} \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\left(-\ell\right) \cdot \ell + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(-\ell\right) \cdot \ell} + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} + \left(-\ell\right) \cdot \ell\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{x - -1}{x - 1}} \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   7. associate-*l/ N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\frac{\left(x - -1\right) \cdot \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\color{blue}{\left(x - -1\right) \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}} + \left(-\ell\right) \cdot \ell\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{\frac{1}{4}}} \]

   10. lower-/.f64 28.4

       \[\leadsto \frac{\sqrt{2} \cdot t}{{\left(\mathsf{fma}\left(x - -1, \color{blue}{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}}, \left(-\ell\right) \cdot \ell\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

7. Applied rewrites 28.4%

   \[\leadsto \frac{\sqrt{2} \cdot t}{{\color{blue}{\left(\mathsf{fma}\left(x - -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x - 1}, \left(-\ell\right) \cdot \ell\right)\right)}}^{0.25} \cdot {\left(\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1}\right)\right)}^{0.25}} \]

8. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 75.9

      \[\leadsto \left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}} \]

10. Applied rewrites 75.9%

    \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025108 
(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)))))