logq (problem 3.4.3)

Average Error: 58.7 → 0.2

Time: 31.8s

Precision: 64

Math

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

↓

\[(\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\frac{\left({\left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)}^{3} - 8\right) \cdot \varepsilon}{(2 \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) + \left((\left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) + 4)_*\right))_*}\right))_*\]

C

double f(double eps) {
        double r14770178 = 1.0;
        double r14770179 = eps;
        double r14770180 = r14770178 - r14770179;
        double r14770181 = r14770178 + r14770179;
        double r14770182 = r14770180 / r14770181;
        double r14770183 = log(r14770182);
        return r14770183;
}

↓

double f(double eps) {
        double r14770184 = eps;
        double r14770185 = 5.0;
        double r14770186 = pow(r14770184, r14770185);
        double r14770187 = -0.4;
        double r14770188 = -0.6666666666666666;
        double r14770189 = r14770188 * r14770184;
        double r14770190 = r14770189 * r14770184;
        double r14770191 = 3.0;
        double r14770192 = pow(r14770190, r14770191);
        double r14770193 = 8.0;
        double r14770194 = r14770192 - r14770193;
        double r14770195 = r14770194 * r14770184;
        double r14770196 = 2.0;
        double r14770197 = 4.0;
        double r14770198 = fma(r14770190, r14770190, r14770197);
        double r14770199 = fma(r14770196, r14770190, r14770198);
        double r14770200 = r14770195 / r14770199;
        double r14770201 = fma(r14770186, r14770187, r14770200);
        return r14770201;
}

Error

Bits error versus eps

Target

Original	58.7
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 58.7

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

2. Taylor expanded around 0 0.2

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

3. Simplified 0.2

   \[\leadsto \color{blue}{(\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \frac{-2}{3}\right) \cdot \varepsilon - 2\right)\right))_*}\]
4. Using strategy rm

5. Applied flip3-- 0.2

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

6. Applied associate-*r/ 0.2

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

7. Simplified 0.2

   \[\leadsto (\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\frac{\varepsilon \cdot \left({\left(\left(\varepsilon \cdot \frac{-2}{3}\right) \cdot \varepsilon\right)}^{3} - {2}^{3}\right)}{\color{blue}{(2 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{-2}{3}\right)\right) + \left((\left(\varepsilon \cdot \left(\varepsilon \cdot \frac{-2}{3}\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{-2}{3}\right)\right) + 4)_*\right))_*}}\right))_*\]

8. Final simplification 0.2

   \[\leadsto (\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\frac{\left({\left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)}^{3} - 8\right) \cdot \varepsilon}{(2 \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) + \left((\left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) + 4)_*\right))_*}\right))_*\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))