logq (problem 3.4.3)

Percentage Accurate: 8.4% → 99.8%

Time: 11.6s

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: 8.4% 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: 99.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.044444444444444446 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.02328042328042328\right)\right)\right) + -0.5} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (/
  eps
  (+
   (*
    (* eps eps)
    (+
     0.16666666666666666
     (*
      eps
      (* eps (+ 0.044444444444444446 (* (* eps eps) 0.02328042328042328))))))
   -0.5)))

C

double code(double eps) {
	return eps / (((eps * eps) * (0.16666666666666666 + (eps * (eps * (0.044444444444444446 + ((eps * eps) * 0.02328042328042328)))))) + -0.5);
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = eps / (((eps * eps) * (0.16666666666666666d0 + (eps * (eps * (0.044444444444444446d0 + ((eps * eps) * 0.02328042328042328d0)))))) + (-0.5d0))
end function

Java

public static double code(double eps) {
	return eps / (((eps * eps) * (0.16666666666666666 + (eps * (eps * (0.044444444444444446 + ((eps * eps) * 0.02328042328042328)))))) + -0.5);
}

Python

def code(eps):
	return eps / (((eps * eps) * (0.16666666666666666 + (eps * (eps * (0.044444444444444446 + ((eps * eps) * 0.02328042328042328)))))) + -0.5)

Julia

function code(eps)
	return Float64(eps / Float64(Float64(Float64(eps * eps) * Float64(0.16666666666666666 + Float64(eps * Float64(eps * Float64(0.044444444444444446 + Float64(Float64(eps * eps) * 0.02328042328042328)))))) + -0.5))
end

MATLAB

function tmp = code(eps)
	tmp = eps / (((eps * eps) * (0.16666666666666666 + (eps * (eps * (0.044444444444444446 + ((eps * eps) * 0.02328042328042328)))))) + -0.5);
end

Wolfram

code[eps_] := N[(eps / N[(N[(N[(eps * eps), $MachinePrecision] * N[(0.16666666666666666 + N[(eps * N[(eps * N[(0.044444444444444446 + N[(N[(eps * eps), $MachinePrecision] * 0.02328042328042328), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

5. Simplified 99.7%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{-2 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.4 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.2857142857142857\right)\right)\right)}}} \]

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + {\varepsilon}^{2} \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right) - \frac{1}{2}\right)}\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + {\varepsilon}^{2} \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + {\varepsilon}^{2} \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \left(\frac{1}{6} + {\varepsilon}^{2} \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\frac{1}{6} + {\varepsilon}^{2} \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{6} + {\varepsilon}^{2} \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({\varepsilon}^{2} \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{2}{45}, \left(\frac{22}{945} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{2}{45}, \left({\varepsilon}^{2} \cdot \frac{22}{945}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{2}{45}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{22}{945}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{2}{45}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{22}{945}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{2}{45}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{22}{945}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   16. metadata-eval 99.7%

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{2}{45}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{22}{945}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

10. Simplified 99.7%

    \[\leadsto \frac{\varepsilon}{\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.044444444444444446 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.02328042328042328\right)\right)\right) + -0.5}} \]
11. Add Preprocessing

Alternative 2: 99.8% accurate, 5.1× speedup

Math

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

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(eps * Float64(eps * Float64(-0.6666666666666666 + Float64(eps * Float64(eps * Float64(-0.4 + Float64(Float64(eps * 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[(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]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

5. Simplified 99.7%

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

Alternative 3: 99.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{-0.5 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.044444444444444446\right)\right)} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (/
  eps
  (+
   -0.5
   (*
    eps
    (* eps (+ 0.16666666666666666 (* (* eps eps) 0.044444444444444446)))))))

C

double code(double eps) {
	return eps / (-0.5 + (eps * (eps * (0.16666666666666666 + ((eps * eps) * 0.044444444444444446)))));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = eps / ((-0.5d0) + (eps * (eps * (0.16666666666666666d0 + ((eps * eps) * 0.044444444444444446d0)))))
end function

Java

public static double code(double eps) {
	return eps / (-0.5 + (eps * (eps * (0.16666666666666666 + ((eps * eps) * 0.044444444444444446)))));
}

Python

def code(eps):
	return eps / (-0.5 + (eps * (eps * (0.16666666666666666 + ((eps * eps) * 0.044444444444444446)))))

Julia

function code(eps)
	return Float64(eps / Float64(-0.5 + Float64(eps * Float64(eps * Float64(0.16666666666666666 + Float64(Float64(eps * eps) * 0.044444444444444446))))))
end

MATLAB

function tmp = code(eps)
	tmp = eps / (-0.5 + (eps * (eps * (0.16666666666666666 + ((eps * eps) * 0.044444444444444446)))));
end

Wolfram

code[eps_] := N[(eps / N[(-0.5 + N[(eps * N[(eps * N[(0.16666666666666666 + N[(N[(eps * eps), $MachinePrecision] * 0.044444444444444446), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

5. Simplified 99.7%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{-2 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.4 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.2857142857142857\right)\right)\right)}}} \]

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + \frac{2}{45} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)}\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + \frac{2}{45} \cdot {\varepsilon}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left({\varepsilon}^{2} \cdot \left(\frac{1}{6} + \frac{2}{45} \cdot {\varepsilon}^{2}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{6} + \frac{2}{45} \cdot {\varepsilon}^{2}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{6} + \frac{2}{45} \cdot {\varepsilon}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{1}{6} + \frac{2}{45} \cdot {\varepsilon}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{6} + \frac{2}{45} \cdot {\varepsilon}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{2}{45} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{6}, \left({\varepsilon}^{2} \cdot \frac{2}{45}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{2}{45}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{2}{45}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{2}{45}\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   12. metadata-eval 99.5%

       \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{2}{45}\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

10. Simplified 99.5%

    \[\leadsto \frac{\varepsilon}{\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.044444444444444446\right)\right) + -0.5}} \]

11. Final simplification 99.5%

    \[\leadsto \frac{\varepsilon}{-0.5 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.044444444444444446\right)\right)} \]
12. Add Preprocessing

Alternative 4: 99.7% accurate, 7.1× speedup

Math

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

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(eps * Float64(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[(eps * N[(eps * N[(-0.6666666666666666 + N[(N[(eps * eps), $MachinePrecision] * -0.4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

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

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

   16. *-commutative N/A

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

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

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

   18. unpow2 N/A

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

   19. *-lowering-*.f64 99.5%

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

5. Simplified 99.5%

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

Alternative 5: 99.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.16666666666666666} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (/ eps (+ -0.5 (* (* eps eps) 0.16666666666666666))))

C

double code(double eps) {
	return eps / (-0.5 + ((eps * eps) * 0.16666666666666666));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = eps / ((-0.5d0) + ((eps * eps) * 0.16666666666666666d0))
end function

Java

public static double code(double eps) {
	return eps / (-0.5 + ((eps * eps) * 0.16666666666666666));
}

Python

def code(eps):
	return eps / (-0.5 + ((eps * eps) * 0.16666666666666666))

Julia

function code(eps)
	return Float64(eps / Float64(-0.5 + Float64(Float64(eps * eps) * 0.16666666666666666)))
end

MATLAB

function tmp = code(eps)
	tmp = eps / (-0.5 + ((eps * eps) * 0.16666666666666666));
end

Wolfram

code[eps_] := N[(eps / N[(-0.5 + N[(N[(eps * eps), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

5. Simplified 99.7%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{1}{-2 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.6666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.4 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.2857142857142857\right)\right)\right)}}} \]

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left(\frac{1}{6} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)}\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \left(\frac{1}{6} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{1}{6} \cdot {\varepsilon}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left({\varepsilon}^{2} \cdot \frac{1}{6}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{1}{6}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{6}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{6}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   7. metadata-eval 99.2%

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{6}\right), \frac{-1}{2}\right)\right) \]

10. Simplified 99.2%

    \[\leadsto \frac{\varepsilon}{\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot 0.16666666666666666 + -0.5}} \]

11. Final simplification 99.2%

    \[\leadsto \frac{\varepsilon}{-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.16666666666666666} \]
12. Add Preprocessing

Alternative 6: 99.5% accurate, 11.9× speedup

Math

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

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(eps * Float64(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[(eps * N[(eps * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 99.1%

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

5. Simplified 99.1%

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

Alternative 7: 99.1% 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 9.3%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 98.3%

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

5. Simplified 98.3%

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

6. Final simplification 98.3%

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

Alternative 8: 5.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (eps) :precision binary64 0.0)

C

double code(double eps) {
	return 0.0;
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double eps) {
	return 0.0;
}

Python

def code(eps):
	return 0.0

Julia

function code(eps)
	return 0.0
end

MATLAB

function tmp = code(eps)
	tmp = 0.0;
end

Wolfram

code[eps_] := 0.0

Derivation

1. Initial program 9.3%

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

4. Applied egg-rr 5.2%

   \[\leadsto \color{blue}{0} \]
5. 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 2024191 
(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))))