Average Error: 52.4 → 37.3
Time: 5.9m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.00755826716117 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\left(i + \alpha\right) + \beta}{\frac{\left(i + i\right) + \left(\alpha + \beta\right)}{i}} \cdot \frac{\left(\alpha + \beta\right) \cdot i + \left(i \cdot i + \beta \cdot \alpha\right)}{\left(i + i\right) + \left(\alpha + \beta\right)}}{\left(\left(i + i\right) + \left(\alpha + \beta\right)\right) \cdot \left(\left(i + i\right) + \left(\alpha + \beta\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.00755826716117e+153

    1. Initial program 50.4

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

    2. Taylor expanded around 0 50.4

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

    3. Applied simplify34.9

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

    if 4.00755826716117e+153 < alpha

    1. Initial program 62.5

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

    2. Taylor expanded around inf 49.1

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 5.9m)Debug logProfile

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(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)))