Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 85.0%

Time: 5.7s

Alternatives: 10

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \mathsf{fma}\left(\ell, \ell, \frac{\ell \cdot \ell}{x}\right)\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.1 \cdot 10^{-147}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(\frac{t\_2 - \left(-t\_2\right)}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_4\right)}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot \frac{1 + x}{x - 1}, 2, \frac{t\_3 - -1 \cdot t\_3}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - \frac{1}{1 + x}} \cdot 1\\ \end{array} \end{array} \end{array} \]

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 (fma l l (/ (* l l) x)))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 4.1e-147)
      (/ t_4 (fma (/ (- t_2 (- t_2)) (* (* (sqrt 2.0) x) t_m)) 0.5 t_4))
      (if (<= t_m 1.6e+69)
        (/
         t_4
         (sqrt
          (fma
           (* (* t_m t_m) (/ (+ 1.0 x) (- x 1.0)))
           2.0
           (/ (- t_3 (* -1.0 t_3)) x))))
        (* (sqrt (- (/ x (+ 1.0 x)) (/ 1.0 (+ 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 t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = fma(l, l, ((l * l) / x));
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 4.1e-147) {
		tmp = t_4 / fma(((t_2 - -t_2) / ((sqrt(2.0) * x) * t_m)), 0.5, t_4);
	} else if (t_m <= 1.6e+69) {
		tmp = t_4 / sqrt(fma(((t_m * t_m) * ((1.0 + x) / (x - 1.0))), 2.0, ((t_3 - (-1.0 * t_3)) / x)));
	} else {
		tmp = sqrt(((x / (1.0 + x)) - (1.0 / (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)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = fma(l, l, Float64(Float64(l * l) / x))
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 4.1e-147)
		tmp = Float64(t_4 / fma(Float64(Float64(t_2 - Float64(-t_2)) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_4));
	elseif (t_m <= 1.6e+69)
		tmp = Float64(t_4 / sqrt(fma(Float64(Float64(t_m * t_m) * Float64(Float64(1.0 + x) / Float64(x - 1.0))), 2.0, Float64(Float64(t_3 - Float64(-1.0 * t_3)) / x))));
	else
		tmp = Float64(sqrt(Float64(Float64(x / Float64(1.0 + x)) - Float64(1.0 / Float64(1.0 + 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[(l * l + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.1e-147], N[(t$95$4 / N[(N[(N[(t$95$2 - (-t$95$2)), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+69], N[(t$95$4 / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(t$95$3 - N[(-1.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.1e-147

   1. Initial program 6.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 61.3%

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

   if 4.1e-147 < t < 1.59999999999999992e69

   1. Initial program 57.8%

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

   2. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 N/A

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 85.1%

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

   if 1.59999999999999992e69 < t

   1. Initial program 27.1%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 95.4

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

   6. Applied rewrites 95.4%

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

Alternative 2: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.1 \cdot 10^{-147}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\frac{t\_2 - \left(-t\_2\right)}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot \frac{1 + x}{x - 1}, 2, \frac{\mathsf{fma}\left(\ell, \ell, 1 \cdot \left(\ell \cdot \ell\right)\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - \frac{1}{1 + x}} \cdot 1\\ \end{array} \end{array} \end{array} \]

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 4.1e-147)
      (/ t_3 (fma (/ (- t_2 (- t_2)) (* (* (sqrt 2.0) x) t_m)) 0.5 t_3))
      (if (<= t_m 1.6e+69)
        (/
         t_3
         (sqrt
          (fma
           (* (* t_m t_m) (/ (+ 1.0 x) (- x 1.0)))
           2.0
           (/ (fma l l (* 1.0 (* l l))) x))))
        (* (sqrt (- (/ x (+ 1.0 x)) (/ 1.0 (+ 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 t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 4.1e-147) {
		tmp = t_3 / fma(((t_2 - -t_2) / ((sqrt(2.0) * x) * t_m)), 0.5, t_3);
	} else if (t_m <= 1.6e+69) {
		tmp = t_3 / sqrt(fma(((t_m * t_m) * ((1.0 + x) / (x - 1.0))), 2.0, (fma(l, l, (1.0 * (l * l))) / x)));
	} else {
		tmp = sqrt(((x / (1.0 + x)) - (1.0 / (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)
	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 <= 4.1e-147)
		tmp = Float64(t_3 / fma(Float64(Float64(t_2 - Float64(-t_2)) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_3));
	elseif (t_m <= 1.6e+69)
		tmp = Float64(t_3 / sqrt(fma(Float64(Float64(t_m * t_m) * Float64(Float64(1.0 + x) / Float64(x - 1.0))), 2.0, Float64(fma(l, l, Float64(1.0 * Float64(l * l))) / x))));
	else
		tmp = Float64(sqrt(Float64(Float64(x / Float64(1.0 + x)) - Float64(1.0 / Float64(1.0 + 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, 4.1e-147], N[(t$95$3 / N[(N[(N[(t$95$2 - (-t$95$2)), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+69], N[(t$95$3 / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(l * l + N[(1.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.1e-147

   1. Initial program 6.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 61.3%

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

   if 4.1e-147 < t < 1.59999999999999992e69

   1. Initial program 57.8%

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

   2. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 N/A

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if 1.59999999999999992e69 < t

   1. Initial program 27.1%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 95.4

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

   6. Applied rewrites 95.4%

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

Alternative 3: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\mathsf{fma}\left(\ell, \ell, 1 \cdot \left(\ell \cdot \ell\right)\right)}{x}\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, t\_2\right)}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot \frac{1 + x}{x - 1}, 2, t\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - \frac{1}{1 + x}} \cdot 1\\ \end{array} \end{array} \end{array} \]

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 l l (* 1.0 (* l l))) x)) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.35e-222)
      (/ t_3 (sqrt (fma (* t_m t_m) 2.0 t_2)))
      (if (<= t_m 1.5e-167)
        (sqrt (/ (- x 1.0) (+ 1.0 x)))
        (if (<= t_m 1.6e+69)
          (/ t_3 (sqrt (fma (* (* t_m t_m) (/ (+ 1.0 x) (- x 1.0))) 2.0 t_2)))
          (* (sqrt (- (/ x (+ 1.0 x)) (/ 1.0 (+ 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 t_2 = fma(l, l, (1.0 * (l * l))) / x;
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.35e-222) {
		tmp = t_3 / sqrt(fma((t_m * t_m), 2.0, t_2));
	} else if (t_m <= 1.5e-167) {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	} else if (t_m <= 1.6e+69) {
		tmp = t_3 / sqrt(fma(((t_m * t_m) * ((1.0 + x) / (x - 1.0))), 2.0, t_2));
	} else {
		tmp = sqrt(((x / (1.0 + x)) - (1.0 / (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)
	t_2 = Float64(fma(l, l, Float64(1.0 * Float64(l * l))) / x)
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 2.35e-222)
		tmp = Float64(t_3 / sqrt(fma(Float64(t_m * t_m), 2.0, t_2)));
	elseif (t_m <= 1.5e-167)
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	elseif (t_m <= 1.6e+69)
		tmp = Float64(t_3 / sqrt(fma(Float64(Float64(t_m * t_m) * Float64(Float64(1.0 + x) / Float64(x - 1.0))), 2.0, t_2)));
	else
		tmp = Float64(sqrt(Float64(Float64(x / Float64(1.0 + x)) - Float64(1.0 / Float64(1.0 + 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[(l * l + N[(1.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.35e-222], N[(t$95$3 / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e-167], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$m, 1.6e+69], N[(t$95$3 / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.3499999999999999e-222

   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. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 N/A

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

   4. Applied rewrites 3.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 54.3

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

   10. Applied rewrites 54.3%

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

   if 2.3499999999999999e-222 < t < 1.4999999999999999e-167

   1. Initial program 3.7%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 50.1

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

   6. Applied rewrites 50.1%

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

   if 1.4999999999999999e-167 < t < 1.59999999999999992e69

   1. Initial program 55.3%

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

   2. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 N/A

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if 1.59999999999999992e69 < t

   1. Initial program 27.1%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 95.4

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

   6. Applied rewrites 95.4%

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

Alternative 4: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\mathsf{fma}\left(\ell, \ell, 1 \cdot \left(\ell \cdot \ell\right)\right)}{x}\right)}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot \frac{1 + x}{x - 1}, 2, 2 \cdot \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - \frac{1}{1 + x}} \cdot 1\\ \end{array} \end{array} \]

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 2.35e-222)
    (/
     (* (sqrt 2.0) t_m)
     (sqrt (fma (* t_m t_m) 2.0 (/ (fma l l (* 1.0 (* l l))) x))))
    (if (<= t_m 1.5e-167)
      (sqrt (/ (- x 1.0) (+ 1.0 x)))
      (if (<= t_m 1.6e+69)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (fma
            (* (* t_m t_m) (/ (+ 1.0 x) (- x 1.0)))
            2.0
            (* 2.0 (/ (* l l) x))))))
        (* (sqrt (- (/ x (+ 1.0 x)) (/ 1.0 (+ 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 (t_m <= 2.35e-222) {
		tmp = (sqrt(2.0) * t_m) / sqrt(fma((t_m * t_m), 2.0, (fma(l, l, (1.0 * (l * l))) / x)));
	} else if (t_m <= 1.5e-167) {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	} else if (t_m <= 1.6e+69) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(((t_m * t_m) * ((1.0 + x) / (x - 1.0))), 2.0, (2.0 * ((l * l) / x)))));
	} else {
		tmp = sqrt(((x / (1.0 + x)) - (1.0 / (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 (t_m <= 2.35e-222)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(fma(Float64(t_m * t_m), 2.0, Float64(fma(l, l, Float64(1.0 * Float64(l * l))) / x))));
	elseif (t_m <= 1.5e-167)
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	elseif (t_m <= 1.6e+69)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(Float64(Float64(t_m * t_m) * Float64(Float64(1.0 + x) / Float64(x - 1.0))), 2.0, Float64(2.0 * Float64(Float64(l * l) / x))))));
	else
		tmp = Float64(sqrt(Float64(Float64(x / Float64(1.0 + x)) - Float64(1.0 / Float64(1.0 + 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, 2.35e-222], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(l * l + N[(1.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e-167], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$m, 1.6e+69], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 2.3499999999999999e-222

   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. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 N/A

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

   4. Applied rewrites 3.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 54.3

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

   10. Applied rewrites 54.3%

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

   if 2.3499999999999999e-222 < t < 1.4999999999999999e-167

   1. Initial program 3.7%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 50.1

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

   6. Applied rewrites 50.1%

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

   if 1.4999999999999999e-167 < t < 1.59999999999999992e69

   1. Initial program 55.3%

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

   2. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 N/A

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 83.5

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

   7. Applied rewrites 83.5%

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

   9. Applied rewrites 83.4%

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

   if 1.59999999999999992e69 < t

   1. Initial program 27.1%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 95.4

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

   6. Applied rewrites 95.4%

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

Alternative 5: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2} \cdot t\_m}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\mathsf{fma}\left(\ell, \ell, 1 \cdot \left(\ell \cdot \ell\right)\right)}{x}\right)}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \mathbf{elif}\;t\_m \leq 0.046:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - \frac{1}{1 + x}} \cdot 1\\ \end{array} \end{array} \end{array} \]

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)
          (sqrt (fma (* t_m t_m) 2.0 (/ (fma l l (* 1.0 (* l l))) x))))))
   (*
    t_s
    (if (<= t_m 2.35e-222)
      t_2
      (if (<= t_m 1.5e-167)
        (sqrt (/ (- x 1.0) (+ 1.0 x)))
        (if (<= t_m 0.046)
          t_2
          (* (sqrt (- (/ x (+ 1.0 x)) (/ 1.0 (+ 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 t_2 = (sqrt(2.0) * t_m) / sqrt(fma((t_m * t_m), 2.0, (fma(l, l, (1.0 * (l * l))) / x)));
	double tmp;
	if (t_m <= 2.35e-222) {
		tmp = t_2;
	} else if (t_m <= 1.5e-167) {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	} else if (t_m <= 0.046) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x / (1.0 + x)) - (1.0 / (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)
	t_2 = Float64(Float64(sqrt(2.0) * t_m) / sqrt(fma(Float64(t_m * t_m), 2.0, Float64(fma(l, l, Float64(1.0 * Float64(l * l))) / x))))
	tmp = 0.0
	if (t_m <= 2.35e-222)
		tmp = t_2;
	elseif (t_m <= 1.5e-167)
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	elseif (t_m <= 0.046)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(Float64(x / Float64(1.0 + x)) - Float64(1.0 / Float64(1.0 + 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[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(l * l + N[(1.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.35e-222], t$95$2, If[LessEqual[t$95$m, 1.5e-167], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$m, 0.046], t$95$2, N[(N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3499999999999999e-222 or 1.4999999999999999e-167 < t < 0.045999999999999999

   1. Initial program 36.8%

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

   2. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower--.f64 N/A

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

   4. Applied rewrites 46.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 74.8

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 73.9

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

   10. Applied rewrites 73.9%

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

   if 2.3499999999999999e-222 < t < 1.4999999999999999e-167

   1. Initial program 3.7%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-sqrt.f64 50.1

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

   6. Applied rewrites 50.1%

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

   if 0.045999999999999999 < t

   1. Initial program 35.1%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 92.5

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

   4. Applied rewrites 92.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 92.5

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

   6. Applied rewrites 92.5%

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

Alternative 6: 78.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - \frac{1}{1 + x}} \cdot 1\\ \end{array} \end{array} \]

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 2.35e-222)
    (/ (* (sqrt 2.0) t_m) (sqrt (* 2.0 (/ (* l l) x))))
    (* (sqrt (- (/ x (+ 1.0 x)) (/ 1.0 (+ 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 (t_m <= 2.35e-222) {
		tmp = (sqrt(2.0) * t_m) / sqrt((2.0 * ((l * l) / x)));
	} else {
		tmp = sqrt(((x / (1.0 + x)) - (1.0 / (1.0 + x)))) * 1.0;
	}
	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) :: tmp
    if (t_m <= 2.35d-222) then
        tmp = (sqrt(2.0d0) * t_m) / sqrt((2.0d0 * ((l * l) / x)))
    else
        tmp = sqrt(((x / (1.0d0 + x)) - (1.0d0 / (1.0d0 + x)))) * 1.0d0
    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 tmp;
	if (t_m <= 2.35e-222) {
		tmp = (Math.sqrt(2.0) * t_m) / Math.sqrt((2.0 * ((l * l) / x)));
	} else {
		tmp = Math.sqrt(((x / (1.0 + x)) - (1.0 / (1.0 + x)))) * 1.0;
	}
	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):
	tmp = 0
	if t_m <= 2.35e-222:
		tmp = (math.sqrt(2.0) * t_m) / math.sqrt((2.0 * ((l * l) / x)))
	else:
		tmp = math.sqrt(((x / (1.0 + x)) - (1.0 / (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 (t_m <= 2.35e-222)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(2.0 * Float64(Float64(l * l) / x))));
	else
		tmp = Float64(sqrt(Float64(Float64(x / Float64(1.0 + x)) - Float64(1.0 / Float64(1.0 + x)))) * 1.0);
	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)
	tmp = 0.0;
	if (t_m <= 2.35e-222)
		tmp = (sqrt(2.0) * t_m) / sqrt((2.0 * ((l * l) / x)));
	else
		tmp = sqrt(((x / (1.0 + x)) - (1.0 / (1.0 + x)))) * 1.0;
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-222], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.3499999999999999e-222

   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

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 6.2%

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

   4. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 4.1

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

   6. Applied rewrites 4.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 53.9

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

   9. Applied rewrites 53.9%

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

   if 2.3499999999999999e-222 < t

   1. Initial program 36.8%

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

   2. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 81.6

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

   4. Applied rewrites 81.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 81.7

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

   6. Applied rewrites 81.7%

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

Alternative 7: 76.9% accurate, 1.8× speedup

Math

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

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 (- (/ x (+ 1.0 x)) (/ 1.0 (+ 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) {
	return t_s * (sqrt(((x / (1.0 + x)) - (1.0 / (1.0 + x)))) * 1.0);
}

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(((x / (1.0d0 + x)) - (1.0d0 / (1.0d0 + x)))) * 1.0d0)
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(((x / (1.0 + x)) - (1.0 / (1.0 + x)))) * 1.0);
}

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(((x / (1.0 + x)) - (1.0 / (1.0 + x)))) * 1.0)

Julia

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

Derivation

1. Initial program 33.4%

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

2. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 76.9

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

4. Applied rewrites 76.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 76.9

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

6. Applied rewrites 76.9%

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

Alternative 8: 76.9% accurate, 3.0× 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 33.4%

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

2. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 76.9

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

4. Applied rewrites 76.9%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-sqrt.f64 76.9

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

6. Applied rewrites 76.9%

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

Alternative 9: 76.3% accurate, 3.4× speedup

Math

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

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

Derivation

1. Initial program 33.4%

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

2. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 76.9

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

4. Applied rewrites 76.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 76.9

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

6. Applied rewrites 76.9%

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

7. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 76.3

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

9. Applied rewrites 76.3%

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

    1. lift-*.f64 N/A

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

    2. *-rgt-identity 76.3

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

    3. sub-div 76.3

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

11. Applied rewrites 76.3%

    \[\leadsto \color{blue}{\sqrt{1 - \frac{2}{x}}} \]
12. Add Preprocessing

Alternative 10: 75.6% accurate, 85.0× speedup

Math

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

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))

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;
}

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
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;
}

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

Julia

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

Derivation

1. Initial program 33.4%

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

2. Taylor expanded in x around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   3. metadata-eval 75.6

      \[\leadsto 1 \]

4. Applied rewrites 75.6%

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

Reproduce

herbie shell --seed 2025093 
(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)))))