Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 84.1%

Time: 6.7s

Alternatives: 8

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: 33.7% 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: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := -t\_2\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-153}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(0.5, \frac{\left(\ell \cdot \ell\right) \cdot 2}{t\_m \cdot \left(x \cdot \sqrt{2}\right)}, t\_m \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(2 \cdot t\_m, t\_m, -\frac{\mathsf{fma}\left(t\_2 - t\_3, -1, \frac{t\_3}{x}\right) - \frac{t\_2}{x}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - \frac{0.5}{x}}{x}, -1, 1\right)\\ \end{array} \end{array} \end{array} \]

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))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 3e-153)
      (/
       t_4
       (fma
        0.5
        (/ (* (* l l) 2.0) (* t_m (* x (sqrt 2.0))))
        (* t_m (sqrt 2.0))))
      (if (<= t_m 1.6e-10)
        (/
         t_4
         (sqrt
          (fma
           (* 2.0 t_m)
           t_m
           (- (/ (- (fma (- t_2 t_3) -1.0 (/ t_3 x)) (/ t_2 x)) x)))))
        (fma (/ (- 1.0 (/ 0.5 x)) x) -1.0 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = -t_2;
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3e-153) {
		tmp = t_4 / fma(0.5, (((l * l) * 2.0) / (t_m * (x * sqrt(2.0)))), (t_m * sqrt(2.0)));
	} else if (t_m <= 1.6e-10) {
		tmp = t_4 / sqrt(fma((2.0 * t_m), t_m, -((fma((t_2 - t_3), -1.0, (t_3 / x)) - (t_2 / x)) / x)));
	} else {
		tmp = fma(((1.0 - (0.5 / x)) / x), -1.0, 1.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(-t_2)
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 3e-153)
		tmp = Float64(t_4 / fma(0.5, Float64(Float64(Float64(l * l) * 2.0) / Float64(t_m * Float64(x * sqrt(2.0)))), Float64(t_m * sqrt(2.0))));
	elseif (t_m <= 1.6e-10)
		tmp = Float64(t_4 / sqrt(fma(Float64(2.0 * t_m), t_m, Float64(-Float64(Float64(fma(Float64(t_2 - t_3), -1.0, Float64(t_3 / x)) - Float64(t_2 / x)) / x)))));
	else
		tmp = fma(Float64(Float64(1.0 - Float64(0.5 / x)) / x), -1.0, 1.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3e-153], N[(t$95$4 / N[(0.5 * N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e-10], N[(t$95$4 / N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m + (-N[(N[(N[(N[(t$95$2 - t$95$3), $MachinePrecision] * -1.0 + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3e-153

   1. Initial program 4.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 6.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   6. Applied rewrites 59.5%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 59.5

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

   9. Applied rewrites 59.5%

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

   if 3e-153 < t < 1.5999999999999999e-10

   1. Initial program 52.5%

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

   2. Taylor expanded in x around -inf

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

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

   4. Applied rewrites 85.5%

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

   if 1.5999999999999999e-10 < t

   1. Initial program 36.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 19.7%

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

   4. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-+.f64 92.4

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

   6. Applied rewrites 92.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. pow2 N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 45.6

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

   8. Applied rewrites 45.6%

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

   9. Taylor expanded in x around -inf

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

       1. sqrt-div N/A

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

       2. pow2 N/A

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

       3. sub-div N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. sqrt-div N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

   11. Applied rewrites 91.8%

       \[\leadsto \mathsf{fma}\left(\frac{1 - \frac{0.5}{x}}{x}, \color{blue}{-1}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-153}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{\left(\ell \cdot \ell\right) \cdot 2}{t\_m \cdot \left(x \cdot \sqrt{2}\right)}, t\_m \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - \frac{0.5}{x}}{x}, -1, 1\right)\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= t_m 3e-153)
      (/
       t_2
       (fma
        0.5
        (/ (* (* l l) 2.0) (* t_m (* x (sqrt 2.0))))
        (* t_m (sqrt 2.0))))
      (if (<= t_m 1.6e-10)
        (/
         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))))
        (fma (/ (- 1.0 (/ 0.5 x)) x) -1.0 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 3e-153) {
		tmp = t_2 / fma(0.5, (((l * l) * 2.0) / (t_m * (x * sqrt(2.0)))), (t_m * sqrt(2.0)));
	} else if (t_m <= 1.6e-10) {
		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 = fma(((1.0 - (0.5 / x)) / x), -1.0, 1.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 3e-153)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(Float64(l * l) * 2.0) / Float64(t_m * Float64(x * sqrt(2.0)))), Float64(t_m * sqrt(2.0))));
	elseif (t_m <= 1.6e-10)
		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 = fma(Float64(Float64(1.0 - Float64(0.5 / x)) / x), -1.0, 1.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3e-153], N[(t$95$2 / N[(0.5 * N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e-10], 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[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3e-153

   1. Initial program 4.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 6.6%

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

   4. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   6. Applied rewrites 59.5%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 59.5

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

   9. Applied rewrites 59.5%

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

   if 3e-153 < t < 1.5999999999999999e-10

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 85.1%

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

   if 1.5999999999999999e-10 < t

   1. Initial program 36.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 19.7%

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

   4. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-+.f64 92.4

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

   6. Applied rewrites 92.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. pow2 N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 45.6

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

   8. Applied rewrites 45.6%

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

   9. Taylor expanded in x around -inf

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

       1. sqrt-div N/A

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

       2. pow2 N/A

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

       3. sub-div N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. sqrt-div N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

   11. Applied rewrites 91.8%

       \[\leadsto \mathsf{fma}\left(\frac{1 - \frac{0.5}{x}}{x}, \color{blue}{-1}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\mathsf{fma}\left(0.5, \frac{\left(\ell \cdot \ell\right) \cdot 2}{t\_m \cdot \left(x \cdot \sqrt{2}\right)}, t\_m \cdot \sqrt{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - \frac{0.5}{x}}{x}, -1, 1\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-28)
    (/
     (* (sqrt 2.0) t_m)
     (fma 0.5 (/ (* (* l l) 2.0) (* t_m (* x (sqrt 2.0)))) (* t_m (sqrt 2.0))))
    (fma (/ (- 1.0 (/ 0.5 x)) x) -1.0 1.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.95e-28) {
		tmp = (sqrt(2.0) * t_m) / fma(0.5, (((l * l) * 2.0) / (t_m * (x * sqrt(2.0)))), (t_m * sqrt(2.0)));
	} else {
		tmp = fma(((1.0 - (0.5 / x)) / x), -1.0, 1.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.95e-28)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / fma(0.5, Float64(Float64(Float64(l * l) * 2.0) / Float64(t_m * Float64(x * sqrt(2.0)))), Float64(t_m * sqrt(2.0))));
	else
		tmp = fma(Float64(Float64(1.0 - Float64(0.5 / x)) / x), -1.0, 1.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.95e-28], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(0.5 * N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.94999999999999999e-28

   1. Initial program 26.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 27.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   6. Applied rewrites 64.1%

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

   7. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.9

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

   9. Applied rewrites 63.9%

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

   if 1.94999999999999999e-28 < t

   1. Initial program 38.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 22.0%

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

   4. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-+.f64 91.6

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

   6. Applied rewrites 91.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. pow2 N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 45.4

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

   8. Applied rewrites 45.4%

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

   9. Taylor expanded in x around -inf

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

       1. sqrt-div N/A

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

       2. pow2 N/A

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

       3. sub-div N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. sqrt-div N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

   11. Applied rewrites 91.0%

       \[\leadsto \mathsf{fma}\left(\frac{1 - \frac{0.5}{x}}{x}, \color{blue}{-1}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{2}{x}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\frac{x}{1 + x} - \frac{1}{1 + x}}\\ \end{array} \end{array} \]

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 1.6e-198)
    (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 2.0 x)) l))
    (* 1.0 (sqrt (- (/ x (+ 1.0 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 <= 1.6e-198) {
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l);
	} else {
		tmp = 1.0 * sqrt(((x / (1.0 + 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 <= 1.6d-198) then
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((2.0d0 / x)) * l)
    else
        tmp = 1.0d0 * sqrt(((x / (1.0d0 + 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 <= 1.6e-198) {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((2.0 / x)) * l);
	} else {
		tmp = 1.0 * Math.sqrt(((x / (1.0 + 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 <= 1.6e-198:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((2.0 / x)) * l)
	else:
		tmp = 1.0 * math.sqrt(((x / (1.0 + 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 <= 1.6e-198)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(2.0 / x)) * l));
	else
		tmp = Float64(1.0 * sqrt(Float64(Float64(x / Float64(1.0 + x)) - Float64(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 <= 1.6e-198)
		tmp = (sqrt(2.0) * t_m) / (sqrt((2.0 / x)) * l);
	else
		tmp = 1.0 * sqrt(((x / (1.0 + 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, 1.6e-198], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.59999999999999997e-198

   1. Initial program 3.7%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift--.f64 3.7

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

   4. Applied rewrites 3.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 45.6

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

   7. Applied rewrites 45.6%

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

   if 1.59999999999999997e-198 < t

   1. Initial program 38.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 26.9%

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

   4. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-+.f64 83.3

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

   6. Applied rewrites 83.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 83.3

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

   8. Applied rewrites 83.3%

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

Alternative 5: 77.0% accurate, 1.8× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (* 1.0 (sqrt (- (/ x (+ 1.0 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 * (1.0 * sqrt(((x / (1.0 + 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 * (1.0d0 * sqrt(((x / (1.0d0 + 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 * (1.0 * Math.sqrt(((x / (1.0 + 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 * (1.0 * math.sqrt(((x / (1.0 + 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 * Float64(1.0 * sqrt(Float64(Float64(x / Float64(1.0 + x)) - Float64(1.0 / Float64(1.0 + x))))))
end

MATLAB

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

Derivation

1. Initial program 33.7%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 24.1%

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

4. Taylor expanded in l around 0

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
5. 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. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-+.f64 77.0

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

6. Applied rewrites 77.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 77.0

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

8. Applied rewrites 77.0%

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

Alternative 6: 76.5% accurate, 2.7× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (fma (/ (- 1.0 (/ 0.5 x)) x) -1.0 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 * fma(((1.0 - (0.5 / x)) / x), -1.0, 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 * fma(Float64(Float64(1.0 - Float64(0.5 / x)) / x), -1.0, 1.0))
end

Wolfram

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

Derivation

1. Initial program 33.7%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 24.1%

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

4. Taylor expanded in l around 0

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
5. 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. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-+.f64 77.0

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

6. Applied rewrites 77.0%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. div-sub N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 36.8

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

8. Applied rewrites 36.8%

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

9. Taylor expanded in x around -inf

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

    1. sqrt-div N/A

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

    2. pow2 N/A

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

    3. sub-div N/A

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

    4. pow2 N/A

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

    5. metadata-eval N/A

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

    6. flip-- N/A

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

    7. sqrt-div N/A

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

    8. +-commutative N/A

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

    9. *-commutative N/A

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

    10. lower-fma.f64 N/A

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

11. Applied rewrites 76.5%

    \[\leadsto \mathsf{fma}\left(\frac{1 - \frac{0.5}{x}}{x}, \color{blue}{-1}, 1\right) \]
12. Add Preprocessing

Alternative 7: 77.0% accurate, 3.0× speedup

Math

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

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 (sqrt (/ (- x 1.0) (+ x 1.0)))))

C

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

Fortran

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

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

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

Java

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

Python

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

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(x + 1.0))))
end

MATLAB

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

Derivation

1. Initial program 33.7%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 24.1%

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

4. Taylor expanded in l around 0

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
5. 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. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-+.f64 77.0

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

6. Applied rewrites 77.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-sub N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 77.0

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

8. Applied rewrites 77.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lower-sqrt.f64 N/A

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

   10. sub-div N/A

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

   11. lower-/.f64 N/A

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

10. Applied rewrites 77.0%

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

Alternative 8: 75.6% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.7%

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

2. Taylor expanded in x around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   3. metadata-eval 75.6

      \[\leadsto 1 \]

4. Applied rewrites 75.6%

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

Reproduce

herbie shell --seed 2025096 
(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)))))