logq (problem 3.4.3)

Percentage Accurate: 8.3% → 100.0%

Time: 3.8s

Alternatives: 5

Speedup: 19.7×

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: 8.3% 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: 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]

Derivation

1. Initial program 8.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. log-div N/A

      \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) - \log \left(1 + \varepsilon\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) - \log \left(1 + \varepsilon\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \log \color{blue}{\left(1 - \varepsilon\right)} - \log \left(1 + \varepsilon\right) \]

   6. sub-negate1 N/A

      \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} - \log \left(1 + \varepsilon\right) \]

   7. lower-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(\varepsilon\right)\right)} - \log \left(1 + \varepsilon\right) \]

   8. lower-neg.f64 N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-\varepsilon}\right) - \log \left(1 + \varepsilon\right) \]

   9. lift-+.f64 N/A

      \[\leadsto \mathsf{log1p}\left(-\varepsilon\right) - \log \color{blue}{\left(1 + \varepsilon\right)} \]

   10. lower-log1p.f64 100.0

       \[\leadsto \mathsf{log1p}\left(-\varepsilon\right) - \color{blue}{\mathsf{log1p}\left(\varepsilon\right)} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)} \]
4. Add Preprocessing

Alternative 2: 99.8% accurate, 3.0× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 8.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon}}}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{\color{blue}{1 - \varepsilon}}}\right) \]

   5. associate-/r/ N/A

      \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \log \left(\color{blue}{\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1} - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   10. lower-*.f64 8.3

       \[\leadsto \log \left(\frac{1 - \varepsilon}{1 - \color{blue}{\varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

3. Applied rewrites 8.3%

   \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \log \left(\color{blue}{\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   2. lift--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   3. sub-negate2 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - 1\right)\right)}} \cdot \left(1 - \varepsilon\right)\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\mathsf{neg}\left(\left(\color{blue}{\varepsilon \cdot \varepsilon} - 1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - \color{blue}{-1 \cdot -1}\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   6. lift--.f64 N/A

      \[\leadsto \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - -1 \cdot -1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   7. sub-negate2 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\mathsf{neg}\left(\left(\varepsilon - 1\right)\right)}}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - -1 \cdot -1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   8. frac-2neg-rev N/A

      \[\leadsto \log \left(\color{blue}{\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon - -1 \cdot -1}} \cdot \left(1 - \varepsilon\right)\right) \]

   9. sub-negate1 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\varepsilon + \left(\mathsf{neg}\left(1\right)\right)}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon + \color{blue}{-1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \log \left(\color{blue}{\frac{\varepsilon + -1}{\varepsilon \cdot \varepsilon - -1 \cdot -1}} \cdot \left(1 - \varepsilon\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   13. sub-negate1 N/A

       \[\leadsto \log \left(\frac{\color{blue}{\varepsilon - 1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   14. lower--.f64 N/A

       \[\leadsto \log \left(\frac{\color{blue}{\varepsilon - 1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon - \color{blue}{1}} \cdot \left(1 - \varepsilon\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon} - 1} \cdot \left(1 - \varepsilon\right)\right) \]

   17. sub-negate1 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)}} \cdot \left(1 - \varepsilon\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon + \color{blue}{-1}} \cdot \left(1 - \varepsilon\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon} + -1} \cdot \left(1 - \varepsilon\right)\right) \]

   20. lower-fma.f64 8.3

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

5. Applied rewrites 8.3%

   \[\leadsto \log \color{blue}{\left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)} \cdot \left(1 - \varepsilon\right)\right)} \]
6. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. log-prod N/A

      \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \log \color{blue}{\left(1 - \varepsilon\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   6. sub-negate1 N/A

      \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   7. lift-neg.f64 N/A

      \[\leadsto \log \left(1 + \color{blue}{\left(-\varepsilon\right)}\right) + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   8. lift-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   9. unpow1 N/A

      \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{1}} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   10. sqr-pow N/A

       \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)}} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \color{blue}{\left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right)} \]

   12. frac-2neg N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \color{blue}{\left(\frac{\mathsf{neg}\left(\left(\varepsilon - 1\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right)} \]

   13. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\mathsf{neg}\left(\color{blue}{\left(\varepsilon - 1\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   14. sub-negate2 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   15. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   16. log-div N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \color{blue}{\left(\log \left(1 - \varepsilon\right) - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right)} \]

   17. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \color{blue}{\left(1 - \varepsilon\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   18. sub-negate1 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   19. lift-neg.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \left(1 + \color{blue}{\left(-\varepsilon\right)}\right) - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   20. lift-log1p.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   21. lift-fma.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \left(\mathsf{neg}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon + -1\right)}\right)\right)\right) \]

   22. +-commutative N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \left(\mathsf{neg}\left(\color{blue}{\left(-1 + \varepsilon \cdot \varepsilon\right)}\right)\right)\right) \]

   23. distribute-neg-in N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right)\right)}\right) \]

7. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{0.5}, {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{0.5}, -\mathsf{log1p}\left(\varepsilon\right)\right)} \]

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

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

Alternative 3: 99.7% accurate, 4.2× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 8.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon}}}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{\color{blue}{1 - \varepsilon}}}\right) \]

   5. associate-/r/ N/A

      \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \log \left(\color{blue}{\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1} - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   10. lower-*.f64 8.3

       \[\leadsto \log \left(\frac{1 - \varepsilon}{1 - \color{blue}{\varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

3. Applied rewrites 8.3%

   \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \log \left(\color{blue}{\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   2. lift--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   3. sub-negate2 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - 1\right)\right)}} \cdot \left(1 - \varepsilon\right)\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\mathsf{neg}\left(\left(\color{blue}{\varepsilon \cdot \varepsilon} - 1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - \color{blue}{-1 \cdot -1}\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   6. lift--.f64 N/A

      \[\leadsto \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - -1 \cdot -1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   7. sub-negate2 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\mathsf{neg}\left(\left(\varepsilon - 1\right)\right)}}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - -1 \cdot -1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   8. frac-2neg-rev N/A

      \[\leadsto \log \left(\color{blue}{\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon - -1 \cdot -1}} \cdot \left(1 - \varepsilon\right)\right) \]

   9. sub-negate1 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\varepsilon + \left(\mathsf{neg}\left(1\right)\right)}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon + \color{blue}{-1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \log \left(\color{blue}{\frac{\varepsilon + -1}{\varepsilon \cdot \varepsilon - -1 \cdot -1}} \cdot \left(1 - \varepsilon\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   13. sub-negate1 N/A

       \[\leadsto \log \left(\frac{\color{blue}{\varepsilon - 1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   14. lower--.f64 N/A

       \[\leadsto \log \left(\frac{\color{blue}{\varepsilon - 1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon - \color{blue}{1}} \cdot \left(1 - \varepsilon\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon} - 1} \cdot \left(1 - \varepsilon\right)\right) \]

   17. sub-negate1 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)}} \cdot \left(1 - \varepsilon\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon + \color{blue}{-1}} \cdot \left(1 - \varepsilon\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon} + -1} \cdot \left(1 - \varepsilon\right)\right) \]

   20. lower-fma.f64 8.3

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

5. Applied rewrites 8.3%

   \[\leadsto \log \color{blue}{\left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)} \cdot \left(1 - \varepsilon\right)\right)} \]
6. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. log-prod N/A

      \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \log \color{blue}{\left(1 - \varepsilon\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   6. sub-negate1 N/A

      \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   7. lift-neg.f64 N/A

      \[\leadsto \log \left(1 + \color{blue}{\left(-\varepsilon\right)}\right) + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   8. lift-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   9. unpow1 N/A

      \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{1}} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   10. sqr-pow N/A

       \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)}} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \color{blue}{\left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right)} \]

   12. frac-2neg N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \color{blue}{\left(\frac{\mathsf{neg}\left(\left(\varepsilon - 1\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right)} \]

   13. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\mathsf{neg}\left(\color{blue}{\left(\varepsilon - 1\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   14. sub-negate2 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   15. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   16. log-div N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \color{blue}{\left(\log \left(1 - \varepsilon\right) - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right)} \]

   17. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \color{blue}{\left(1 - \varepsilon\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   18. sub-negate1 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   19. lift-neg.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \left(1 + \color{blue}{\left(-\varepsilon\right)}\right) - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   20. lift-log1p.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   21. lift-fma.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \left(\mathsf{neg}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon + -1\right)}\right)\right)\right) \]

   22. +-commutative N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \left(\mathsf{neg}\left(\color{blue}{\left(-1 + \varepsilon \cdot \varepsilon\right)}\right)\right)\right) \]

   23. distribute-neg-in N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right)\right)}\right) \]

7. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{0.5}, {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{0.5}, -\mathsf{log1p}\left(\varepsilon\right)\right)} \]

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-negate1 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-negate1 N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 4: 99.5% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon}}}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{\color{blue}{1 - \varepsilon}}}\right) \]

   5. associate-/r/ N/A

      \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \log \left(\color{blue}{\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1} - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   10. lower-*.f64 8.3

       \[\leadsto \log \left(\frac{1 - \varepsilon}{1 - \color{blue}{\varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

3. Applied rewrites 8.3%

   \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \log \left(\color{blue}{\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   2. lift--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   3. sub-negate2 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - 1\right)\right)}} \cdot \left(1 - \varepsilon\right)\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\mathsf{neg}\left(\left(\color{blue}{\varepsilon \cdot \varepsilon} - 1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - \color{blue}{-1 \cdot -1}\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   6. lift--.f64 N/A

      \[\leadsto \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - -1 \cdot -1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   7. sub-negate2 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\mathsf{neg}\left(\left(\varepsilon - 1\right)\right)}}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - -1 \cdot -1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   8. frac-2neg-rev N/A

      \[\leadsto \log \left(\color{blue}{\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon - -1 \cdot -1}} \cdot \left(1 - \varepsilon\right)\right) \]

   9. sub-negate1 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\varepsilon + \left(\mathsf{neg}\left(1\right)\right)}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon + \color{blue}{-1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \log \left(\color{blue}{\frac{\varepsilon + -1}{\varepsilon \cdot \varepsilon - -1 \cdot -1}} \cdot \left(1 - \varepsilon\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   13. sub-negate1 N/A

       \[\leadsto \log \left(\frac{\color{blue}{\varepsilon - 1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   14. lower--.f64 N/A

       \[\leadsto \log \left(\frac{\color{blue}{\varepsilon - 1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon - \color{blue}{1}} \cdot \left(1 - \varepsilon\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon} - 1} \cdot \left(1 - \varepsilon\right)\right) \]

   17. sub-negate1 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)}} \cdot \left(1 - \varepsilon\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon + \color{blue}{-1}} \cdot \left(1 - \varepsilon\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon} + -1} \cdot \left(1 - \varepsilon\right)\right) \]

   20. lower-fma.f64 8.3

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

5. Applied rewrites 8.3%

   \[\leadsto \log \color{blue}{\left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)} \cdot \left(1 - \varepsilon\right)\right)} \]
6. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. log-prod N/A

      \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \log \color{blue}{\left(1 - \varepsilon\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   6. sub-negate1 N/A

      \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   7. lift-neg.f64 N/A

      \[\leadsto \log \left(1 + \color{blue}{\left(-\varepsilon\right)}\right) + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   8. lift-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   9. unpow1 N/A

      \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{1}} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   10. sqr-pow N/A

       \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)}} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \color{blue}{\left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right)} \]

   12. frac-2neg N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \color{blue}{\left(\frac{\mathsf{neg}\left(\left(\varepsilon - 1\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right)} \]

   13. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\mathsf{neg}\left(\color{blue}{\left(\varepsilon - 1\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   14. sub-negate2 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   15. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   16. log-div N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \color{blue}{\left(\log \left(1 - \varepsilon\right) - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right)} \]

   17. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \color{blue}{\left(1 - \varepsilon\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   18. sub-negate1 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   19. lift-neg.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \left(1 + \color{blue}{\left(-\varepsilon\right)}\right) - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   20. lift-log1p.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   21. lift-fma.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \left(\mathsf{neg}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon + -1\right)}\right)\right)\right) \]

   22. +-commutative N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \left(\mathsf{neg}\left(\color{blue}{\left(-1 + \varepsilon \cdot \varepsilon\right)}\right)\right)\right) \]

   23. distribute-neg-in N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right)\right)}\right) \]

7. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{0.5}, {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{0.5}, -\mathsf{log1p}\left(\varepsilon\right)\right)} \]

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-negate1 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.5

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

10. Applied rewrites 99.5%

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

Alternative 5: 99.1% accurate, 19.7× 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 8.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon}}}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{\color{blue}{1 - \varepsilon}}}\right) \]

   5. associate-/r/ N/A

      \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \log \left(\color{blue}{\frac{1 - \varepsilon}{1 \cdot 1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1} - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   10. lower-*.f64 8.3

       \[\leadsto \log \left(\frac{1 - \varepsilon}{1 - \color{blue}{\varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

3. Applied rewrites 8.3%

   \[\leadsto \log \color{blue}{\left(\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon} \cdot \left(1 - \varepsilon\right)\right)} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \log \left(\color{blue}{\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   2. lift--.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{1 - \varepsilon \cdot \varepsilon}} \cdot \left(1 - \varepsilon\right)\right) \]

   3. sub-negate2 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\color{blue}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - 1\right)\right)}} \cdot \left(1 - \varepsilon\right)\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\mathsf{neg}\left(\left(\color{blue}{\varepsilon \cdot \varepsilon} - 1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \log \left(\frac{1 - \varepsilon}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - \color{blue}{-1 \cdot -1}\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   6. lift--.f64 N/A

      \[\leadsto \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - -1 \cdot -1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   7. sub-negate2 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\mathsf{neg}\left(\left(\varepsilon - 1\right)\right)}}{\mathsf{neg}\left(\left(\varepsilon \cdot \varepsilon - -1 \cdot -1\right)\right)} \cdot \left(1 - \varepsilon\right)\right) \]

   8. frac-2neg-rev N/A

      \[\leadsto \log \left(\color{blue}{\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon - -1 \cdot -1}} \cdot \left(1 - \varepsilon\right)\right) \]

   9. sub-negate1 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\varepsilon + \left(\mathsf{neg}\left(1\right)\right)}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon + \color{blue}{-1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \log \left(\color{blue}{\frac{\varepsilon + -1}{\varepsilon \cdot \varepsilon - -1 \cdot -1}} \cdot \left(1 - \varepsilon\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   13. sub-negate1 N/A

       \[\leadsto \log \left(\frac{\color{blue}{\varepsilon - 1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   14. lower--.f64 N/A

       \[\leadsto \log \left(\frac{\color{blue}{\varepsilon - 1}}{\varepsilon \cdot \varepsilon - -1 \cdot -1} \cdot \left(1 - \varepsilon\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon - \color{blue}{1}} \cdot \left(1 - \varepsilon\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon} - 1} \cdot \left(1 - \varepsilon\right)\right) \]

   17. sub-negate1 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)}} \cdot \left(1 - \varepsilon\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\varepsilon \cdot \varepsilon + \color{blue}{-1}} \cdot \left(1 - \varepsilon\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \log \left(\frac{\varepsilon - 1}{\color{blue}{\varepsilon \cdot \varepsilon} + -1} \cdot \left(1 - \varepsilon\right)\right) \]

   20. lower-fma.f64 8.3

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

5. Applied rewrites 8.3%

   \[\leadsto \log \color{blue}{\left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)} \cdot \left(1 - \varepsilon\right)\right)} \]
6. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. log-prod N/A

      \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \log \color{blue}{\left(1 - \varepsilon\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   6. sub-negate1 N/A

      \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   7. lift-neg.f64 N/A

      \[\leadsto \log \left(1 + \color{blue}{\left(-\varepsilon\right)}\right) + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   8. lift-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   9. unpow1 N/A

      \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{1}} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   10. sqr-pow N/A

       \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)}} + \log \left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \color{blue}{\left(\frac{\varepsilon - 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}\right)} \]

   12. frac-2neg N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \color{blue}{\left(\frac{\mathsf{neg}\left(\left(\varepsilon - 1\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right)} \]

   13. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\mathsf{neg}\left(\color{blue}{\left(\varepsilon - 1\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   14. sub-negate2 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   15. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \log \left(\frac{\color{blue}{1 - \varepsilon}}{\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)}\right) \]

   16. log-div N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \color{blue}{\left(\log \left(1 - \varepsilon\right) - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right)} \]

   17. lift--.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \color{blue}{\left(1 - \varepsilon\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   18. sub-negate1 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   19. lift-neg.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\log \left(1 + \color{blue}{\left(-\varepsilon\right)}\right) - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   20. lift-log1p.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\color{blue}{\mathsf{log1p}\left(-\varepsilon\right)} - \log \left(\mathsf{neg}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)\right)\right)\right) \]

   21. lift-fma.f64 N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \left(\mathsf{neg}\left(\color{blue}{\left(\varepsilon \cdot \varepsilon + -1\right)}\right)\right)\right) \]

   22. +-commutative N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \left(\mathsf{neg}\left(\color{blue}{\left(-1 + \varepsilon \cdot \varepsilon\right)}\right)\right)\right) \]

   23. distribute-neg-in N/A

       \[\leadsto {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{\left(\frac{1}{2}\right)} + \left(\mathsf{log1p}\left(-\varepsilon\right) - \log \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right)\right)}\right) \]

7. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{0.5}, {\left(\mathsf{log1p}\left(-\varepsilon\right)\right)}^{0.5}, -\mathsf{log1p}\left(\varepsilon\right)\right)} \]

8. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.1

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

10. Applied rewrites 99.1%

    \[\leadsto \color{blue}{-2 \cdot \varepsilon} \]
11. 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 2025107 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64
  :pre (< (fabs eps) 1.0)

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

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