Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 83.8%

Time: 6.8s

Alternatives: 9

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.3% 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: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := x \cdot \sqrt{2}\\ t_4 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_5 := -t\_4\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-231}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{elif}\;t\_m \leq 2.15 \cdot 10^{-168}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(2, \frac{t\_m}{t\_3}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{t\_m \cdot t\_3}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.06 \cdot 10^{+30}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, -\frac{\mathsf{fma}\left(t\_4 - t\_5, -1, \frac{t\_5}{x}\right) - \frac{t\_4}{x}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m))
        (t_3 (* x (sqrt 2.0)))
        (t_4 (fma (* t_m t_m) 2.0 (* l l)))
        (t_5 (- t_4)))
   (*
    t_s
    (if (<= t_m 1.8e-231)
      (/
       t_2
       (sqrt
        (fma 2.0 (/ (fma 2.0 (* t_m t_m) (* l l)) x) (* 2.0 (* t_m t_m)))))
      (if (<= t_m 2.15e-168)
        (/
         t_2
         (fma 2.0 (/ t_m t_3) (fma t_m (sqrt 2.0) (/ (* l l) (* t_m t_3)))))
        (if (<= t_m 1.06e+30)
          (/
           t_2
           (sqrt
            (fma
             (* 2.0 t_m)
             t_m
             (- (/ (- (fma (- t_4 t_5) -1.0 (/ t_5 x)) (/ t_4 x)) 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 = sqrt(2.0) * t_m;
	double t_3 = x * sqrt(2.0);
	double t_4 = fma((t_m * t_m), 2.0, (l * l));
	double t_5 = -t_4;
	double tmp;
	if (t_m <= 1.8e-231) {
		tmp = t_2 / sqrt(fma(2.0, (fma(2.0, (t_m * t_m), (l * l)) / x), (2.0 * (t_m * t_m))));
	} else if (t_m <= 2.15e-168) {
		tmp = t_2 / fma(2.0, (t_m / t_3), fma(t_m, sqrt(2.0), ((l * l) / (t_m * t_3))));
	} else if (t_m <= 1.06e+30) {
		tmp = t_2 / sqrt(fma((2.0 * t_m), t_m, -((fma((t_4 - t_5), -1.0, (t_5 / x)) - (t_4 / x)) / 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(sqrt(2.0) * t_m)
	t_3 = Float64(x * sqrt(2.0))
	t_4 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_5 = Float64(-t_4)
	tmp = 0.0
	if (t_m <= 1.8e-231)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(fma(2.0, Float64(t_m * t_m), Float64(l * l)) / x), Float64(2.0 * Float64(t_m * t_m)))));
	elseif (t_m <= 2.15e-168)
		tmp = Float64(t_2 / fma(2.0, Float64(t_m / t_3), fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(t_m * t_3)))));
	elseif (t_m <= 1.06e+30)
		tmp = Float64(t_2 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(-Float64(Float64(fma(Float64(t_4 - t_5), -1.0, Float64(t_5 / x)) - Float64(t_4 / x)) / 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[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, N[(t$95$s * If[LessEqual[t$95$m, 1.8e-231], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.15e-168], N[(t$95$2 / N[(2.0 * N[(t$95$m / t$95$3), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t$95$m * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.06e+30], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + (-N[(N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] * -1.0 + N[(t$95$5 / x), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 / x), $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 4 regimes

2. if t < 1.79999999999999987e-231

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 3.9

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

   4. Applied rewrites 3.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   if 1.79999999999999987e-231 < t < 2.14999999999999998e-168

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 3.2

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

   4. Applied rewrites 3.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 63.3

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

   7. Applied rewrites 63.3%

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

   if 2.14999999999999998e-168 < t < 1.06e30

   1. Initial program 51.6%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 83.1%

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

   if 1.06e30 < t

   1. Initial program 32.4%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 93.2

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

   7. Applied rewrites 93.2%

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

      1. flip-+ 93.2

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

      2. pow2 93.2

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

      3. metadata-eval 93.2

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

      4. pow2 93.2

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

      5. lift-*.f64 N/A

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

   9. Applied rewrites 93.1%

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

Alternative 2: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := x \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-231}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{elif}\;t\_m \leq 2.15 \cdot 10^{-168}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(2, \frac{t\_m}{t\_3}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{t\_m \cdot t\_3}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)) (t_3 (* x (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.8e-231)
      (/
       t_2
       (sqrt
        (fma 2.0 (/ (fma 2.0 (* t_m t_m) (* l l)) x) (* 2.0 (* t_m t_m)))))
      (if (<= t_m 2.15e-168)
        (/
         t_2
         (fma 2.0 (/ t_m t_3) (fma t_m (sqrt 2.0) (/ (* l l) (* t_m t_3)))))
        (if (<= t_m 9e+29)
          (/
           t_2
           (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 = sqrt(2.0) * t_m;
	double t_3 = x * sqrt(2.0);
	double tmp;
	if (t_m <= 1.8e-231) {
		tmp = t_2 / sqrt(fma(2.0, (fma(2.0, (t_m * t_m), (l * l)) / x), (2.0 * (t_m * t_m))));
	} else if (t_m <= 2.15e-168) {
		tmp = t_2 / fma(2.0, (t_m / t_3), fma(t_m, sqrt(2.0), ((l * l) / (t_m * t_3))));
	} else if (t_m <= 9e+29) {
		tmp = t_2 / 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(sqrt(2.0) * t_m)
	t_3 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.8e-231)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(fma(2.0, Float64(t_m * t_m), Float64(l * l)) / x), Float64(2.0 * Float64(t_m * t_m)))));
	elseif (t_m <= 2.15e-168)
		tmp = Float64(t_2 / fma(2.0, Float64(t_m / t_3), fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(t_m * t_3)))));
	elseif (t_m <= 9e+29)
		tmp = Float64(t_2 / 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[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.8e-231], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.15e-168], N[(t$95$2 / N[(2.0 * N[(t$95$m / t$95$3), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t$95$m * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e+29], N[(t$95$2 / 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 4 regimes

2. if t < 1.79999999999999987e-231

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 3.9

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

   4. Applied rewrites 3.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   if 1.79999999999999987e-231 < t < 2.14999999999999998e-168

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 3.2

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

   4. Applied rewrites 3.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 63.3

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

   7. Applied rewrites 63.3%

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

   if 2.14999999999999998e-168 < t < 9.0000000000000005e29

   1. Initial program 51.6%

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

   3. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 82.6%

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

   if 9.0000000000000005e29 < t

   1. Initial program 32.4%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 93.2

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

   7. Applied rewrites 93.2%

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

      1. flip-+ 93.2

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

      2. pow2 93.2

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

      3. metadata-eval 93.2

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

      4. pow2 93.2

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

      5. lift-*.f64 N/A

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

   9. Applied rewrites 93.1%

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

Alternative 3: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ t_4 := x \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-231}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 2.15 \cdot 10^{-168}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(2, \frac{t\_m}{t\_4}, \mathsf{fma}\left(t\_m, \sqrt{2}, \frac{\ell \cdot \ell}{t\_m \cdot t\_4}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+29}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

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))
        (t_3
         (/
          t_2
          (sqrt
           (fma 2.0 (/ (fma 2.0 (* t_m t_m) (* l l)) x) (* 2.0 (* t_m t_m))))))
        (t_4 (* x (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.8e-231)
      t_3
      (if (<= t_m 2.15e-168)
        (/
         t_2
         (fma 2.0 (/ t_m t_4) (fma t_m (sqrt 2.0) (/ (* l l) (* t_m t_4)))))
        (if (<= t_m 9e+29) t_3 (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;
	double t_3 = t_2 / sqrt(fma(2.0, (fma(2.0, (t_m * t_m), (l * l)) / x), (2.0 * (t_m * t_m))));
	double t_4 = x * sqrt(2.0);
	double tmp;
	if (t_m <= 1.8e-231) {
		tmp = t_3;
	} else if (t_m <= 2.15e-168) {
		tmp = t_2 / fma(2.0, (t_m / t_4), fma(t_m, sqrt(2.0), ((l * l) / (t_m * t_4))));
	} else if (t_m <= 9e+29) {
		tmp = t_3;
	} 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) * t_m)
	t_3 = Float64(t_2 / sqrt(fma(2.0, Float64(fma(2.0, Float64(t_m * t_m), Float64(l * l)) / x), Float64(2.0 * Float64(t_m * t_m)))))
	t_4 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.8e-231)
		tmp = t_3;
	elseif (t_m <= 2.15e-168)
		tmp = Float64(t_2 / fma(2.0, Float64(t_m / t_4), fma(t_m, sqrt(2.0), Float64(Float64(l * l) / Float64(t_m * t_4)))));
	elseif (t_m <= 9e+29)
		tmp = t_3;
	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] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.8e-231], t$95$3, If[LessEqual[t$95$m, 2.15e-168], N[(t$95$2 / N[(2.0 * N[(t$95$m / t$95$4), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t$95$m * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e+29], t$95$3, 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 < 1.79999999999999987e-231 or 2.14999999999999998e-168 < t < 9.0000000000000005e29

   1. Initial program 40.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 22.5

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

   4. Applied rewrites 22.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if 1.79999999999999987e-231 < t < 2.14999999999999998e-168

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 3.2

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

   4. Applied rewrites 3.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 63.3

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

   7. Applied rewrites 63.3%

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

   if 9.0000000000000005e29 < t

   1. Initial program 32.4%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 93.2

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

   7. Applied rewrites 93.2%

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

      1. flip-+ 93.2

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

      2. pow2 93.2

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

      3. metadata-eval 93.2

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

      4. pow2 93.2

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

      5. lift-*.f64 N/A

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

   9. Applied rewrites 93.1%

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

Alternative 4: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\mathsf{fma}\left(2, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{x}, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

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 9e+29)
    (/
     (* (sqrt 2.0) t_m)
     (sqrt (fma 2.0 (/ (fma 2.0 (* t_m t_m) (* l l)) x) (* 2.0 (* t_m t_m)))))
    (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 <= 9e+29) {
		tmp = (sqrt(2.0) * t_m) / sqrt(fma(2.0, (fma(2.0, (t_m * t_m), (l * l)) / x), (2.0 * (t_m * t_m))));
	} 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)
	tmp = 0.0
	if (t_m <= 9e+29)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(fma(2.0, Float64(fma(2.0, Float64(t_m * t_m), Float64(l * l)) / x), Float64(2.0 * Float64(t_m * t_m)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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, 9e+29], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.0000000000000005e29

   1. Initial program 34.3%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 19.2

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

   4. Applied rewrites 19.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if 9.0000000000000005e29 < t

   1. Initial program 32.4%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 93.1

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

   5. Applied rewrites 93.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 93.2

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

   7. Applied rewrites 93.2%

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

      1. flip-+ 93.2

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

      2. pow2 93.2

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

      3. metadata-eval 93.2

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

      4. pow2 93.2

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

      5. lift-*.f64 N/A

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

   9. Applied rewrites 93.1%

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

Alternative 5: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-217}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{\mathsf{fma}\left(\ell, \ell, 1 \cdot \left(\ell \cdot \ell\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

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-217)
    (/ (* (sqrt 2.0) t_m) (sqrt (/ (fma l l (* 1.0 (* 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-217) {
		tmp = (sqrt(2.0) * t_m) / sqrt((fma(l, l, (1.0 * (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)
	tmp = 0.0
	if (t_m <= 3.9e-217)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(fma(l, l, Float64(1.0 * 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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.9e-217], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(l * l + N[(1.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 3.9000000000000001e-217

   1. Initial program 3.5%

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

   3. Taylor expanded in l around 0

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift--.f64 36.4

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in t around 0

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

      1. flip-+ N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower--.f64 N/A

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

   8. Applied rewrites 5.7%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

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

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

       3. pow2 N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 51.5

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

   11. Applied rewrites 51.5%

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

   if 3.9000000000000001e-217 < t

   1. Initial program 37.0%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 81.6

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

   5. Applied rewrites 81.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 81.6

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

   7. Applied rewrites 81.6%

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

      1. flip-+ 81.6

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

      2. pow2 81.6

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

      3. metadata-eval 81.6

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

      4. pow2 81.6

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

      5. lift-*.f64 N/A

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

   9. Applied rewrites 81.6%

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

Alternative 6: 75.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.6 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - 1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

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 8.6e-218)
    (* (sqrt (- (/ x (+ 1.0 x)) 1.0)) 1.0)
    (sqrt (/ (- x 1.0) (+ 1.0 x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 8.6e-218) {
		tmp = sqrt(((x / (1.0 + x)) - 1.0)) * 1.0;
	} 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 <= 8.6d-218) then
        tmp = sqrt(((x / (1.0d0 + x)) - 1.0d0)) * 1.0d0
    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 <= 8.6e-218) {
		tmp = Math.sqrt(((x / (1.0 + x)) - 1.0)) * 1.0;
	} 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 <= 8.6e-218:
		tmp = math.sqrt(((x / (1.0 + x)) - 1.0)) * 1.0
	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 <= 8.6e-218)
		tmp = Float64(sqrt(Float64(Float64(x / Float64(1.0 + x)) - 1.0)) * 1.0);
	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 <= 8.6e-218)
		tmp = sqrt(((x / (1.0 + x)) - 1.0)) * 1.0;
	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, 8.6e-218], N[(N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision] * 1.0), $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 < 8.6e-218

   1. Initial program 3.5%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 36.4

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

   5. Applied rewrites 36.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 36.4

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 23.8%

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

   if 8.6e-218 < t

   1. Initial program 36.9%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 81.6

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

   5. Applied rewrites 81.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 81.6

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

   7. Applied rewrites 81.6%

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

      1. flip-+ 81.6

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

      2. pow2 81.6

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

      3. metadata-eval 81.6

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

      4. pow2 81.6

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

      5. lift-*.f64 N/A

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

   9. Applied rewrites 81.6%

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

Alternative 7: 76.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

3. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 76.7

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

5. Applied rewrites 76.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 76.7

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

7. Applied rewrites 76.7%

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

   1. flip-+ 76.7

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

   2. pow2 76.7

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

   3. metadata-eval 76.7

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

   4. pow2 76.7

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

   5. lift-*.f64 N/A

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

9. Applied rewrites 76.7%

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

Alternative 8: 75.4% accurate, 3.4× speedup

Math

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

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) 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) / 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) / 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) / 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) / 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) / 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) / 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] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

3. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 76.7

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

5. Applied rewrites 76.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. flip-+ N/A

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

   6. pow2 N/A

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

   7. metadata-eval N/A

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

   8. pow2 N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. metadata-eval N/A

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

   14. flip-+ N/A

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

   15. lower-+.f64 36.8

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

7. Applied rewrites 36.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 35.6%

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

    1. lift-*.f64 N/A

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

    2. *-rgt-identity 35.6

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

    3. lift-+.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift--.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. pow2 N/A

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

    8. pow2 N/A

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

    9. metadata-eval N/A

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

    10. flip-- N/A

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

    11. lift--.f64 75.4

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

12. Applied rewrites 75.4%

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

Alternative 9: 75.4% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

3. Taylor expanded in x around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   3. metadata-eval 75.4

      \[\leadsto 1 \]

5. Applied rewrites 75.4%

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

Reproduce

herbie shell --seed 2025091 
(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)))))