logq (problem 3.4.3)

Percentage Accurate: 8.5% → 99.7%

Time: 4.8s

Alternatives: 8

Speedup: 3.9×

Specification

\[\left|\varepsilon\right| < 1\]

Math

\[\begin{array}{l} \\ \log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))

C

double code(double eps) {
	return log(((1.0 - eps) / (1.0 + eps)));
}

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(eps)
use fmin_fmax_functions
    real(8), intent (in) :: eps
    code = log(((1.0d0 - eps) / (1.0d0 + eps)))
end function

Java

public static double code(double eps) {
	return Math.log(((1.0 - eps) / (1.0 + eps)));
}

Python

def code(eps):
	return math.log(((1.0 - eps) / (1.0 + eps)))

Julia

function code(eps)
	return log(Float64(Float64(1.0 - eps) / Float64(1.0 + eps)))
end

MATLAB

function tmp = code(eps)
	tmp = log(((1.0 - eps) / (1.0 + eps)));
end

Wolfram

code[eps_] := N[Log[N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 8.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))

C

double code(double eps) {
	return log(((1.0 - eps) / (1.0 + eps)));
}

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(eps)
use fmin_fmax_functions
    real(8), intent (in) :: eps
    code = log(((1.0d0 - eps) / (1.0d0 + eps)))
end function

Java

public static double code(double eps) {
	return Math.log(((1.0 - eps) / (1.0 + eps)));
}

Python

def code(eps):
	return math.log(((1.0 - eps) / (1.0 + eps)))

Julia

function code(eps)
	return log(Float64(Float64(1.0 - eps) / Float64(1.0 + eps)))
end

MATLAB

function tmp = code(eps)
	tmp = log(((1.0 - eps) / (1.0 + eps)));
end

Wolfram

code[eps_] := N[Log[N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.2857142857142857, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (-
  (*
   (fma
    (*
     (fma
      (fma (* eps eps) -0.2857142857142857 -0.4)
      (* eps eps)
      -0.6666666666666666)
     eps)
    eps
    -1.0)
   eps)
  eps))

C

double code(double eps) {
	return (fma((fma(fma((eps * eps), -0.2857142857142857, -0.4), (eps * eps), -0.6666666666666666) * eps), eps, -1.0) * eps) - eps;
}

Julia

function code(eps)
	return Float64(Float64(fma(Float64(fma(fma(Float64(eps * eps), -0.2857142857142857, -0.4), Float64(eps * eps), -0.6666666666666666) * eps), eps, -1.0) * eps) - eps)
end

Wolfram

code[eps_] := N[(N[(N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -0.2857142857142857 + -0.4), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] * eps), $MachinePrecision] * eps + -1.0), $MachinePrecision] * eps), $MachinePrecision] - eps), $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   5. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   10. lower-pow.f64 99.7

       \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   3. sub-flip N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + -2\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)\right) \cdot \varepsilon + \color{blue}{-2 \cdot \varepsilon} \]

   6. lift-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)\right) \cdot \varepsilon + -2 \cdot \color{blue}{\varepsilon} \]

   7. lower-fma.f64 99.8

      \[\leadsto \mathsf{fma}\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right), \color{blue}{\varepsilon}, -2 \cdot \varepsilon\right) \]

6. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.2857142857142857, \varepsilon \cdot \varepsilon, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon\right) \cdot \varepsilon, \color{blue}{\varepsilon}, -2 \cdot \varepsilon\right) \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon + \color{blue}{-2 \cdot \varepsilon} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon + -2 \cdot \color{blue}{\varepsilon} \]

   3. distribute-rgt-out N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -2\right)} \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + \left(-1 + \color{blue}{-1}\right)\right) \]

   5. associate-+l+ N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -1\right) + \color{blue}{-1}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -1\right) + -1\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) + -1\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon + \color{blue}{-1 \cdot \varepsilon} \]

   9. add-flip N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \varepsilon\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\varepsilon} \]

8. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.2857142857142857, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\varepsilon} \]
9. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.2857142857142857, \varepsilon \cdot \varepsilon, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (*
  (fma
   (*
    (fma
     (fma -0.2857142857142857 (* eps eps) -0.4)
     (* eps eps)
     -0.6666666666666666)
    eps)
   eps
   -2.0)
  eps))

C

double code(double eps) {
	return fma((fma(fma(-0.2857142857142857, (eps * eps), -0.4), (eps * eps), -0.6666666666666666) * eps), eps, -2.0) * eps;
}

Julia

function code(eps)
	return Float64(fma(Float64(fma(fma(-0.2857142857142857, Float64(eps * eps), -0.4), Float64(eps * eps), -0.6666666666666666) * eps), eps, -2.0) * eps)
end

Wolfram

code[eps_] := N[(N[(N[(N[(N[(-0.2857142857142857 * N[(eps * eps), $MachinePrecision] + -0.4), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] * eps), $MachinePrecision] * eps + -2.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   5. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   10. lower-pow.f64 99.7

       \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. *-commutative N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \cdot \color{blue}{\varepsilon} \]

   3. lower-*.f64 99.7

      \[\leadsto \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right) \cdot \color{blue}{\varepsilon} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.2857142857142857, \varepsilon \cdot \varepsilon, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon} \]
7. Add Preprocessing

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.4, \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (-
  (* (fma (* (fma -0.4 (* eps eps) -0.6666666666666666) eps) eps -1.0) eps)
  eps))

C

double code(double eps) {
	return (fma((fma(-0.4, (eps * eps), -0.6666666666666666) * eps), eps, -1.0) * eps) - eps;
}

Julia

function code(eps)
	return Float64(Float64(fma(Float64(fma(-0.4, Float64(eps * eps), -0.6666666666666666) * eps), eps, -1.0) * eps) - eps)
end

Wolfram

code[eps_] := N[(N[(N[(N[(N[(-0.4 * N[(eps * eps), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] * eps), $MachinePrecision] * eps + -1.0), $MachinePrecision] * eps), $MachinePrecision] - eps), $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   5. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   10. lower-pow.f64 99.7

       \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   3. sub-flip N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + -2\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)\right) \cdot \varepsilon + \color{blue}{-2 \cdot \varepsilon} \]

   6. lift-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)\right) \cdot \varepsilon + -2 \cdot \color{blue}{\varepsilon} \]

   7. lower-fma.f64 99.8

      \[\leadsto \mathsf{fma}\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right), \color{blue}{\varepsilon}, -2 \cdot \varepsilon\right) \]

6. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.2857142857142857, \varepsilon \cdot \varepsilon, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon\right) \cdot \varepsilon, \color{blue}{\varepsilon}, -2 \cdot \varepsilon\right) \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon + \color{blue}{-2 \cdot \varepsilon} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon + -2 \cdot \color{blue}{\varepsilon} \]

   3. distribute-rgt-out N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -2\right)} \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + \left(-1 + \color{blue}{-1}\right)\right) \]

   5. associate-+l+ N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -1\right) + \color{blue}{-1}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -1\right) + -1\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) + -1\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon + \color{blue}{-1 \cdot \varepsilon} \]

   9. add-flip N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \varepsilon\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\varepsilon} \]

8. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.2857142857142857, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\varepsilon} \]

9. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{5}, \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 99.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.4, \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \]
12. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.4, \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (* (fma (* (fma -0.4 (* eps eps) -0.6666666666666666) eps) eps -2.0) eps))

C

double code(double eps) {
	return fma((fma(-0.4, (eps * eps), -0.6666666666666666) * eps), eps, -2.0) * eps;
}

Julia

function code(eps)
	return Float64(fma(Float64(fma(-0.4, Float64(eps * eps), -0.6666666666666666) * eps), eps, -2.0) * eps)
end

Wolfram

code[eps_] := N[(N[(N[(N[(-0.4 * N[(eps * eps), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] * eps), $MachinePrecision] * eps + -2.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   5. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   10. lower-pow.f64 99.7

       \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right)} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   2. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \left(1 + \color{blue}{1}\right)\right) \]

   3. associate--r+ N/A

      \[\leadsto \varepsilon \cdot \left(\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 1\right) - \color{blue}{1}\right) \]

   4. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 1\right) - \color{blue}{1}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.2857142857142857, \varepsilon \cdot \varepsilon, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) - \color{blue}{1}\right) \]

7. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{5}, \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) - 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.6%

   \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.4, \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) - 1\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{5}, \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) - 1\right)} \]

    2. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{5}, \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) - 1\right) \cdot \color{blue}{\varepsilon} \]

    3. lower-*.f64 99.6

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.4, \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) - 1\right) \cdot \color{blue}{\varepsilon} \]

    4. lift--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{5}, \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) - 1\right) \cdot \varepsilon \]

    5. lift-fma.f64 N/A

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{-2}{5}, \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -1\right) - 1\right) \cdot \varepsilon \]

    6. associate--l+ N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{-2}{5}, \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + \left(-1 - 1\right)\right) \cdot \varepsilon \]

    7. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{fma}\left(\frac{-2}{5}, \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -2\right) \cdot \varepsilon \]

    8. lower-fma.f64 99.7

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.4, \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon \]

11. Applied rewrites 99.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.4, \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon} \]
12. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, -0.6666666666666666, -2 \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (fma (* (* eps eps) eps) -0.6666666666666666 (* -2.0 eps)))

C

double code(double eps) {
	return fma(((eps * eps) * eps), -0.6666666666666666, (-2.0 * eps));
}

Julia

function code(eps)
	return fma(Float64(Float64(eps * eps) * eps), -0.6666666666666666, Float64(-2.0 * eps))
end

Wolfram

code[eps_] := N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * -0.6666666666666666 + N[(-2.0 * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - \color{blue}{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right) \]

   4. lower-pow.f64 99.5

      \[\leadsto \varepsilon \cdot \left(-0.6666666666666666 \cdot {\varepsilon}^{2} - 2\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(-0.6666666666666666 \cdot {\varepsilon}^{2} - 2\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - \color{blue}{2}\right) \]

   3. sub-flip N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + -2\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left(\frac{-2}{3} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + \color{blue}{-2 \cdot \varepsilon} \]

   6. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2}\right) + \color{blue}{-2} \cdot \varepsilon \]

   7. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2}\right) + -2 \cdot \varepsilon \]

   8. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \frac{-2}{3}\right) + -2 \cdot \varepsilon \]

   9. associate-*r* N/A

      \[\leadsto \left(\varepsilon \cdot {\varepsilon}^{2}\right) \cdot \frac{-2}{3} + \color{blue}{-2} \cdot \varepsilon \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot {\varepsilon}^{2}\right) \cdot \frac{-2}{3} + -2 \cdot \color{blue}{\varepsilon} \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot {\varepsilon}^{2}, \color{blue}{\frac{-2}{3}}, -2 \cdot \varepsilon\right) \]

   12. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot {\varepsilon}^{2}, \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   14. cube-unmult N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{3}, \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{\left(2 + 1\right)}, \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   16. pow-plus N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{2} \cdot \varepsilon, \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   17. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{2} \cdot \varepsilon, \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   18. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{2} \cdot \varepsilon, -0.6666666666666666, -2 \cdot \varepsilon\right) \]

   19. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{2} \cdot \varepsilon, \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   21. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, -0.6666666666666666, -2 \cdot \varepsilon\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \color{blue}{-0.6666666666666666}, -2 \cdot \varepsilon\right) \]
7. Add Preprocessing

Alternative 6: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.6666666666666666 \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (- (* (fma (* -0.6666666666666666 eps) eps -1.0) eps) eps))

C

double code(double eps) {
	return (fma((-0.6666666666666666 * eps), eps, -1.0) * eps) - eps;
}

Julia

function code(eps)
	return Float64(Float64(fma(Float64(-0.6666666666666666 * eps), eps, -1.0) * eps) - eps)
end

Wolfram

code[eps_] := N[(N[(N[(N[(-0.6666666666666666 * eps), $MachinePrecision] * eps + -1.0), $MachinePrecision] * eps), $MachinePrecision] - eps), $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   5. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \]

   10. lower-pow.f64 99.7

       \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right) - 2\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - \color{blue}{2}\right) \]

   3. sub-flip N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + -2\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)\right) \cdot \varepsilon + \color{blue}{-2 \cdot \varepsilon} \]

   6. lift-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)\right) \cdot \varepsilon + -2 \cdot \color{blue}{\varepsilon} \]

   7. lower-fma.f64 99.8

      \[\leadsto \mathsf{fma}\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(-0.2857142857142857 \cdot {\varepsilon}^{2} - 0.4\right) - 0.6666666666666666\right), \color{blue}{\varepsilon}, -2 \cdot \varepsilon\right) \]

6. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.2857142857142857, \varepsilon \cdot \varepsilon, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon\right) \cdot \varepsilon, \color{blue}{\varepsilon}, -2 \cdot \varepsilon\right) \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon + \color{blue}{-2 \cdot \varepsilon} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon + -2 \cdot \color{blue}{\varepsilon} \]

   3. distribute-rgt-out N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -2\right)} \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + \left(-1 + \color{blue}{-1}\right)\right) \]

   5. associate-+l+ N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -1\right) + \color{blue}{-1}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon + -1\right) + -1\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) + -1\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon + \color{blue}{-1 \cdot \varepsilon} \]

   9. add-flip N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \varepsilon\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{7}, \varepsilon \cdot \varepsilon, \frac{-2}{5}\right), \varepsilon \cdot \varepsilon, \frac{-2}{3}\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\varepsilon} \]

8. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.2857142857142857, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \color{blue}{\varepsilon} \]

9. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\frac{-2}{3} \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(-0.6666666666666666 \cdot \varepsilon, \varepsilon, -1\right) \cdot \varepsilon - \varepsilon \]
12. Add Preprocessing

Alternative 7: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.6666666666666666, \varepsilon \cdot \varepsilon, -2\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (* (fma -0.6666666666666666 (* eps eps) -2.0) eps))

C

double code(double eps) {
	return fma(-0.6666666666666666, (eps * eps), -2.0) * eps;
}

Julia

function code(eps)
	return Float64(fma(-0.6666666666666666, Float64(eps * eps), -2.0) * eps)
end

Wolfram

code[eps_] := N[(N[(-0.6666666666666666 * N[(eps * eps), $MachinePrecision] + -2.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - \color{blue}{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right) \]

   4. lower-pow.f64 99.5

      \[\leadsto \varepsilon \cdot \left(-0.6666666666666666 \cdot {\varepsilon}^{2} - 2\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(-0.6666666666666666 \cdot {\varepsilon}^{2} - 2\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right) \cdot \color{blue}{\varepsilon} \]

   3. lower-*.f64 99.5

      \[\leadsto \left(-0.6666666666666666 \cdot {\varepsilon}^{2} - 2\right) \cdot \color{blue}{\varepsilon} \]

   4. lift--.f64 N/A

      \[\leadsto \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right) \cdot \varepsilon \]

   5. sub-flip N/A

      \[\leadsto \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot \varepsilon \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot \varepsilon \]

   7. metadata-eval N/A

      \[\leadsto \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + -2\right) \cdot \varepsilon \]

   8. lower-fma.f64 99.5

      \[\leadsto \mathsf{fma}\left(-0.6666666666666666, {\varepsilon}^{2}, -2\right) \cdot \varepsilon \]

   9. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-2}{3}, {\varepsilon}^{2}, -2\right) \cdot \varepsilon \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-2}{3}, \varepsilon \cdot \varepsilon, -2\right) \cdot \varepsilon \]

   11. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(-0.6666666666666666, \varepsilon \cdot \varepsilon, -2\right) \cdot \varepsilon \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(-0.6666666666666666, \varepsilon \cdot \varepsilon, -2\right) \cdot \color{blue}{\varepsilon} \]
7. Add Preprocessing

Alternative 8: 98.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \varepsilon \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (* -2.0 eps))

C

double code(double eps) {
	return -2.0 * eps;
}

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(eps)
use fmin_fmax_functions
    real(8), intent (in) :: eps
    code = (-2.0d0) * eps
end function

Java

public static double code(double eps) {
	return -2.0 * eps;
}

Python

def code(eps):
	return -2.0 * eps

Julia

function code(eps)
	return Float64(-2.0 * eps)
end

MATLAB

function tmp = code(eps)
	tmp = -2.0 * eps;
end

Wolfram

code[eps_] := N[(-2.0 * eps), $MachinePrecision]

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 98.9

      \[\leadsto -2 \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{-2 \cdot \varepsilon} \]
5. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right) \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (- (log1p (- eps)) (log1p eps)))

C

double code(double eps) {
	return log1p(-eps) - log1p(eps);
}

Java

public static double code(double eps) {
	return Math.log1p(-eps) - Math.log1p(eps);
}

Python

def code(eps):
	return math.log1p(-eps) - math.log1p(eps)

Julia

function code(eps)
	return Float64(log1p(Float64(-eps)) - log1p(eps))
end

Wolfram

code[eps_] := N[(N[Log[1 + (-eps)], $MachinePrecision] - N[Log[1 + eps], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025155 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64
  :pre (< (fabs eps) 1.0)

  :alt
  (! :herbie-platform c (- (log1p (- eps)) (log1p eps)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))