logq (problem 3.4.3)

Percentage Accurate: 8.3% → 100.0%

Time: 9.0s

Alternatives: 9

Speedup: 19.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. flip-- N/A

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

   2. lift-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 8.1

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

4. Applied egg-rr 8.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. flip-- N/A

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

   10. lift--.f64 N/A

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

   11. remove-double-div N/A

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

   12. lift-/.f64 N/A

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

   13. associate-/r/ N/A

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

   14. associate-/r/ N/A

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

6. Applied egg-rr 100.0%

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

Alternative 2: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.1

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

5. Simplified 99.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. pow3 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

7. Applied egg-rr 99.1%

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

Alternative 3: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.1

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

5. Simplified 99.1%

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

Alternative 4: 99.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.0

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

5. Simplified 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. pow3 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. cube-unmult N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 99.0

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.0

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

7. Applied egg-rr 99.0%

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

Alternative 5: 99.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.0

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

5. Simplified 99.0%

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

Alternative 6: 99.5% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.9

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

5. Simplified 98.9%

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

   1. lift-*.f64 N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lower-*.f64 98.9

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 98.9

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

7. Applied egg-rr 98.9%

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

Alternative 7: 99.5% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.9

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

5. Simplified 98.9%

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

Alternative 8: 99.0% accurate, 19.7× 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 8.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 98.6

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 9: 5.4% accurate, 118.0× 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 8.1%

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

4. Applied egg-rr 5.4%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(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))))