Average Error: 52.3 → 34.1
Time: 2.4m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 6.69854170595899 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}{\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*\right) \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(3 \cdot \left(\frac{\beta}{i} + \frac{\alpha}{i}\right) + 8)_*}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 6.69854170595899e+153

    1. Initial program 42.7

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

    2. Applied simplify15.5

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

    4. Applied clear-num15.5

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

    if 6.69854170595899e+153 < i

    1. Initial program 62.1

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

    2. Applied simplify62.1

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

    4. Applied clear-num62.1

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

    5. Taylor expanded around 0 53.1

      \[\leadsto \frac{1}{\color{blue}{8 + \left(3 \cdot \frac{\beta}{i} + 3 \cdot \frac{\alpha}{i}\right)}}\]

    6. Applied simplify53.1

      \[\leadsto \color{blue}{\frac{1}{(3 \cdot \left(\frac{\beta}{i} + \frac{\alpha}{i}\right) + 8)_*}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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)))