Toniolo and Linder, Equation (7)

Percentage Accurate: 33.1% → 84.8%

Time: 7.3s

Alternatives: 10

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.1% 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: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\frac{t\_2 - -2 \cdot \left(t\_m \cdot t\_m\right)}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{t\_2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l l))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 7.5e-159)
      (/
       t_3
       (fma (/ (- t_2 (* -2.0 (* t_m t_m))) (* (* (sqrt 2.0) x) t_m)) 0.5 t_3))
      (if (<= t_m 2.5e+82)
        (/
         t_3
         (sqrt
          (+
           (fma (/ (* t_m t_m) x) 2.0 (fma (* t_m t_m) 2.0 (/ (* l l) x)))
           (/ t_2 x))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 7.5e-159) {
		tmp = t_3 / fma(((t_2 - (-2.0 * (t_m * t_m))) / ((sqrt(2.0) * x) * t_m)), 0.5, t_3);
	} else if (t_m <= 2.5e+82) {
		tmp = t_3 / sqrt((fma(((t_m * t_m) / x), 2.0, fma((t_m * t_m), 2.0, ((l * l) / x))) + (t_2 / x)));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 7.5e-159)
		tmp = Float64(t_3 / fma(Float64(Float64(t_2 - Float64(-2.0 * Float64(t_m * t_m))) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_3));
	elseif (t_m <= 2.5e+82)
		tmp = Float64(t_3 / 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(t_2 / x))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.5e-159], N[(t$95$3 / N[(N[(N[(t$95$2 - N[(-2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+82], N[(t$95$3 / 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[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.5e-159

   1. Initial program 33.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 18.1

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

   8. Applied rewrites 18.1%

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

   if 7.5e-159 < t < 2.50000000000000008e82

   1. Initial program 63.7%

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

   3. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 91.7%

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

   if 2.50000000000000008e82 < t

   1. Initial program 17.8%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

   7. Applied rewrites 98.3%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) - -2 \cdot \left(t \cdot t\right)}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \mathsf{fma}\left(t \cdot t, 2, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\frac{t\_2 + t\_2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{t\_3}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l l))) (t_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 7.2e-65)
      (/ t_3 (fma (/ (+ t_2 t_2) (* (* (sqrt 2.0) x) t_m)) 0.5 t_3))
      (if (<= t_m 4.8e-24)
        (/ t_3 (sqrt (* 2.0 (* l (/ l x)))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 7.2e-65) {
		tmp = t_3 / fma(((t_2 + t_2) / ((sqrt(2.0) * x) * t_m)), 0.5, t_3);
	} else if (t_m <= 4.8e-24) {
		tmp = t_3 / sqrt((2.0 * (l * (l / x))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 7.2e-65)
		tmp = Float64(t_3 / fma(Float64(Float64(t_2 + t_2) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_3));
	elseif (t_m <= 4.8e-24)
		tmp = Float64(t_3 / sqrt(Float64(2.0 * Float64(l * Float64(l / x)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-65], N[(t$95$3 / N[(N[(N[(t$95$2 + t$95$2), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e-24], N[(t$95$3 / N[Sqrt[N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.1999999999999996e-65

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 24.2%

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

   if 7.1999999999999996e-65 < t < 4.7999999999999996e-24

   1. Initial program 30.0%

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

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

   4. Applied rewrites 30.2%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 69.7

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

   12. Applied rewrites 69.7%

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

   if 4.7999999999999996e-24 < t

   1. Initial program 36.3%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.4

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

   5. Applied rewrites 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

   7. Applied rewrites 95.4%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{t\_2}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 7.2e-65)
      (/ t_2 (fma (/ (* (* l l) 2.0) (* (* (sqrt 2.0) x) t_m)) 0.5 t_2))
      (if (<= t_m 4.8e-24)
        (/ t_2 (sqrt (* 2.0 (* l (/ l x)))))
        (sqrt (/ (- x 1.0) (+ 1.0 x))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 7.2e-65) {
		tmp = t_2 / fma((((l * l) * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else if (t_m <= 4.8e-24) {
		tmp = t_2 / sqrt((2.0 * (l * (l / x))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 7.2e-65)
		tmp = Float64(t_2 / fma(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	elseif (t_m <= 4.8e-24)
		tmp = Float64(t_2 / sqrt(Float64(2.0 * Float64(l * Float64(l / x)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-65], N[(t$95$2 / N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e-24], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.1999999999999996e-65

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 24.2%

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

   6. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 24.3

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

   8. Applied rewrites 24.3%

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

   if 7.1999999999999996e-65 < t < 4.7999999999999996e-24

   1. Initial program 30.0%

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

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

   4. Applied rewrites 30.2%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 69.7

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

   12. Applied rewrites 69.7%

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

   if 4.7999999999999996e-24 < t

   1. Initial program 36.3%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 95.4

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

   5. Applied rewrites 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

   7. Applied rewrites 95.4%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-220)
    (/ (* (sqrt 2.0) t_m) (sqrt (* 2.0 (* l (/ l x)))))
    (sqrt (/ (- x 1.0) (+ 1.0 x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 5.2e-220) {
		tmp = (sqrt(2.0) * t_m) / sqrt((2.0 * (l * (l / x))));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, x, l, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 5.2d-220) then
        tmp = (sqrt(2.0d0) * t_m) / sqrt((2.0d0 * (l * (l / x))))
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 5.2e-220) {
		tmp = (Math.sqrt(2.0) * t_m) / Math.sqrt((2.0 * (l * (l / x))));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 5.2e-220:
		tmp = (math.sqrt(2.0) * t_m) / math.sqrt((2.0 * (l * (l / x))))
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 5.2e-220)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(2.0 * Float64(l * Float64(l / x)))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 5.2e-220)
		tmp = (sqrt(2.0) * t_m) / sqrt((2.0 * (l * (l / x))));
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-220], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.2e-220

   1. Initial program 36.3%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 25.8%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 3.4

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 19.6

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

   10. Applied rewrites 19.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 25.1

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

   12. Applied rewrites 25.1%

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

   if 5.2e-220 < t

   1. Initial program 36.2%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

   7. Applied rewrites 86.1%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{\ell \cdot \ell}{x} \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-220)
    (* (sqrt 2.0) (/ t_m (sqrt (* (/ (* l l) x) 2.0))))
    (sqrt (/ (- x 1.0) (+ 1.0 x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 5.2e-220) {
		tmp = sqrt(2.0) * (t_m / sqrt((((l * l) / x) * 2.0)));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, x, l, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 5.2d-220) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((l * l) / x) * 2.0d0)))
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 5.2e-220) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((l * l) / x) * 2.0)));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 5.2e-220:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((l * l) / x) * 2.0)))
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 5.2e-220)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(l * l) / x) * 2.0))));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 5.2e-220)
		tmp = sqrt(2.0) * (t_m / sqrt((((l * l) / x) * 2.0)));
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-220], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.2e-220

   1. Initial program 36.3%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 25.8%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 3.4

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 19.6

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

   10. Applied rewrites 19.6%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

   12. Applied rewrites 19.7%

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

   if 5.2e-220 < t

   1. Initial program 36.2%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

   7. Applied rewrites 86.1%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e-222)
    (sqrt (- (/ x (+ 1.0 x)) 1.0))
    (sqrt (/ (- x 1.0) (+ 1.0 x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 7.6e-222) {
		tmp = sqrt(((x / (1.0 + x)) - 1.0));
	} else {
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, x, l, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 7.6d-222) then
        tmp = sqrt(((x / (1.0d0 + x)) - 1.0d0))
    else
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 7.6e-222) {
		tmp = Math.sqrt(((x / (1.0 + x)) - 1.0));
	} else {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 7.6e-222:
		tmp = math.sqrt(((x / (1.0 + x)) - 1.0))
	else:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 7.6e-222)
		tmp = sqrt(Float64(Float64(x / Float64(1.0 + x)) - 1.0));
	else
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 7.6e-222)
		tmp = sqrt(((x / (1.0 + x)) - 1.0));
	else
		tmp = sqrt(((x - 1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e-222], N[Sqrt[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.59999999999999993e-222

   1. Initial program 36.3%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 2.9

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 11.7%

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

   if 7.59999999999999993e-222 < t

   1. Initial program 36.2%

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

   3. Taylor expanded in l around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-+.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-rgt-identity N/A

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

   7. Applied rewrites 86.1%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\frac{x}{1 + x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 41.6

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

5. Applied rewrites 41.6%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-rgt-identity N/A

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

7. Applied rewrites 41.6%

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

8. Final simplification 41.6%

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

Alternative 8: 75.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 41.6

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

5. Applied rewrites 41.6%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-rgt-identity N/A

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

7. Applied rewrites 41.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 41.2%

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

11. Final simplification 41.2%

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

Alternative 9: 75.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in l around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-+.f64 41.6

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

5. Applied rewrites 41.6%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-rgt-identity N/A

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

7. Applied rewrites 41.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 41.2%

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

11. Final simplification 41.2%

    \[\leadsto \sqrt{\frac{x}{1 + x}} \]
12. Add Preprocessing

Alternative 10: 75.4% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in x around inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   3. metadata-eval 41.2

      \[\leadsto 1 \]

5. Applied rewrites 41.2%

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

Reproduce

herbie shell --seed 2025071 
(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)))))