Average Error: 52.8 → 38.8
Time: 3.6m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.537524988239782 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{(\left(i + \left(\beta + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{\frac{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_*}{i \cdot \left(i + \left(\beta + \alpha\right)\right)}}}{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_* - 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 5.537524988239782e+93

    1. Initial program 50.2

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

    2. Simplified50.2

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

    4. Applied associate-/l*35.3

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

    if 5.537524988239782e+93 < alpha

    1. Initial program 61.2

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

    2. Simplified61.2

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

    3. Taylor expanded around inf 50.1

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification38.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.537524988239782 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{(\left(i + \left(\beta + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{\frac{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_*}{i \cdot \left(i + \left(\beta + \alpha\right)\right)}}}{(2 \cdot i + \left(\beta + \alpha\right))_* \cdot (2 \cdot i + \left(\beta + \alpha\right))_* - 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))