Average Error: 22.9 → 11.7
Time: 5.4m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\log \left(e^{(\left(\log_* (1 + \sqrt[3]{\sqrt[3]{\left((e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^*\right) \cdot \left(\left((e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^*\right) \cdot \left((e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^*\right)\right)} \cdot (e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^*})\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*}\right)}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 22.9

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

  2. Initial simplification11.9

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

  4. Applied add-log-exp11.9

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

  6. Applied log1p-expm1-u11.9

    \[\leadsto \frac{\log \left(e^{(\color{blue}{\left(\log_* (1 + (e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^*)\right)} \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}\right)}{2.0}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube11.7

    \[\leadsto \frac{\log \left(e^{(\left(\log_* (1 + \color{blue}{\sqrt[3]{\left((e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^*\right) \cdot (e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^*}})\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}\right)}{2.0}\]
  9. Using strategy rm

  10. Applied add-cbrt-cube11.7

    \[\leadsto \frac{\log \left(e^{(\left(\log_* (1 + \sqrt[3]{\color{blue}{\sqrt[3]{\left(\left((e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^*\right) \cdot \left((e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^*\right)\right) \cdot \left((e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^*\right)}} \cdot (e^{\frac{\beta - \alpha}{\left(\beta + 2.0\right) + (2 \cdot i + \alpha)_*}} - 1)^*})\right) \cdot \left(\frac{\beta + \alpha}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}\right)}{2.0}\]

  11. Final simplification11.7

    \[\leadsto \frac{\log \left(e^{(\left(\log_* (1 + \sqrt[3]{\sqrt[3]{\left((e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^*\right) \cdot \left(\left((e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^*\right) \cdot \left((e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^* \cdot (e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^*\right)\right)} \cdot (e^{\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}} - 1)^*})\right) \cdot \left(\frac{\alpha + \beta}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*}\right)}{2.0}\]

Runtime

Time bar (total: 5.4m)Debug logProfile

herbie shell --seed 2018234 +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))