logq (problem 3.4.3)

Percentage Accurate: 8.5% → 99.8%

Time: 9.1s

Alternatives: 7

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: 8.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

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

   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. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), -0.2857142857142857, -0.4\right), -0.6666666666666666\right), -2\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)\right) + \frac{-2}{3}\right)\right)\right) + \varepsilon \cdot -2} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)\right) + \frac{-2}{3}\right)\right)} + \varepsilon \cdot -2 \]

   3. *-commutative N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)\right) + \frac{-2}{3}\right)\right) + \color{blue}{-2 \cdot \varepsilon} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)\right) + \frac{-2}{3}\right), -2 \cdot \varepsilon\right)} \]

   5. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon + 0}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)\right) + \frac{-2}{3}\right), -2 \cdot \varepsilon\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right)}, \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)\right) + \frac{-2}{3}\right), -2 \cdot \varepsilon\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)\right) + \frac{-2}{3}\right)}, -2 \cdot \varepsilon\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right), \frac{-2}{3}\right)}, -2 \cdot \varepsilon\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)}, \frac{-2}{3}\right), -2 \cdot \varepsilon\right) \]

   10. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-2}{7} + \frac{-2}{5}\right), \frac{-2}{3}\right), -2 \cdot \varepsilon\right) \]

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

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-2}{7}, \frac{-2}{5}\right)}, \frac{-2}{3}\right), -2 \cdot \varepsilon\right) \]

   12. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon + 0}, \frac{-2}{7}, \frac{-2}{5}\right), \frac{-2}{3}\right), -2 \cdot \varepsilon\right) \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right)}, \frac{-2}{7}, \frac{-2}{5}\right), \frac{-2}{3}\right), -2 \cdot \varepsilon\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \frac{-2}{7}, \frac{-2}{5}\right), \frac{-2}{3}\right), \color{blue}{\varepsilon \cdot -2}\right) \]

   15. *-lowering-*.f64 99.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), -0.2857142857142857, -0.4\right), -0.6666666666666666\right), \color{blue}{\varepsilon \cdot -2}\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), -0.2857142857142857, -0.4\right), -0.6666666666666666\right), \varepsilon \cdot -2\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)\right) \cdot \varepsilon} + \frac{-2}{3}\right), \varepsilon \cdot -2\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon + 0\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right), \varepsilon, \frac{-2}{3}\right)}, \varepsilon \cdot -2\right) \]

   3. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-2}{7} + \frac{-2}{5}\right), \varepsilon, \frac{-2}{3}\right), \varepsilon \cdot -2\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-2}{7} + \frac{-2}{5}\right)}, \varepsilon, \frac{-2}{3}\right), \varepsilon \cdot -2\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{-2}{7}\right)} + \frac{-2}{5}\right), \varepsilon, \frac{-2}{3}\right), \varepsilon \cdot -2\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-2}{7}, \frac{-2}{5}\right)}, \varepsilon, \frac{-2}{3}\right), \varepsilon \cdot -2\right) \]

   7. *-lowering-*.f64 99.9

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot -0.2857142857142857}, -0.4\right), \varepsilon, -0.6666666666666666\right), \varepsilon \cdot -2\right) \]

9. Applied egg-rr 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.2857142857142857, -0.4\right), \varepsilon, -0.6666666666666666\right)}, \varepsilon \cdot -2\right) \]
10. Step-by-step derivation

    1. +-rgt-identity N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

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

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

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

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

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

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

    7. *-commutative N/A

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

    8. accelerator-lowering-fma.f64 N/A

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

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

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

    10. accelerator-lowering-fma.f64 N/A

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

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

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

    12. *-lowering-*.f64 99.9

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 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 8.0%

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

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

   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. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), -0.2857142857142857, -0.4\right), -0.6666666666666666\right), -2\right)} \]

6. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

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

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

8. Simplified 99.9%

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

Alternative 3: 99.7% accurate, 4.2× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

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

   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. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-commutative N/A

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

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

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

   10. sub-neg N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

   16. *-commutative N/A

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

   17. *-lowering-*.f64 99.8

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

5. Simplified 99.8%

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

Alternative 4: 99.5% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

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

   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. accelerator-lowering-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. *-lowering-*.f64 99.8

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

5. Simplified 99.8%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

   5. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon + 0}, \varepsilon \cdot \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right)}, \varepsilon \cdot \frac{-2}{3}, -2 \cdot \varepsilon\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \color{blue}{\varepsilon \cdot \frac{-2}{3}}, -2 \cdot \varepsilon\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot \frac{-2}{3}, \color{blue}{\varepsilon \cdot -2}\right) \]

   9. *-lowering-*.f64 99.8

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot -0.6666666666666666, \color{blue}{\varepsilon \cdot -2}\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon, 0\right), \varepsilon \cdot -0.6666666666666666, \varepsilon \cdot -2\right)} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 99.8

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

9. Applied egg-rr 99.8%

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

Alternative 5: 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 8.0%

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

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

   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. accelerator-lowering-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. *-lowering-*.f64 99.8

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

5. Simplified 99.8%

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

Alternative 6: 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 8.0%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 99.4

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 7: 5.3% 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 8.0%

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

4. Applied egg-rr 5.2%

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

Developer Target 1: 100.0% accurate, 0.6× 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 2024198 
(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))))