Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 84.6%

Time: 7.7s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.6% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := t\_2 \cdot -1\\ t_4 := t\_2 - t\_3\\ t_5 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{t\_5}{\mathsf{fma}\left(\frac{t\_4}{\left({2}^{0.5} \cdot x\right) \cdot t\_m}, 0.5, {2}^{0.5} \cdot t\_m\right)}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{t\_5}{\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_4, -1, \frac{t\_3}{x}\right) - \frac{t\_2}{x}}{x}, -1, \left(t\_m \cdot t\_m\right) \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\\ \end{array} \end{array} \end{array} \]

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 (fma (* t_m t_m) 2.0 (* l l)))
        (t_3 (* t_2 -1.0))
        (t_4 (- t_2 t_3))
        (t_5 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.8e-155)
      (/
       t_5
       (fma (/ t_4 (* (* (pow 2.0 0.5) x) t_m)) 0.5 (* (pow 2.0 0.5) t_m)))
      (if (<= t_m 4.6e+17)
        (/
         t_5
         (sqrt
          (fma
           (/ (- (fma t_4 -1.0 (/ t_3 x)) (/ t_2 x)) x)
           -1.0
           (* (* t_m t_m) 2.0))))
        (- (+ 1.0 (/ 0.5 (pow x 2.0))) (/ 1.0 x)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = t_2 * -1.0;
	double t_4 = t_2 - t_3;
	double t_5 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.8e-155) {
		tmp = t_5 / fma((t_4 / ((pow(2.0, 0.5) * x) * t_m)), 0.5, (pow(2.0, 0.5) * t_m));
	} else if (t_m <= 4.6e+17) {
		tmp = t_5 / sqrt(fma(((fma(t_4, -1.0, (t_3 / x)) - (t_2 / x)) / x), -1.0, ((t_m * t_m) * 2.0)));
	} else {
		tmp = (1.0 + (0.5 / pow(x, 2.0))) - (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 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(t_2 * -1.0)
	t_4 = Float64(t_2 - t_3)
	t_5 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 2.8e-155)
		tmp = Float64(t_5 / fma(Float64(t_4 / Float64(Float64((2.0 ^ 0.5) * x) * t_m)), 0.5, Float64((2.0 ^ 0.5) * t_m)));
	elseif (t_m <= 4.6e+17)
		tmp = Float64(t_5 / sqrt(fma(Float64(Float64(fma(t_4, -1.0, Float64(t_3 / x)) - Float64(t_2 / x)) / x), -1.0, Float64(Float64(t_m * t_m) * 2.0))));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 / (x ^ 2.0))) - 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[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.8e-155], N[(t$95$5 / N[(N[(t$95$4 / N[(N[(N[Power[2.0, 0.5], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Power[2.0, 0.5], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e+17], N[(t$95$5 / N[Sqrt[N[(N[(N[(N[(t$95$4 * -1.0 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.8e-155

   1. Initial program 24.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 17.0%

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

   if 2.8e-155 < t < 4.6e17

   1. Initial program 58.8%

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

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 85.9%

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

   if 4.6e17 < t

   1. Initial program 32.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-+.f64 41.2

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

   5. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\frac{{\left(x - 1\right)}^{0.5}}{{\left(1 + x\right)}^{0.5}} \cdot 1} \]

   6. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 91.2

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

   8. Applied rewrites 91.2%

      \[\leadsto \left(\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) - \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -1}{\left({2}^{0.5} \cdot x\right) \cdot t}, 0.5, {2}^{0.5} \cdot t\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) - \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -1, -1, \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -1}{x}\right) - \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}{x}, -1, \left(t \cdot t\right) \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\mathsf{fma}\left(\frac{t\_2 - t\_2 \cdot -1}{\left({2}^{0.5} \cdot x\right) \cdot t\_m}, 0.5, {2}^{0.5} \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\\ \end{array} \end{array} \end{array} \]

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 (fma (* t_m t_m) 2.0 (* l l))))
   (*
    t_s
    (if (<= t_m 5.2e+22)
      (/
       (* (sqrt 2.0) t_m)
       (fma
        (/ (- t_2 (* t_2 -1.0)) (* (* (pow 2.0 0.5) x) t_m))
        0.5
        (* (pow 2.0 0.5) t_m)))
      (- (+ 1.0 (/ 0.5 (pow x 2.0))) (/ 1.0 x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double tmp;
	if (t_m <= 5.2e+22) {
		tmp = (sqrt(2.0) * t_m) / fma(((t_2 - (t_2 * -1.0)) / ((pow(2.0, 0.5) * x) * t_m)), 0.5, (pow(2.0, 0.5) * t_m));
	} else {
		tmp = (1.0 + (0.5 / pow(x, 2.0))) - (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 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	tmp = 0.0
	if (t_m <= 5.2e+22)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / fma(Float64(Float64(t_2 - Float64(t_2 * -1.0)) / Float64(Float64((2.0 ^ 0.5) * x) * t_m)), 0.5, Float64((2.0 ^ 0.5) * t_m)));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 / (x ^ 2.0))) - 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[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.2e+22], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[(t$95$2 - N[(t$95$2 * -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[2.0, 0.5], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Power[2.0, 0.5], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.2e22

   1. Initial program 29.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 25.5%

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

   if 5.2e22 < t

   1. Initial program 32.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-+.f64 41.2

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

   5. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\frac{{\left(x - 1\right)}^{0.5}}{{\left(1 + x\right)}^{0.5}} \cdot 1} \]

   6. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 91.2

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

   8. Applied rewrites 91.2%

      \[\leadsto \left(\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) - \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot -1}{\left({2}^{0.5} \cdot x\right) \cdot t}, 0.5, {2}^{0.5} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, {t\_m}^{2}, {\ell}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\frac{\mathsf{fma}\left(0.5, \frac{t\_2 - -1 \cdot t\_2}{t\_m \cdot \sqrt{2}}, t\_m \cdot \left(x \cdot \sqrt{2}\right)\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\\ \end{array} \end{array} \end{array} \]

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 (fma 2.0 (pow t_m 2.0) (pow l 2.0))))
   (*
    t_s
    (if (<= t_m 2.9e-53)
      (/
       (* (sqrt 2.0) t_m)
       (/
        (fma
         0.5
         (/ (- t_2 (* -1.0 t_2)) (* t_m (sqrt 2.0)))
         (* t_m (* x (sqrt 2.0))))
        x))
      (- (+ 1.0 (/ 0.5 (pow x 2.0))) (/ 1.0 x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma(2.0, pow(t_m, 2.0), pow(l, 2.0));
	double tmp;
	if (t_m <= 2.9e-53) {
		tmp = (sqrt(2.0) * t_m) / (fma(0.5, ((t_2 - (-1.0 * t_2)) / (t_m * sqrt(2.0))), (t_m * (x * sqrt(2.0)))) / x);
	} else {
		tmp = (1.0 + (0.5 / pow(x, 2.0))) - (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 = fma(2.0, (t_m ^ 2.0), (l ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.9e-53)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(fma(0.5, Float64(Float64(t_2 - Float64(-1.0 * t_2)) / Float64(t_m * sqrt(2.0))), Float64(t_m * Float64(x * sqrt(2.0)))) / x));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 / (x ^ 2.0))) - 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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-53], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(0.5 * N[(N[(t$95$2 - N[(-1.0 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8999999999999998e-53

   1. Initial program 26.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 21.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 21.7%

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

   if 2.8999999999999998e-53 < t

   1. Initial program 39.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-+.f64 41.8

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

   5. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{{\left(x - 1\right)}^{0.5}}{{\left(1 + x\right)}^{0.5}} \cdot 1} \]

   6. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 91.2

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

   8. Applied rewrites 91.2%

      \[\leadsto \left(\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right) - -1 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \sqrt{2}}, t \cdot \left(x \cdot \sqrt{2}\right)\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {4}^{0.25}\\ t_3 := \frac{t\_2}{{0.5}^{0.5}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+186}:\\ \;\;\;\;\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.25, t\_3 \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0625, t\_3 \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t\_2}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t\_m \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)}{\ell}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (pow 4.0 0.25))) (t_3 (/ t_2 (pow 0.5 0.5))))
   (*
    t_s
    (if (<= l 1.3e+186)
      (- (+ 1.0 (/ 0.5 (pow x 2.0))) (/ 1.0 x))
      (/
       (*
        x
        (fma
         -0.25
         (* t_3 (pow (pow (* (* x x) x) -1.0) 0.5))
         (fma
          -0.0625
          (* t_3 (pow (pow (pow x 5.0) -1.0) 0.5))
          (fma
           -0.0078125
           (* (/ t_2 (pow (pow 0.5 0.5) 5.0)) (pow (pow (pow x 7.0) -1.0) 0.5))
           (* t_m (pow (pow x -1.0) 0.5))))))
       l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * pow(4.0, 0.25);
	double t_3 = t_2 / pow(0.5, 0.5);
	double tmp;
	if (l <= 1.3e+186) {
		tmp = (1.0 + (0.5 / pow(x, 2.0))) - (1.0 / x);
	} else {
		tmp = (x * fma(-0.25, (t_3 * pow(pow(((x * x) * x), -1.0), 0.5)), fma(-0.0625, (t_3 * pow(pow(pow(x, 5.0), -1.0), 0.5)), fma(-0.0078125, ((t_2 / pow(pow(0.5, 0.5), 5.0)) * pow(pow(pow(x, 7.0), -1.0), 0.5)), (t_m * pow(pow(x, -1.0), 0.5)))))) / l;
	}
	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(t_m * (4.0 ^ 0.25))
	t_3 = Float64(t_2 / (0.5 ^ 0.5))
	tmp = 0.0
	if (l <= 1.3e+186)
		tmp = Float64(Float64(1.0 + Float64(0.5 / (x ^ 2.0))) - Float64(1.0 / x));
	else
		tmp = Float64(Float64(x * fma(-0.25, Float64(t_3 * ((Float64(Float64(x * x) * x) ^ -1.0) ^ 0.5)), fma(-0.0625, Float64(t_3 * (((x ^ 5.0) ^ -1.0) ^ 0.5)), fma(-0.0078125, Float64(Float64(t_2 / ((0.5 ^ 0.5) ^ 5.0)) * (((x ^ 7.0) ^ -1.0) ^ 0.5)), Float64(t_m * ((x ^ -1.0) ^ 0.5)))))) / l);
	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[(t$95$m * N[Power[4.0, 0.25], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Power[0.5, 0.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 1.3e+186], N[(N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-0.25 * N[(t$95$3 * N[Power[N[Power[N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(t$95$3 * N[Power[N[Power[N[Power[x, 5.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.0078125 * N[(N[(t$95$2 / N[Power[N[Power[0.5, 0.5], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 7.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Power[N[Power[x, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e186

   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. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-+.f64 19.2

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

   5. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\frac{{\left(x - 1\right)}^{0.5}}{{\left(1 + x\right)}^{0.5}} \cdot 1} \]

   6. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 39.7

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

   8. Applied rewrites 39.7%

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

   if 1.3e186 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 25.6%

      \[\leadsto \color{blue}{{\left({\left(x - 1\right)}^{-1} + \left(\frac{x}{x - 1} - 1\right)\right)}^{-0.5} \cdot \frac{{2}^{0.5} \cdot t}{\ell}} \]

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-sqrt.f64 52.6

         \[\leadsto \frac{t \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{x} \]

   8. Applied rewrites 52.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{32} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{3}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + \frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

   11. Applied rewrites 42.5%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-0.25, \frac{t \cdot {2}^{0.5}}{\ell \cdot {0.5}^{0.5}} \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.03125, \frac{t \cdot {2}^{0.5}}{\ell \cdot \left({0.5}^{0.5} \cdot 0.5\right)} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{\ell \cdot {\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, \frac{t \cdot 1}{\ell} \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)} \]

   12. Taylor expanded in l around 0

       \[\leadsto \frac{x \cdot \left(\frac{-1}{4} \cdot \left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{16} \cdot \left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{{\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + t \cdot \sqrt{\frac{1}{x}}\right)\right)\right)}{\ell} \]

   14. Applied rewrites 61.1%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(-0.25, \frac{t \cdot {4}^{0.25}}{{0.5}^{0.5}} \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0625, \frac{t \cdot {4}^{0.25}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {4}^{0.25}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+186}:\\ \;\;\;\;\left(1 + \frac{0.5}{{x}^{2}}\right) - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.25, \frac{t \cdot {4}^{0.25}}{{0.5}^{0.5}} \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0625, \frac{t \cdot {4}^{0.25}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {4}^{0.25}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 13.9% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{t\_m \cdot \left(\sqrt{0.5} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)}{\ell} \cdot \sqrt{x}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (* (/ (* t_m (* (sqrt 0.5) (* (pow 2.0 0.25) (pow 2.0 0.25)))) l) (sqrt 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 * (((t_m * (sqrt(0.5) * (pow(2.0, 0.25) * pow(2.0, 0.25)))) / l) * sqrt(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 * (((t_m * (sqrt(0.5d0) * ((2.0d0 ** 0.25d0) * (2.0d0 ** 0.25d0)))) / l) * sqrt(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 * (((t_m * (Math.sqrt(0.5) * (Math.pow(2.0, 0.25) * Math.pow(2.0, 0.25)))) / l) * Math.sqrt(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 * (((t_m * (math.sqrt(0.5) * (math.pow(2.0, 0.25) * math.pow(2.0, 0.25)))) / l) * math.sqrt(x))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(Float64(t_m * Float64(sqrt(0.5) * Float64((2.0 ^ 0.25) * (2.0 ^ 0.25)))) / l) * sqrt(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 * (((t_m * (sqrt(0.5) * ((2.0 ^ 0.25) * (2.0 ^ 0.25)))) / l) * sqrt(x));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * N[(N[(N[(t$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in l around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 9.0%

   \[\leadsto \color{blue}{{\left({\left(x - 1\right)}^{-1} + \left(\frac{x}{x - 1} - 1\right)\right)}^{-0.5} \cdot \frac{{2}^{0.5} \cdot t}{\ell}} \]

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lower-sqrt.f64 15.5

      \[\leadsto \frac{t \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{x} \]

8. Applied rewrites 15.5%

   \[\leadsto \frac{t \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{\ell} \cdot \color{blue}{\sqrt{x}} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-pow.f64 15.6

      \[\leadsto \frac{t \cdot \left(\sqrt{0.5} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)}{\ell} \cdot \sqrt{x} \]

10. Applied rewrites 15.6%

    \[\leadsto \frac{t \cdot \left(\sqrt{0.5} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)}{\ell} \cdot \sqrt{x} \]
11. Add Preprocessing

Alternative 6: 13.8% accurate, N/A× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (* (/ (* t_m (* (sqrt 0.5) (sqrt 2.0))) l) (sqrt 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 * (((t_m * (sqrt(0.5) * sqrt(2.0))) / l) * sqrt(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 * (((t_m * (sqrt(0.5d0) * sqrt(2.0d0))) / l) * sqrt(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 * (((t_m * (Math.sqrt(0.5) * Math.sqrt(2.0))) / l) * Math.sqrt(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 * (((t_m * (math.sqrt(0.5) * math.sqrt(2.0))) / l) * math.sqrt(x))

Julia

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

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * N[(N[(N[(t$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in l around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 9.0%

   \[\leadsto \color{blue}{{\left({\left(x - 1\right)}^{-1} + \left(\frac{x}{x - 1} - 1\right)\right)}^{-0.5} \cdot \frac{{2}^{0.5} \cdot t}{\ell}} \]

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lower-sqrt.f64 15.5

      \[\leadsto \frac{t \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{x} \]

8. Applied rewrites 15.5%

   \[\leadsto \frac{t \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{\ell} \cdot \color{blue}{\sqrt{x}} \]
9. Add Preprocessing

Alternative 7: 13.6% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {4}^{0.25}\\ t_3 := \frac{t\_2}{{0.5}^{0.5}}\\ t\_s \cdot \frac{x \cdot \mathsf{fma}\left(-0.25, t\_3 \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0625, t\_3 \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t\_2}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t\_m \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)}{\ell} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (pow 4.0 0.25))) (t_3 (/ t_2 (pow 0.5 0.5))))
   (*
    t_s
    (/
     (*
      x
      (fma
       -0.25
       (* t_3 (pow (pow (* (* x x) x) -1.0) 0.5))
       (fma
        -0.0625
        (* t_3 (pow (pow (pow x 5.0) -1.0) 0.5))
        (fma
         -0.0078125
         (* (/ t_2 (pow (pow 0.5 0.5) 5.0)) (pow (pow (pow x 7.0) -1.0) 0.5))
         (* t_m (pow (pow x -1.0) 0.5))))))
     l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * pow(4.0, 0.25);
	double t_3 = t_2 / pow(0.5, 0.5);
	return t_s * ((x * fma(-0.25, (t_3 * pow(pow(((x * x) * x), -1.0), 0.5)), fma(-0.0625, (t_3 * pow(pow(pow(x, 5.0), -1.0), 0.5)), fma(-0.0078125, ((t_2 / pow(pow(0.5, 0.5), 5.0)) * pow(pow(pow(x, 7.0), -1.0), 0.5)), (t_m * pow(pow(x, -1.0), 0.5)))))) / l);
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * (4.0 ^ 0.25))
	t_3 = Float64(t_2 / (0.5 ^ 0.5))
	return Float64(t_s * Float64(Float64(x * fma(-0.25, Float64(t_3 * ((Float64(Float64(x * x) * x) ^ -1.0) ^ 0.5)), fma(-0.0625, Float64(t_3 * (((x ^ 5.0) ^ -1.0) ^ 0.5)), fma(-0.0078125, Float64(Float64(t_2 / ((0.5 ^ 0.5) ^ 5.0)) * (((x ^ 7.0) ^ -1.0) ^ 0.5)), Float64(t_m * ((x ^ -1.0) ^ 0.5)))))) / l))
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[(t$95$m * N[Power[4.0, 0.25], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Power[0.5, 0.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * N[(N[(x * N[(-0.25 * N[(t$95$3 * N[Power[N[Power[N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(t$95$3 * N[Power[N[Power[N[Power[x, 5.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.0078125 * N[(N[(t$95$2 / N[Power[N[Power[0.5, 0.5], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Power[x, 7.0], $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Power[N[Power[x, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in l around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 9.0%

   \[\leadsto \color{blue}{{\left({\left(x - 1\right)}^{-1} + \left(\frac{x}{x - 1} - 1\right)\right)}^{-0.5} \cdot \frac{{2}^{0.5} \cdot t}{\ell}} \]

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lower-sqrt.f64 15.5

      \[\leadsto \frac{t \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{x} \]

8. Applied rewrites 15.5%

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

9. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{32} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{3}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + \frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)} \]

11. Applied rewrites 13.0%

    \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-0.25, \frac{t \cdot {2}^{0.5}}{\ell \cdot {0.5}^{0.5}} \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.03125, \frac{t \cdot {2}^{0.5}}{\ell \cdot \left({0.5}^{0.5} \cdot 0.5\right)} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {2}^{0.5}}{\ell \cdot {\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, \frac{t \cdot 1}{\ell} \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)} \]

12. Taylor expanded in l around 0

    \[\leadsto \frac{x \cdot \left(\frac{-1}{4} \cdot \left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + \left(\frac{-1}{16} \cdot \left(\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{{x}^{5}}}\right) + \left(\frac{-1}{128} \cdot \left(\frac{t \cdot \sqrt{2}}{{\left(\sqrt{\frac{1}{2}}\right)}^{5}} \cdot \sqrt{\frac{1}{{x}^{7}}}\right) + t \cdot \sqrt{\frac{1}{x}}\right)\right)\right)}{\ell} \]

14. Applied rewrites 14.3%

    \[\leadsto \frac{x \cdot \mathsf{fma}\left(-0.25, \frac{t \cdot {4}^{0.25}}{{0.5}^{0.5}} \cdot {\left({\left(\left(x \cdot x\right) \cdot x\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0625, \frac{t \cdot {4}^{0.25}}{{0.5}^{0.5}} \cdot {\left({\left({x}^{5}\right)}^{-1}\right)}^{0.5}, \mathsf{fma}\left(-0.0078125, \frac{t \cdot {4}^{0.25}}{{\left({0.5}^{0.5}\right)}^{5}} \cdot {\left({\left({x}^{7}\right)}^{-1}\right)}^{0.5}, t \cdot {\left({x}^{-1}\right)}^{0.5}\right)\right)\right)}{\ell} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))))