Average Error: 24.2 → 12.4
Time: 1.1m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\log \left(\sqrt{e^{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}}\right) + \log \left(\sqrt{e^{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}}\right)}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.2

    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

  2. Applied simplify12.4

    \[\leadsto \color{blue}{\frac{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}{2.0}}\]
  3. Using strategy rm

  4. Applied add-log-exp12.4

    \[\leadsto \frac{\color{blue}{\log \left(e^{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}\right)}}{2.0}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt12.4

    \[\leadsto \frac{\log \color{blue}{\left(\sqrt{e^{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}} \cdot \sqrt{e^{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}}\right)}}{2.0}\]

  7. Applied log-prod12.4

    \[\leadsto \frac{\color{blue}{\log \left(\sqrt{e^{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}}\right) + \log \left(\sqrt{e^{(\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right) + 1.0)_*}}\right)}}{2.0}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))