Average Error: 58.5 → 0.0
Time: 3.3m
Precision: 64
Internal Precision: 1344
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log_* (1 + \left(-\varepsilon\right)) + \left(-\log_* (1 + \varepsilon)\right)\]

Error

Bits error versus eps

Target

Original58.5
Target0.2
Herbie0.0
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.5

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied div-inv58.5

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

  4. Applied log-prod58.5

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

  5. Applied simplify50.4

    \[\leadsto \log \left(1 - \varepsilon\right) + \color{blue}{\left(-\log_* (1 + \varepsilon)\right)}\]
  6. Using strategy rm

  7. Applied log1p-expm1-u50.4

    \[\leadsto \color{blue}{\log_* (1 + (e^{\log \left(1 - \varepsilon\right)} - 1)^*)} + \left(-\log_* (1 + \varepsilon)\right)\]

  8. Applied simplify0.0

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +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))))