Average Error: 23.2 → 12.1
Time: 1.6m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\log \left(e^{(\left(\frac{1}{\sqrt{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}} \cdot \frac{\beta + \alpha}{\sqrt{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}\right) + 1.0)_*}\right)}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.2

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

  2. Applied simplify12.0

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

  4. Applied add-log-exp12.0

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

  6. Applied add-sqr-sqrt12.1

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

  7. Applied *-un-lft-identity12.1

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

  8. Applied times-frac12.1

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018170 +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))