Average Error: 22.9 → 12.1
Time: 35.1s
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\log \left((\left(\frac{\beta + \alpha}{\left(\beta + \alpha\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\log \left(e^{\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}}\right)\right) + 1.0)_*\right)}}{2.0}\]

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. Applied simplify12.1

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

  4. Applied add-exp-log12.1

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

  6. Applied add-log-exp12.1

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

Runtime

Time bar (total: 35.1s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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))