Average Error: 52.7 → 37.6
Time: 1.7m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 2.2950245454369153 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\frac{\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*\right) \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}} \cdot \sqrt{\frac{\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*\right) \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.2950245454369153e+151

    1. Initial program 50.7

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

    2. Applied simplify35.1

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

    4. Applied add-sqr-sqrt35.1

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

    if 2.2950245454369153e+151 < beta

    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. Applied simplify56.3

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

    3. Taylor expanded around inf 49.3

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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)))