Average Error: 52.6 → 35.6
Time: 1.7m
Precision: 64
Internal precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
⬇
\[\frac{{\left(i \cdot \left(\left(\beta + \left(i + \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) + \left(i + i\right)}\right)\right)}^2}{{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right)}^2 - 1.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 52.6

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

  2. Applied taylor 52.5

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

  3. Taylor expanded around inf 52.5

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

  4. Applied simplify 35.5

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

  6. Applied div-inv 35.6

    \[\leadsto \frac{{\left(i \cdot \color{blue}{\left(\left(\beta + \left(i + \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) + \left(i + i\right)}\right)}\right)}^2}{{\left(\left(\beta + \alpha\right) + \left(i + i\right)\right)}^2 - 1.0}\]
  7. Removed slow pow expressions

Runtime

Time bar (total: 1.7m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3874281246 3958136478 2430629073 520591626 1792551642 3257977733)'
(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)))