Average Error: 58.6 → 0.1
Time: 2.5min
Precision: binary64
Cost: 26880
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\left(\sqrt[3]{\varepsilon}\right)}^{6} \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\left(\sqrt[3]{\varepsilon}\right)}^{6} \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right)\right)
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps)
 :precision binary64
 (+
  (* eps -2.0)
  (+
   (* (pow eps 7.0) -0.2857142857142857)
   (+
    (* (pow eps 5.0) -0.4)
    (* (pow (cbrt eps) 6.0) (* eps -0.6666666666666666))))))
double code(double eps) {
	return log((1.0 - eps) / (1.0 + eps));
}
↓
double code(double eps) {
	return (eps * -2.0) + ((pow(eps, 7.0) * -0.2857142857142857) + ((pow(eps, 5.0) * -0.4) + (pow(cbrt(eps), 6.0) * (eps * -0.6666666666666666))));
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.2
Herbie0.1
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Alternatives

Alternative 1
Error0.1
Cost14144
\[\varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + \left(\varepsilon \cdot -0.6666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\]
Alternative 2
Error0.2
Cost13632
\[{\varepsilon}^{5} \cdot -0.4 + \left(\varepsilon \cdot -2 - 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\]
Alternative 3
Error0.3
Cost13632
\[\varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + -0.6666666666666666 \cdot {\varepsilon}^{3}\right)\]
Alternative 4
Error0.3
Cost6912
\[\varepsilon \cdot -2 - 0.6666666666666666 \cdot {\varepsilon}^{3}\]
Alternative 5
Error0.6
Cost192
\[\varepsilon \cdot -2\]
Alternative 6
Error52.0
Cost128
\[-\varepsilon\]
Alternative 7
Error60.6
Cost64
\[0\]
Alternative 8
Error60.5
Cost385
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.63963353512124 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 9
Error61.5
Cost64
\[1\]

Error

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{-\left(0.4 \cdot {\varepsilon}^{5} + \left(0.2857142857142857 \cdot {\varepsilon}^{7} + \left(2 \cdot \varepsilon + 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right)}\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot -2 - {\varepsilon}^{3} \cdot 0.6666666666666666\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}}\]
  4. Using strategy rm

  5. Applied sub-neg_binary64_7530.1

    \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot -2 + \left(-{\varepsilon}^{3} \cdot 0.6666666666666666\right)\right)} - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]

  6. Applied associate--l+_binary64_6970.1

    \[\leadsto \color{blue}{\left(\varepsilon \cdot -2 + \left(\left(-{\varepsilon}^{3} \cdot 0.6666666666666666\right) - 0.4 \cdot {\varepsilon}^{5}\right)\right)} - 0.2857142857142857 \cdot {\varepsilon}^{7}\]

  7. Applied associate--l+_binary64_6970.1

    \[\leadsto \color{blue}{\varepsilon \cdot -2 + \left(\left(\left(-{\varepsilon}^{3} \cdot 0.6666666666666666\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\right)}\]

  8. Simplified0.1

    \[\leadsto \varepsilon \cdot -2 + \color{blue}{\left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\varepsilon}^{3} \cdot -0.6666666666666666\right)\right)}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt_binary64_7950.1

    \[\leadsto \varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\color{blue}{\left(\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \sqrt[3]{\varepsilon}\right)}}^{3} \cdot -0.6666666666666666\right)\right)\]

  11. Applied unpow-prod-down_binary64_8390.1

    \[\leadsto \varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + \color{blue}{\left({\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{3} \cdot {\left(\sqrt[3]{\varepsilon}\right)}^{3}\right)} \cdot -0.6666666666666666\right)\right)\]

  12. Applied associate-*l*_binary64_7010.1

    \[\leadsto \varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + \color{blue}{{\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{3} \cdot \left({\left(\sqrt[3]{\varepsilon}\right)}^{3} \cdot -0.6666666666666666\right)}\right)\right)\]

  13. Simplified0.1

    \[\leadsto \varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{3} \cdot \color{blue}{\left(\varepsilon \cdot -0.6666666666666666\right)}\right)\right)\]
  14. Using strategy rm

  15. Applied pow2_binary64_8410.1

    \[\leadsto \varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\color{blue}{\left({\left(\sqrt[3]{\varepsilon}\right)}^{2}\right)}}^{3} \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right)\right)\]

  16. Applied pow-pow_binary64_8320.1

    \[\leadsto \varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + \color{blue}{{\left(\sqrt[3]{\varepsilon}\right)}^{\left(2 \cdot 3\right)}} \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right)\right)\]

  17. Simplified0.1

    \[\leadsto \varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\left(\sqrt[3]{\varepsilon}\right)}^{\color{blue}{6}} \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right)\right)\]

  18. Simplified0.1

    \[\leadsto \color{blue}{\varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\left(\sqrt[3]{\varepsilon}\right)}^{6} \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right)\right)}\]

  19. Final simplification0.1

    \[\leadsto \varepsilon \cdot -2 + \left({\varepsilon}^{7} \cdot -0.2857142857142857 + \left({\varepsilon}^{5} \cdot -0.4 + {\left(\sqrt[3]{\varepsilon}\right)}^{6} \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right)\right)\]

Reproduce

herbie shell --seed 2021014 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

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