Toniolo and Linder, Equation (7)

Percentage Accurate: 33.6% → 86.4%

Time: 7.0s

Alternatives: 7

Speedup: 6.5×

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: 33.6% 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: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(\left(t\_m + t\_m\right) \cdot t\_m\right) \cdot \frac{-1 - x}{1 - x}\\ t_3 := \sqrt{2} \cdot t\_m\\ t_4 := -1 \cdot \ell - \ell\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-247}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2 - \frac{\ell \cdot t\_4}{x}}}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{1 - 2 \cdot \frac{1}{x}}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+108}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2 - \ell \cdot \frac{t\_4}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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 (* (* (+ t_m t_m) t_m) (/ (- -1.0 x) (- 1.0 x))))
        (t_3 (* (sqrt 2.0) t_m))
        (t_4 (- (* -1.0 l) l)))
   (*
    t_s
    (if (<= t_m 2.4e-247)
      (/ t_3 (sqrt (- t_2 (/ (* l t_4) x))))
      (if (<= t_m 2.55e-176)
        (sqrt (- 1.0 (* 2.0 (/ 1.0 x))))
        (if (<= t_m 6e+108)
          (/ t_3 (sqrt (- t_2 (* l (/ t_4 x)))))
          (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 t_2 = ((t_m + t_m) * t_m) * ((-1.0 - x) / (1.0 - x));
	double t_3 = sqrt(2.0) * t_m;
	double t_4 = (-1.0 * l) - l;
	double tmp;
	if (t_m <= 2.4e-247) {
		tmp = t_3 / sqrt((t_2 - ((l * t_4) / x)));
	} else if (t_m <= 2.55e-176) {
		tmp = sqrt((1.0 - (2.0 * (1.0 / x))));
	} else if (t_m <= 6e+108) {
		tmp = t_3 / sqrt((t_2 - (l * (t_4 / x))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

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
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = ((t_m + t_m) * t_m) * (((-1.0d0) - x) / (1.0d0 - x))
    t_3 = sqrt(2.0d0) * t_m
    t_4 = ((-1.0d0) * l) - l
    if (t_m <= 2.4d-247) then
        tmp = t_3 / sqrt((t_2 - ((l * t_4) / x)))
    else if (t_m <= 2.55d-176) then
        tmp = sqrt((1.0d0 - (2.0d0 * (1.0d0 / x))))
    else if (t_m <= 6d+108) then
        tmp = t_3 / sqrt((t_2 - (l * (t_4 / x))))
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
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) {
	double t_2 = ((t_m + t_m) * t_m) * ((-1.0 - x) / (1.0 - x));
	double t_3 = Math.sqrt(2.0) * t_m;
	double t_4 = (-1.0 * l) - l;
	double tmp;
	if (t_m <= 2.4e-247) {
		tmp = t_3 / Math.sqrt((t_2 - ((l * t_4) / x)));
	} else if (t_m <= 2.55e-176) {
		tmp = Math.sqrt((1.0 - (2.0 * (1.0 / x))));
	} else if (t_m <= 6e+108) {
		tmp = t_3 / Math.sqrt((t_2 - (l * (t_4 / x))));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = ((t_m + t_m) * t_m) * ((-1.0 - x) / (1.0 - x))
	t_3 = math.sqrt(2.0) * t_m
	t_4 = (-1.0 * l) - l
	tmp = 0
	if t_m <= 2.4e-247:
		tmp = t_3 / math.sqrt((t_2 - ((l * t_4) / x)))
	elif t_m <= 2.55e-176:
		tmp = math.sqrt((1.0 - (2.0 * (1.0 / x))))
	elif t_m <= 6e+108:
		tmp = t_3 / math.sqrt((t_2 - (l * (t_4 / x))))
	else:
		tmp = math.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)
	t_2 = Float64(Float64(Float64(t_m + t_m) * t_m) * Float64(Float64(-1.0 - x) / Float64(1.0 - x)))
	t_3 = Float64(sqrt(2.0) * t_m)
	t_4 = Float64(Float64(-1.0 * l) - l)
	tmp = 0.0
	if (t_m <= 2.4e-247)
		tmp = Float64(t_3 / sqrt(Float64(t_2 - Float64(Float64(l * t_4) / x))));
	elseif (t_m <= 2.55e-176)
		tmp = sqrt(Float64(1.0 - Float64(2.0 * Float64(1.0 / x))));
	elseif (t_m <= 6e+108)
		tmp = Float64(t_3 / sqrt(Float64(t_2 - Float64(l * Float64(t_4 / x)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = ((t_m + t_m) * t_m) * ((-1.0 - x) / (1.0 - x));
	t_3 = sqrt(2.0) * t_m;
	t_4 = (-1.0 * l) - l;
	tmp = 0.0;
	if (t_m <= 2.4e-247)
		tmp = t_3 / sqrt((t_2 - ((l * t_4) / x)));
	elseif (t_m <= 2.55e-176)
		tmp = sqrt((1.0 - (2.0 * (1.0 / x))));
	elseif (t_m <= 6e+108)
		tmp = t_3 / sqrt((t_2 - (l * (t_4 / x))));
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = 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[(N[(t$95$m + t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 * l), $MachinePrecision] - l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.4e-247], N[(t$95$3 / N[Sqrt[N[(t$95$2 - N[(N[(l * t$95$4), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e-176], N[Sqrt[N[(1.0 - N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$m, 6e+108], N[(t$95$3 / N[Sqrt[N[(t$95$2 - N[(l * N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.40000000000000011e-247

   1. Initial program 33.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 41.1%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 53.6

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

   6. Applied rewrites 53.6%

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

   if 2.40000000000000011e-247 < t < 2.5500000000000001e-176

   1. Initial program 33.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r* N/A

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

      12. sqrt-div N/A

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

      13. lower-unsound-sqrt.f64 N/A

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

      14. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 77.0%

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

   8. Applied rewrites 77.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto \sqrt{1 - 2 \cdot \frac{1}{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 76.4

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

   11. Applied rewrites 76.4%

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

   if 2.5500000000000001e-176 < t < 5.99999999999999968e108

   1. Initial program 33.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 41.1%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 5.99999999999999968e108 < t

   1. Initial program 33.6%

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

   3. Applied rewrites 28.6%

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

   4. Taylor expanded in t around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 77.0

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

   6. Applied rewrites 77.0%

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

Alternative 2: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2} \cdot t\_m}{\sqrt{\left(\left(t\_m + t\_m\right) \cdot t\_m\right) \cdot \frac{-1 - x}{1 - x} - \frac{\ell \cdot \left(-1 \cdot \ell - \ell\right)}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-247}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{1 - 2 \cdot \frac{1}{x}}\\ \mathbf{elif}\;t\_m \leq 1.22 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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)
          (sqrt
           (-
            (* (* (+ t_m t_m) t_m) (/ (- -1.0 x) (- 1.0 x)))
            (/ (* l (- (* -1.0 l) l)) x))))))
   (*
    t_s
    (if (<= t_m 2.4e-247)
      t_2
      (if (<= t_m 2.55e-176)
        (sqrt (- 1.0 (* 2.0 (/ 1.0 x))))
        (if (<= t_m 1.22e+45) t_2 (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 t_2 = (sqrt(2.0) * t_m) / sqrt(((((t_m + t_m) * t_m) * ((-1.0 - x) / (1.0 - x))) - ((l * ((-1.0 * l) - l)) / x)));
	double tmp;
	if (t_m <= 2.4e-247) {
		tmp = t_2;
	} else if (t_m <= 2.55e-176) {
		tmp = sqrt((1.0 - (2.0 * (1.0 / x))));
	} else if (t_m <= 1.22e+45) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

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
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (sqrt(2.0d0) * t_m) / sqrt(((((t_m + t_m) * t_m) * (((-1.0d0) - x) / (1.0d0 - x))) - ((l * (((-1.0d0) * l) - l)) / x)))
    if (t_m <= 2.4d-247) then
        tmp = t_2
    else if (t_m <= 2.55d-176) then
        tmp = sqrt((1.0d0 - (2.0d0 * (1.0d0 / x))))
    else if (t_m <= 1.22d+45) then
        tmp = t_2
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
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) {
	double t_2 = (Math.sqrt(2.0) * t_m) / Math.sqrt(((((t_m + t_m) * t_m) * ((-1.0 - x) / (1.0 - x))) - ((l * ((-1.0 * l) - l)) / x)));
	double tmp;
	if (t_m <= 2.4e-247) {
		tmp = t_2;
	} else if (t_m <= 2.55e-176) {
		tmp = Math.sqrt((1.0 - (2.0 * (1.0 / x))));
	} else if (t_m <= 1.22e+45) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = (math.sqrt(2.0) * t_m) / math.sqrt(((((t_m + t_m) * t_m) * ((-1.0 - x) / (1.0 - x))) - ((l * ((-1.0 * l) - l)) / x)))
	tmp = 0
	if t_m <= 2.4e-247:
		tmp = t_2
	elif t_m <= 2.55e-176:
		tmp = math.sqrt((1.0 - (2.0 * (1.0 / x))))
	elif t_m <= 1.22e+45:
		tmp = t_2
	else:
		tmp = math.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)
	t_2 = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(Float64(Float64(Float64(t_m + t_m) * t_m) * Float64(Float64(-1.0 - x) / Float64(1.0 - x))) - Float64(Float64(l * Float64(Float64(-1.0 * l) - l)) / x))))
	tmp = 0.0
	if (t_m <= 2.4e-247)
		tmp = t_2;
	elseif (t_m <= 2.55e-176)
		tmp = sqrt(Float64(1.0 - Float64(2.0 * Float64(1.0 / x))));
	elseif (t_m <= 1.22e+45)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = (sqrt(2.0) * t_m) / sqrt(((((t_m + t_m) * t_m) * ((-1.0 - x) / (1.0 - x))) - ((l * ((-1.0 * l) - l)) / x)));
	tmp = 0.0;
	if (t_m <= 2.4e-247)
		tmp = t_2;
	elseif (t_m <= 2.55e-176)
		tmp = sqrt((1.0 - (2.0 * (1.0 / x))));
	elseif (t_m <= 1.22e+45)
		tmp = t_2;
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = 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[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(-1.0 - x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(l * N[(N[(-1.0 * l), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.4e-247], t$95$2, If[LessEqual[t$95$m, 2.55e-176], N[Sqrt[N[(1.0 - N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$m, 1.22e+45], t$95$2, N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.40000000000000011e-247 or 2.5500000000000001e-176 < t < 1.21999999999999997e45

   1. Initial program 33.6%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. associate--r+ N/A

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

      7. sub-negate N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 41.1%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 53.6

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

   6. Applied rewrites 53.6%

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

   if 2.40000000000000011e-247 < t < 2.5500000000000001e-176

   1. Initial program 33.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r* N/A

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

      12. sqrt-div N/A

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

      13. lower-unsound-sqrt.f64 N/A

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

      14. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 77.0%

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

   8. Applied rewrites 77.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto \sqrt{1 - 2 \cdot \frac{1}{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 76.4

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

   11. Applied rewrites 76.4%

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

   if 1.21999999999999997e45 < t

   1. Initial program 33.6%

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

   3. Applied rewrites 28.6%

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

   4. Taylor expanded in t around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 77.0

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

   6. Applied rewrites 77.0%

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

Alternative 3: 77.0% accurate, 0.8× speedup

Math

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

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 (<= (/ (+ x 1.0) (- x 1.0)) 1.0)
    (sqrt (/ (/ -2.0 (- -1.0 x)) (/ -2.0 (- 1.0 x))))
    (/
     (* (sqrt 2.0) t_m)
     (sqrt
      (fma
       (- l)
       l
       (/ (* (fma (+ t_m t_m) t_m (* l l)) (- x -1.0)) (- 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 (((x + 1.0) / (x - 1.0)) <= 1.0) {
		tmp = sqrt(((-2.0 / (-1.0 - x)) / (-2.0 / (1.0 - x))));
	} else {
		tmp = (sqrt(2.0) * t_m) / sqrt(fma(-l, l, ((fma((t_m + t_m), t_m, (l * l)) * (x - -1.0)) / (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 (Float64(Float64(x + 1.0) / Float64(x - 1.0)) <= 1.0)
		tmp = sqrt(Float64(Float64(-2.0 / Float64(-1.0 - x)) / Float64(-2.0 / Float64(1.0 - x))));
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(fma(Float64(-l), l, Float64(Float64(fma(Float64(t_m + t_m), t_m, Float64(l * l)) * Float64(x - -1.0)) / 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[N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision], 1.0], N[Sqrt[N[(N[(-2.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-2.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[((-l) * l + N[(N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] * t$95$m + N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) < 1

   1. Initial program 33.6%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-/r* N/A

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

      12. sqrt-div N/A

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

      13. lower-unsound-sqrt.f64 N/A

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

      14. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 77.0%

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

   8. Applied rewrites 77.0%

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

   if 1 < (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))

   1. Initial program 33.6%

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

   3. Applied rewrites 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 28.2

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

   5. Applied rewrites 28.2%

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

Alternative 4: 77.0% accurate, 2.2× speedup

Math

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

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 (/ (/ -2.0 (- -1.0 x)) (/ -2.0 (- 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(((-2.0 / (-1.0 - x)) / (-2.0 / (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((((-2.0d0) / ((-1.0d0) - x)) / ((-2.0d0) / (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(((-2.0 / (-1.0 - x)) / (-2.0 / (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(((-2.0 / (-1.0 - x)) / (-2.0 / (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 * sqrt(Float64(Float64(-2.0 / Float64(-1.0 - x)) / Float64(-2.0 / 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(((-2.0 / (-1.0 - x)) / (-2.0 / (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[Sqrt[N[(N[(-2.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-2.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 77.0

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

4. Applied rewrites 77.0%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. associate-/r* N/A

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

   12. sqrt-div N/A

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

   13. lower-unsound-sqrt.f64 N/A

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

   14. lower-unsound-/.f64 N/A

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

6. Applied rewrites 77.0%

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

8. Applied rewrites 77.0%

   \[\leadsto \sqrt{\frac{\frac{-2}{-1 - x}}{\frac{-2}{1 - x}}} \]
9. Add Preprocessing

Alternative 5: 77.0% accurate, 3.5× speedup

Math

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

FPCore

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 (/ (- 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(((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(((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(((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(((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 * 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(((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[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Applied rewrites 28.6%

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

4. Taylor expanded in t around inf

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-+.f64 77.0

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

6. Applied rewrites 77.0%

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

Alternative 6: 76.4% accurate, 5.7× speedup

Math

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

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 (- 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 * (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 * (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 * (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 * (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(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 * (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[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 77.0

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

4. Applied rewrites 77.0%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. associate-/r* N/A

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

   12. sqrt-div N/A

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

   13. lower-unsound-sqrt.f64 N/A

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

   14. lower-unsound-/.f64 N/A

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

6. Applied rewrites 77.0%

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

8. Applied rewrites 77.0%

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

9. Taylor expanded in x around inf

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

    1. lower--.f64 N/A

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

    2. lower-/.f64 76.4

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

11. Applied rewrites 76.4%

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

Alternative 7: 75.8% accurate, 6.5× speedup

Math

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

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 (/ 2.0 2.0))))

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((2.0 / 2.0));
}

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((2.0d0 / 2.0d0))
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((2.0 / 2.0));
}

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((2.0 / 2.0))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * sqrt(Float64(2.0 / 2.0)))
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((2.0 / 2.0));
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[Sqrt[N[(2.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower--.f64 77.0

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

4. Applied rewrites 77.0%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 75.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 75.8

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

9. Applied rewrites 75.8%

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

Reproduce

herbie shell --seed 2025157 
(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)))))