Average Error: 52.4 → 38.2
Time: 2.0m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0768447842882578 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*\right) \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right) \cdot \frac{1}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.0768447842882578e+138

    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. Applied simplify35.2

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

    4. Applied div-inv35.3

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

    if 1.0768447842882578e+138 < alpha

    1. Initial program 62.3

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

    2. Applied simplify55.4

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

    3. Taylor expanded around inf 50.9

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +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)))