logq (problem 3.4.3)

Percentage Accurate: 8.3% → 100.0%

Time: 11.4s

Alternatives: 8

Speedup: 35.7×

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

real(8) function code(eps)
    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

real(8) function code(eps)
    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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(0 - \varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right) \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (- (log1p (- 0.0 eps)) (log1p eps)))

C

double code(double eps) {
	return log1p((0.0 - eps)) - log1p(eps);
}

Java

public static double code(double eps) {
	return Math.log1p((0.0 - eps)) - Math.log1p(eps);
}

Python

def code(eps):
	return math.log1p((0.0 - eps)) - math.log1p(eps)

Julia

function code(eps)
	return Float64(log1p(Float64(0.0 - eps)) - log1p(eps))
end

Wolfram

code[eps_] := N[(N[Log[1 + N[(0.0 - eps), $MachinePrecision]], $MachinePrecision] - N[Log[1 + eps], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

   1. log-div N/A

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

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 - \varepsilon\right), \color{blue}{\log \left(1 + \varepsilon\right)}\right) \]

   3. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(1 - \varepsilon\right)\right), \log \color{blue}{\left(1 + \varepsilon\right)}\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right)\right), \log \left(\color{blue}{1} + \varepsilon\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right)\right), \left(\mathsf{log1p}\left(\varepsilon\right)\right)\right) \]

   6. log1p-lowering-log1p.f64 21.6%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right)\right), \mathsf{log1p.f64}\left(\varepsilon\right)\right) \]

4. Applied egg-rr 21.6%

   \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)} \]
5. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \mathsf{log1p.f64}\left(\varepsilon\right)\right) \]

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \mathsf{log1p.f64}\left(\color{blue}{\varepsilon}\right)\right) \]

   3. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \mathsf{log1p.f64}\left(\color{blue}{\varepsilon}\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\left(0 - \varepsilon\right)\right), \mathsf{log1p.f64}\left(\varepsilon\right)\right) \]

   5. --lowering--.f64 100.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, \varepsilon\right)\right), \mathsf{log1p.f64}\left(\varepsilon\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0 - \varepsilon\right)} - \mathsf{log1p}\left(\varepsilon\right) \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \mathsf{log1p.f64}\left(\varepsilon\right)\right) \]

   2. neg-lowering-neg.f64 100.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(\varepsilon\right)\right), \mathsf{log1p.f64}\left(\varepsilon\right)\right) \]

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \mathsf{log1p}\left(0 - \varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right) \]
10. Add Preprocessing

Alternative 2: 99.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(0.4444444444444444 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.5333333333333333\right) - 4}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.4 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.2857142857142857\right)\right)\right)\right) - -2} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (*
  eps
  (/
   (-
    (*
     (* eps (* eps (* eps eps)))
     (+ 0.4444444444444444 (* (* eps eps) 0.5333333333333333)))
    4.0)
   (-
    (*
     eps
     (*
      eps
      (+
       -0.6666666666666666
       (* eps (* eps (+ -0.4 (* (* eps eps) -0.2857142857142857)))))))
    -2.0))))

C

double code(double eps) {
	return eps * ((((eps * (eps * (eps * eps))) * (0.4444444444444444 + ((eps * eps) * 0.5333333333333333))) - 4.0) / ((eps * (eps * (-0.6666666666666666 + (eps * (eps * (-0.4 + ((eps * eps) * -0.2857142857142857))))))) - -2.0));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = eps * ((((eps * (eps * (eps * eps))) * (0.4444444444444444d0 + ((eps * eps) * 0.5333333333333333d0))) - 4.0d0) / ((eps * (eps * ((-0.6666666666666666d0) + (eps * (eps * ((-0.4d0) + ((eps * eps) * (-0.2857142857142857d0)))))))) - (-2.0d0)))
end function

Java

public static double code(double eps) {
	return eps * ((((eps * (eps * (eps * eps))) * (0.4444444444444444 + ((eps * eps) * 0.5333333333333333))) - 4.0) / ((eps * (eps * (-0.6666666666666666 + (eps * (eps * (-0.4 + ((eps * eps) * -0.2857142857142857))))))) - -2.0));
}

Python

def code(eps):
	return eps * ((((eps * (eps * (eps * eps))) * (0.4444444444444444 + ((eps * eps) * 0.5333333333333333))) - 4.0) / ((eps * (eps * (-0.6666666666666666 + (eps * (eps * (-0.4 + ((eps * eps) * -0.2857142857142857))))))) - -2.0))

Julia

function code(eps)
	return Float64(eps * Float64(Float64(Float64(Float64(eps * Float64(eps * Float64(eps * eps))) * Float64(0.4444444444444444 + Float64(Float64(eps * eps) * 0.5333333333333333))) - 4.0) / Float64(Float64(eps * Float64(eps * Float64(-0.6666666666666666 + Float64(eps * Float64(eps * Float64(-0.4 + Float64(Float64(eps * eps) * -0.2857142857142857))))))) - -2.0)))
end

MATLAB

function tmp = code(eps)
	tmp = eps * ((((eps * (eps * (eps * eps))) * (0.4444444444444444 + ((eps * eps) * 0.5333333333333333))) - 4.0) / ((eps * (eps * (-0.6666666666666666 + (eps * (eps * (-0.4 + ((eps * eps) * -0.2857142857142857))))))) - -2.0));
end

Wolfram

code[eps_] := N[(eps * N[(N[(N[(N[(eps * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.4444444444444444 + N[(N[(eps * eps), $MachinePrecision] * 0.5333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / N[(N[(eps * N[(eps * N[(-0.6666666666666666 + N[(eps * N[(eps * N[(-0.4 + N[(N[(eps * eps), $MachinePrecision] * -0.2857142857142857), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

   1. log-div N/A

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

   2. flip-- N/A

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

   3. log-div N/A

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

   4. associate--l- N/A

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

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 \cdot 1 - \varepsilon \cdot \varepsilon\right), \color{blue}{\left(\log \left(1 + \varepsilon\right) + \log \left(1 + \varepsilon\right)\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 - \varepsilon \cdot \varepsilon\right), \left(\log \left(\color{blue}{1} + \varepsilon\right) + \log \left(1 + \varepsilon\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + \left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\log \color{blue}{\left(1 + \varepsilon\right)} + \log \left(1 + \varepsilon\right)\right)\right) \]

   8. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\color{blue}{\log \left(1 + \varepsilon\right)} + \log \left(1 + \varepsilon\right)\right)\right) \]

   9. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\color{blue}{\log \left(1 + \varepsilon\right)} + \log \left(1 + \varepsilon\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\left(0 - \varepsilon \cdot \varepsilon\right)\right), \left(\log \color{blue}{\left(1 + \varepsilon\right)} + \log \left(1 + \varepsilon\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\log \color{blue}{\left(1 + \varepsilon\right)} + \log \left(1 + \varepsilon\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(\log \left(1 + \color{blue}{\varepsilon}\right) + \log \left(1 + \varepsilon\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\log \left(1 + \varepsilon\right), \color{blue}{\log \left(1 + \varepsilon\right)}\right)\right) \]

   14. log1p-define N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(\varepsilon\right)\right), \log \color{blue}{\left(1 + \varepsilon\right)}\right)\right) \]

   15. log1p-lowering-log1p.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\varepsilon\right), \log \color{blue}{\left(1 + \varepsilon\right)}\right)\right) \]

   16. log1p-define N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\varepsilon\right), \left(\mathsf{log1p}\left(\varepsilon\right)\right)\right)\right) \]

   17. log1p-lowering-log1p.f64 100.0%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\varepsilon\right), \mathsf{log1p.f64}\left(\varepsilon\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0 - \varepsilon \cdot \varepsilon\right) - \left(\mathsf{log1p}\left(\varepsilon\right) + \mathsf{log1p}\left(\varepsilon\right)\right)} \]

5. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + -2\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(-2 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)} - \frac{2}{3}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)} - \frac{2}{3}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) + \frac{-2}{3}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{3} + \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)}\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)\right)}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)}\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{-2}{7} \cdot {\varepsilon}^{2}} - \frac{2}{5}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{-2}{7} \cdot {\varepsilon}^{2}} - \frac{2}{5}\right)\right)\right)\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{7} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{5}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{7} \cdot {\varepsilon}^{2} + \frac{-2}{5}\right)\right)\right)\right)\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{5} + \color{blue}{\frac{-2}{7} \cdot {\varepsilon}^{2}}\right)\right)\right)\right)\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \color{blue}{\left(\frac{-2}{7} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{-2}{7}}\right)\right)\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{-2}{7}}\right)\right)\right)\right)\right)\right)\right) \]

   22. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right) \]

   23. *-lowering-*.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.6%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{5} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{7}\right)\right) + \color{blue}{-2}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{5} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{7}\right)\right)\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{5} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{7}\right)\right)\right) - -2 \cdot -2}{\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{5} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{7}\right)\right) - -2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{5} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{7}\right)\right)\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{5} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{7}\right)\right)\right) - -2 \cdot -2\right), \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{5} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{7}\right)\right) - -2\right)}\right)\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto \varepsilon \cdot \color{blue}{\frac{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.4 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.2857142857142857\right)\right)\right) \cdot \left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.4 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.2857142857142857\right)\right)\right)\right) - 4}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.4 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.2857142857142857\right)\right)\right)\right) - -2}} \]

10. Taylor expanded in eps around 0

    \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left({\varepsilon}^{4} \cdot \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right)}, 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{4}\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{\left(2 \cdot 2\right)}\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    3. pow-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    7. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{3}\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{3}\right)\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    9. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(\frac{4}{9} + \frac{8}{15} \cdot {\varepsilon}^{2}\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    14. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\frac{4}{9}, \left(\frac{8}{15} \cdot {\varepsilon}^{2}\right)\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\frac{4}{9}, \left({\varepsilon}^{2} \cdot \frac{8}{15}\right)\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\frac{4}{9}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{8}{15}\right)\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\frac{4}{9}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{8}{15}\right)\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

    18. *-lowering-*.f64 99.6%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\frac{4}{9}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{8}{15}\right)\right)\right), 4\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{7}\right)\right)\right)\right)\right)\right)\right), -2\right)\right)\right) \]

12. Simplified 99.6%

    \[\leadsto \varepsilon \cdot \frac{\color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(0.4444444444444444 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.5333333333333333\right)} - 4}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.4 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.2857142857142857\right)\right)\right)\right) - -2} \]
13. Add Preprocessing

Alternative 3: 99.8% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = eps * ((-2.0d0) + ((eps * eps) * ((-0.6666666666666666d0) + ((eps * eps) * ((-0.4d0) + (eps * (eps * (-0.2857142857142857d0))))))))
end function

Java

public static double code(double eps) {
	return eps * (-2.0 + ((eps * eps) * (-0.6666666666666666 + ((eps * eps) * (-0.4 + (eps * (eps * -0.2857142857142857)))))));
}

Python

def code(eps):
	return eps * (-2.0 + ((eps * eps) * (-0.6666666666666666 + ((eps * eps) * (-0.4 + (eps * (eps * -0.2857142857142857)))))))

Julia

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

MATLAB

function tmp = code(eps)
	tmp = eps * (-2.0 + ((eps * eps) * (-0.6666666666666666 + ((eps * eps) * (-0.4 + (eps * (eps * -0.2857142857142857)))))));
end

Wolfram

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

Derivation

1. Initial program 9.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) + -2\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(-2 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)} - \frac{2}{3}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)} - \frac{2}{3}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) + \frac{-2}{3}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{3} + \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)}\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)\right)}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right)}\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{-2}{7} \cdot {\varepsilon}^{2}} - \frac{2}{5}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{-2}{7} \cdot {\varepsilon}^{2}} - \frac{2}{5}\right)\right)\right)\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{7} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{5}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{7} \cdot {\varepsilon}^{2} + \frac{-2}{5}\right)\right)\right)\right)\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{5} + \color{blue}{\frac{-2}{7} \cdot {\varepsilon}^{2}}\right)\right)\right)\right)\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \color{blue}{\left(\frac{-2}{7} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \left(\frac{-2}{7} \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right)\right)\right) \]

   21. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \left(\left(\frac{-2}{7} \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right)\right) \]

   22. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \left(\varepsilon \cdot \color{blue}{\left(\frac{-2}{7} \cdot \varepsilon\right)}\right)\right)\right)\right)\right)\right)\right) \]

   23. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{5}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-2}{7} \cdot \varepsilon\right)}\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(-2 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.6666666666666666 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.4 + \varepsilon \cdot \left(\varepsilon \cdot -0.2857142857142857\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 4: 99.7% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = ((eps * (eps * eps)) * ((-0.6666666666666666d0) + (eps * (eps * (-0.4d0))))) + (eps * (-2.0d0))
end function

Java

public static double code(double eps) {
	return ((eps * (eps * eps)) * (-0.6666666666666666 + (eps * (eps * -0.4)))) + (eps * -2.0);
}

Python

def code(eps):
	return ((eps * (eps * eps)) * (-0.6666666666666666 + (eps * (eps * -0.4)))) + (eps * -2.0)

Julia

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

MATLAB

function tmp = code(eps)
	tmp = ((eps * (eps * eps)) * (-0.6666666666666666 + (eps * (eps * -0.4)))) + (eps * -2.0);
end

Wolfram

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

Derivation

1. Initial program 9.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) - 2\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) + -2\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(-2 + \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}} - \frac{2}{3}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}} - \frac{2}{3}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{5} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{5} \cdot {\varepsilon}^{2} + \frac{-2}{3}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{3} + \color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}}\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \color{blue}{\left(\frac{-2}{5} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{-2}{5}}\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{-2}{5}}\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-2}{5}\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{5}\right)\right)\right)\right)\right) \]

5. Simplified 99.4%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{5}\right)\right) \cdot \varepsilon\right), \color{blue}{\left(-2 \cdot \varepsilon\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{5}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right), \left(-2 \cdot \varepsilon\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{5}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right), \left(\color{blue}{-2} \cdot \varepsilon\right)\right) \]

   6. unpow3 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{5}\right) \cdot {\varepsilon}^{3}\right), \left(-2 \cdot \varepsilon\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2}{3} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{5}\right), \left({\varepsilon}^{3}\right)\right), \left(\color{blue}{-2} \cdot \varepsilon\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{5}\right)\right), \left({\varepsilon}^{3}\right)\right), \left(-2 \cdot \varepsilon\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{-2}{5}\right)\right)\right), \left({\varepsilon}^{3}\right)\right), \left(-2 \cdot \varepsilon\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{-2}{5}\right)\right)\right), \left({\varepsilon}^{3}\right)\right), \left(-2 \cdot \varepsilon\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-2}{5}\right)\right)\right), \left({\varepsilon}^{3}\right)\right), \left(-2 \cdot \varepsilon\right)\right) \]

   12. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-2}{5}\right)\right)\right), \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(-2 \cdot \varepsilon\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-2}{5}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(-2 \cdot \varepsilon\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-2}{5}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(-2 \cdot \varepsilon\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-2}{5}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(\varepsilon \cdot \color{blue}{-2}\right)\right) \]

   16. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-2}{5}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{-2}\right)\right) \]

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

   \[\leadsto \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.4\right)\right) + \varepsilon \cdot -2 \]
9. Add Preprocessing

Alternative 5: 99.7% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = eps * ((-2.0d0) + ((eps * eps) * ((-0.6666666666666666d0) + ((eps * eps) * (-0.4d0)))))
end function

Java

public static double code(double eps) {
	return eps * (-2.0 + ((eps * eps) * (-0.6666666666666666 + ((eps * eps) * -0.4))));
}

Python

def code(eps):
	return eps * (-2.0 + ((eps * eps) * (-0.6666666666666666 + ((eps * eps) * -0.4))))

Julia

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

MATLAB

function tmp = code(eps)
	tmp = eps * (-2.0 + ((eps * eps) * (-0.6666666666666666 + ((eps * eps) * -0.4))));
end

Wolfram

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

Derivation

1. Initial program 9.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) - 2\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) + -2\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(-2 + \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}} - \frac{2}{3}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}} - \frac{2}{3}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{5} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{5} \cdot {\varepsilon}^{2} + \frac{-2}{3}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{3} + \color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}}\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \color{blue}{\left(\frac{-2}{5} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{-2}{5}}\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{-2}{5}}\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-2}{5}\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{5}\right)\right)\right)\right)\right) \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(-2 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.6666666666666666 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.4\right)\right)} \]
6. Add Preprocessing

Alternative 6: 99.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps * (-2.0d0)) + (eps * ((eps * eps) * (-0.6666666666666666d0)))
end function

Java

public static double code(double eps) {
	return (eps * -2.0) + (eps * ((eps * eps) * -0.6666666666666666));
}

Python

def code(eps):
	return (eps * -2.0) + (eps * ((eps * eps) * -0.6666666666666666))

Julia

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

MATLAB

function tmp = code(eps)
	tmp = (eps * -2.0) + (eps * ((eps * eps) * -0.6666666666666666));
end

Wolfram

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

Derivation

1. Initial program 9.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) - 2\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) + -2\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(-2 + \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}} - \frac{2}{3}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}} - \frac{2}{3}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{5} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{5} \cdot {\varepsilon}^{2} + \frac{-2}{3}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-2}{3} + \color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}}\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \color{blue}{\left(\frac{-2}{5} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{-2}{5}}\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{-2}{5}}\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-2}{5}\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{5}\right)\right)\right)\right)\right) \]

5. Simplified 99.4%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \color{blue}{\frac{-2}{3}}\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 99.2%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{3}\right) \cdot \varepsilon\right), \color{blue}{\left(\varepsilon \cdot -2\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{3}\right)\right), \left(\color{blue}{\varepsilon} \cdot -2\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{3}\right)\right), \left(\color{blue}{\varepsilon} \cdot -2\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}\right)\right), \left(\varepsilon \cdot -2\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{3}\right)\right), \left(\varepsilon \cdot -2\right)\right) \]

   9. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-2}{3}\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{-2}\right)\right) \]

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

    \[\leadsto \varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.6666666666666666\right) \]
12. Add Preprocessing

Alternative 7: 99.5% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = eps * ((-2.0d0) + ((eps * eps) * (-0.6666666666666666d0)))
end function

Java

public static double code(double eps) {
	return eps * (-2.0 + ((eps * eps) * -0.6666666666666666));
}

Python

def code(eps):
	return eps * (-2.0 + ((eps * eps) * -0.6666666666666666))

Julia

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

MATLAB

function tmp = code(eps)
	tmp = eps * (-2.0 + ((eps * eps) * -0.6666666666666666));
end

Wolfram

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

Derivation

1. Initial program 9.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{-2}{3} \cdot {\varepsilon}^{2} + -2\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(-2 + \color{blue}{\frac{-2}{3} \cdot {\varepsilon}^{2}}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\frac{-2}{3}, \color{blue}{\left({\varepsilon}^{2}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\frac{-2}{3}, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

5. Simplified 99.2%

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

6. Final simplification 99.2%

   \[\leadsto \varepsilon \cdot \left(-2 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.6666666666666666\right) \]
7. Add Preprocessing

Alternative 8: 99.0% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(eps):
	return eps * -2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 98.8%

      \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{\varepsilon}\right) \]

5. Simplified 98.8%

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

6. Final simplification 98.8%

   \[\leadsto \varepsilon \cdot -2 \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.5× 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 2024138 
(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))))