Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 84.6%

Time: 6.2s

Alternatives: 9

Speedup: 39.7×

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: 33.5% 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: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_4 := -t\_3\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\mathsf{fma}\left(2, \frac{t\_m}{t\_2}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{t\_m \cdot t\_2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 27000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, -\frac{\left(-\frac{\left(\frac{t\_3}{x} + \mathsf{fma}\left(2 \cdot t\_m, t\_m, \ell \cdot \ell - t\_4\right)\right) - \frac{t\_4}{x}}{x}\right) + \left(t\_4 - t\_3\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* x (sqrt 2.0)))
        (t_3 (fma (* t_m t_m) 2.0 (* l l)))
        (t_4 (- t_3)))
   (*
    t_s
    (if (<= t_m 1.9e-160)
      (*
       (sqrt 2.0)
       (/
        t_m
        (fma 2.0 (/ t_m t_2) (fma t_m (sqrt 2.0) (/ (* l l) (* t_m t_2))))))
      (if (<= t_m 27000.0)
        (/
         (* (sqrt 2.0) t_m)
         (sqrt
          (fma
           (* 2.0 t_m)
           t_m
           (-
            (/
             (+
              (-
               (/
                (-
                 (+ (/ t_3 x) (fma (* 2.0 t_m) t_m (- (* l l) t_4)))
                 (/ t_4 x))
                x))
              (- t_4 t_3))
             x)))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = x * sqrt(2.0);
	double t_3 = fma((t_m * t_m), 2.0, (l * l));
	double t_4 = -t_3;
	double tmp;
	if (t_m <= 1.9e-160) {
		tmp = sqrt(2.0) * (t_m / fma(2.0, (t_m / t_2), fma(t_m, sqrt(2.0), ((l * l) / (t_m * t_2)))));
	} else if (t_m <= 27000.0) {
		tmp = (sqrt(2.0) * t_m) / sqrt(fma((2.0 * t_m), t_m, -((-((((t_3 / x) + fma((2.0 * t_m), t_m, ((l * l) - t_4))) - (t_4 / x)) / x) + (t_4 - t_3)) / x)));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(x * sqrt(2.0))
	t_3 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_4 = Float64(-t_3)
	tmp = 0.0
	if (t_m <= 1.9e-160)
		tmp = Float64(sqrt(2.0) * Float64(t_m / fma(2.0, Float64(t_m / t_2), fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(t_m * t_2))))));
	elseif (t_m <= 27000.0)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(Float64(t_3 / x) + fma(Float64(2.0 * t_m), t_m, Float64(Float64(l * l) - t_4))) - Float64(t_4 / x)) / x)) + Float64(t_4 - t_3)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-160], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(2.0 * N[(t$95$m / t$95$2), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 27000.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + (-N[(N[((-N[(N[(N[(N[(t$95$3 / x), $MachinePrecision] + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(l * l), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]) + N[(t$95$4 - t$95$3), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8999999999999999e-160

   1. Initial program 3.8%

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

   3. Applied rewrites 3.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 58.9

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

   8. Applied rewrites 58.9%

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

   if 1.8999999999999999e-160 < t < 27000

   1. Initial program 51.7%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 85.4%

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

   if 27000 < t

   1. Initial program 34.9%

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

   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

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

      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 92.5

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

   6. Applied rewrites 92.5%

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

Alternative 2: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\mathsf{fma}\left(2, \frac{t\_m}{t\_2}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{t\_m \cdot t\_2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 27000:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* x (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.9e-160)
      (*
       (sqrt 2.0)
       (/
        t_m
        (fma 2.0 (/ t_m t_2) (fma t_m (sqrt 2.0) (/ (* l l) (* t_m t_2))))))
      (if (<= t_m 27000.0)
        (/
         (* (sqrt 2.0) t_m)
         (sqrt
          (-
           (fma (/ (* t_m t_m) x) 2.0 (fma (* t_m t_m) 2.0 (/ (* l l) x)))
           (/ (- (fma (* t_m t_m) 2.0 (* l l))) x))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = x * sqrt(2.0);
	double tmp;
	if (t_m <= 1.9e-160) {
		tmp = sqrt(2.0) * (t_m / fma(2.0, (t_m / t_2), fma(t_m, sqrt(2.0), ((l * l) / (t_m * t_2)))));
	} else if (t_m <= 27000.0) {
		tmp = (sqrt(2.0) * t_m) / sqrt((fma(((t_m * t_m) / x), 2.0, fma((t_m * t_m), 2.0, ((l * l) / x))) - (-fma((t_m * t_m), 2.0, (l * l)) / x)));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.9e-160)
		tmp = Float64(sqrt(2.0) * Float64(t_m / fma(2.0, Float64(t_m / t_2), fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(t_m * t_2))))));
	elseif (t_m <= 27000.0)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(fma(Float64(Float64(t_m * t_m) / x), 2.0, fma(Float64(t_m * t_m), 2.0, Float64(Float64(l * l) / x))) - Float64(Float64(-fma(Float64(t_m * t_m), 2.0, Float64(l * l))) / x))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-160], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(2.0 * N[(t$95$m / t$95$2), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 27000.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[((-N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]) / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8999999999999999e-160

   1. Initial program 3.8%

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

   3. Applied rewrites 3.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 58.9

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

   8. Applied rewrites 58.9%

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

   if 1.8999999999999999e-160 < t < 27000

   1. Initial program 51.7%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 84.9%

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

   if 27000 < t

   1. Initial program 34.9%

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

   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

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

      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 92.5

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

   6. Applied rewrites 92.5%

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

Alternative 3: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\mathsf{fma}\left(2, \frac{t\_m}{t\_2}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{t\_m \cdot t\_2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 27000:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \mathsf{fma}\left(t\_m + t\_m, t\_m, \frac{\ell \cdot \ell}{x}\right)\right) - \left(-\frac{\mathsf{fma}\left(t\_m + t\_m, t\_m, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* x (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.9e-160)
      (*
       (sqrt 2.0)
       (/
        t_m
        (fma 2.0 (/ t_m t_2) (fma t_m (sqrt 2.0) (/ (* l l) (* t_m t_2))))))
      (if (<= t_m 27000.0)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (-
            (fma 2.0 (/ (* t_m t_m) x) (fma (+ t_m t_m) t_m (/ (* l l) x)))
            (- (/ (fma (+ t_m t_m) t_m (* l l)) x))))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = x * sqrt(2.0);
	double tmp;
	if (t_m <= 1.9e-160) {
		tmp = sqrt(2.0) * (t_m / fma(2.0, (t_m / t_2), fma(t_m, sqrt(2.0), ((l * l) / (t_m * t_2)))));
	} else if (t_m <= 27000.0) {
		tmp = sqrt(2.0) * (t_m / sqrt((fma(2.0, ((t_m * t_m) / x), fma((t_m + t_m), t_m, ((l * l) / x))) - -(fma((t_m + t_m), t_m, (l * l)) / x))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.9e-160)
		tmp = Float64(sqrt(2.0) * Float64(t_m / fma(2.0, Float64(t_m / t_2), fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(t_m * t_2))))));
	elseif (t_m <= 27000.0)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), fma(Float64(t_m + t_m), t_m, Float64(Float64(l * l) / x))) - Float64(-Float64(fma(Float64(t_m + t_m), t_m, Float64(l * l)) / x))))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-160], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(2.0 * N[(t$95$m / t$95$2), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 27000.0], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t$95$m + t$95$m), $MachinePrecision] * t$95$m + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - (-N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] * t$95$m + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8999999999999999e-160

   1. Initial program 3.8%

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

   3. Applied rewrites 3.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 58.9

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

   8. Applied rewrites 58.9%

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

   if 1.8999999999999999e-160 < t < 27000

   1. Initial program 51.7%

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

   3. Applied rewrites 51.6%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. mul-1-neg N/A

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

   6. Applied rewrites 84.8%

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

   if 27000 < t

   1. Initial program 34.9%

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

   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

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

      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 92.5

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

   6. Applied rewrites 92.5%

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

Alternative 4: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\mathsf{fma}\left(2, \frac{t\_m}{t\_2}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{t\_m \cdot t\_2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 27000:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x}, \left(t\_m \cdot t\_m\right) \cdot \left(2 + \frac{4}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* x (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.9e-160)
      (*
       (sqrt 2.0)
       (/
        t_m
        (fma 2.0 (/ t_m t_2) (fma t_m (sqrt 2.0) (/ (* l l) (* t_m t_2))))))
      (if (<= t_m 27000.0)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt (fma 2.0 (/ (* l l) x) (* (* t_m t_m) (+ 2.0 (/ 4.0 x)))))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = x * sqrt(2.0);
	double tmp;
	if (t_m <= 1.9e-160) {
		tmp = sqrt(2.0) * (t_m / fma(2.0, (t_m / t_2), fma(t_m, sqrt(2.0), ((l * l) / (t_m * t_2)))));
	} else if (t_m <= 27000.0) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, ((l * l) / x), ((t_m * t_m) * (2.0 + (4.0 / x))))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.9e-160)
		tmp = Float64(sqrt(2.0) * Float64(t_m / fma(2.0, Float64(t_m / t_2), fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(t_m * t_2))))));
	elseif (t_m <= 27000.0)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64(Float64(l * l) / x), Float64(Float64(t_m * t_m) * Float64(2.0 + Float64(4.0 / x)))))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-160], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(2.0 * N[(t$95$m / t$95$2), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 27000.0], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8999999999999999e-160

   1. Initial program 3.8%

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

   3. Applied rewrites 3.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 58.9

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

   8. Applied rewrites 58.9%

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

   if 1.8999999999999999e-160 < t < 27000

   1. Initial program 51.7%

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

   3. Applied rewrites 51.6%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. mul-1-neg N/A

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in t around 0

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 84.8

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

   9. Applied rewrites 84.8%

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

   if 27000 < t

   1. Initial program 34.9%

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

   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

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

      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 92.5

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

   6. Applied rewrites 92.5%

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

Alternative 5: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x}, \left(t\_m \cdot t\_m\right) \cdot \left(2 + \frac{4}{x}\right)\right)}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-259}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 10^{-166}:\\ \;\;\;\;\sqrt{1 - 2 \cdot \frac{1}{x}}\\ \mathbf{elif}\;t\_m \leq 27000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2
         (*
          (sqrt 2.0)
          (/
           t_m
           (sqrt (fma 2.0 (/ (* l l) x) (* (* t_m t_m) (+ 2.0 (/ 4.0 x)))))))))
   (*
    t_s
    (if (<= t_m 3.9e-259)
      t_2
      (if (<= t_m 1e-166)
        (sqrt (- 1.0 (* 2.0 (/ 1.0 x))))
        (if (<= t_m 27000.0) t_2 (sqrt (/ (- x 1.0) (+ 1.0 x)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = sqrt(2.0) * (t_m / sqrt(fma(2.0, ((l * l) / x), ((t_m * t_m) * (2.0 + (4.0 / x))))));
	double tmp;
	if (t_m <= 3.9e-259) {
		tmp = t_2;
	} else if (t_m <= 1e-166) {
		tmp = sqrt((1.0 - (2.0 * (1.0 / x))));
	} else if (t_m <= 27000.0) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64(Float64(l * l) / x), Float64(Float64(t_m * t_m) * Float64(2.0 + Float64(4.0 / x)))))))
	tmp = 0.0
	if (t_m <= 3.9e-259)
		tmp = t_2;
	elseif (t_m <= 1e-166)
		tmp = sqrt(Float64(1.0 - Float64(2.0 * Float64(1.0 / x))));
	elseif (t_m <= 27000.0)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.9e-259], t$95$2, If[LessEqual[t$95$m, 1e-166], N[Sqrt[N[(1.0 - N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$m, 27000.0], t$95$2, N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.90000000000000016e-259 or 1.00000000000000004e-166 < t < 27000

   1. Initial program 41.7%

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

   3. Applied rewrites 41.7%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. mul-1-neg N/A

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

   6. Applied rewrites 78.7%

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

   7. Taylor expanded in t around 0

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 78.7

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

   9. Applied rewrites 78.7%

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

   if 3.90000000000000016e-259 < t < 1.00000000000000004e-166

   1. Initial program 3.6%

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

   2. Taylor expanded in 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 46.2

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

   4. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}} \cdot 1} \]
   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 46.2

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

   6. Applied rewrites 46.2%

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

   7. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 45.9

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

   9. Applied rewrites 45.9%

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

   if 27000 < t

   1. Initial program 34.9%

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

   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

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

      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 92.5

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

   6. Applied rewrites 92.5%

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

Alternative 6: 78.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.9e-259)
    (* (sqrt 2.0) (/ t_m (sqrt (* 2.0 (/ (* l l) x)))))
    (sqrt (/ (- x 1.0) (+ 1.0 x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.9e-259) {
		tmp = sqrt(2.0) * (t_m / sqrt((2.0 * ((l * l) / x))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, x, l, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.9d-259) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((2.0d0 * ((l * l) / x))))
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.9e-259) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((2.0 * ((l * l) / x))));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.9e-259:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((2.0 * ((l * l) / x))))
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.9e-259)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(2.0 * Float64(Float64(l * l) / x)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.9e-259)
		tmp = sqrt(2.0) * (t_m / sqrt((2.0 * ((l * l) / x))));
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.9e-259], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.90000000000000016e-259

   1. Initial program 3.7%

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

   3. Applied rewrites 3.7%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. mul-1-neg N/A

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

   6. Applied rewrites 56.4%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 56.0

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

   9. Applied rewrites 56.0%

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

   if 3.90000000000000016e-259 < t

   1. Initial program 35.4%

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

   2. 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 79.6

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

   4. Applied rewrites 79.6%

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

      1. lift-*.f64 N/A

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

      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 79.6

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

   6. Applied rewrites 79.6%

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

Alternative 7: 76.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

2. Taylor expanded in 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 76.6

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

4. Applied rewrites 76.6%

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

   1. lift-*.f64 N/A

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

   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 76.6

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

6. Applied rewrites 76.6%

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

Alternative 8: 75.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

2. Taylor expanded in 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 76.6

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

4. Applied rewrites 76.6%

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

5. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 75.9

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

7. Applied rewrites 75.9%

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

Alternative 9: 75.2% accurate, 39.7× 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 33.5%

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

2. Taylor expanded in 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 75.2

      \[\leadsto 1 \]

4. Applied rewrites 75.2%

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

Reproduce

herbie shell --seed 2025110 
(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)))))