Average Error: 22.8 → 12.0
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{(\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\sqrt[3]{{\left(\frac{\beta - \alpha}{\beta + (i \cdot 2 + \alpha)_*}\right)}^{3}}\right) + 1.0)_*}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 22.8

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

  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-cbrt-cube25.0

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

  5. Applied add-cbrt-cube30.4

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

  6. Applied cbrt-undiv30.4

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

  7. Applied simplify12.0

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018195 +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))