Average Error: 52.6 → 12.5
Time: 4.4m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 5.021059441134798 \cdot 10^{+42}:\\ \;\;\;\;\frac{\sqrt{\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}}{\sqrt{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}}{\sqrt{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{i}{\beta + (2 \cdot i + \alpha)_*}}\right) \cdot \frac{\beta + \left(\alpha + i\right)}{\beta + (2 \cdot i + \alpha)_*}}{4}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 5.021059441134798e+42

    1. Initial program 20.9

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

    3. Applied associate-/l*7.4

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

    5. Applied add-sqr-sqrt7.5

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

    6. Applied add-sqr-sqrt7.5

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

    7. Applied times-frac7.5

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

    8. Simplified7.5

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

    9. Simplified7.4

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

    if 5.021059441134798e+42 < i

    1. Initial program 57.4

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

    3. Applied associate-/l*43.4

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

    5. Applied associate-/r/43.4

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

    6. Applied associate-/l*43.4

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

    7. Simplified43.4

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

    8. Taylor expanded around 0 15.5

      \[\leadsto \frac{\frac{i}{(2 \cdot i + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(2 \cdot i + \alpha)_* + \beta}}{\color{blue}{4}}\]
    9. Using strategy rm

    10. Applied add-log-exp13.3

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{i}{(2 \cdot i + \alpha)_* + \beta}}\right)} \cdot \frac{\left(\alpha + i\right) + \beta}{(2 \cdot i + \alpha)_* + \beta}}{4}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 5.021059441134798 \cdot 10^{+42}:\\ \;\;\;\;\frac{\sqrt{\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}}{\sqrt{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}}{\sqrt{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{i}{\beta + (2 \cdot i + \alpha)_*}}\right) \cdot \frac{\beta + \left(\alpha + i\right)}{\beta + (2 \cdot i + \alpha)_*}}{4}\\ \end{array}\]

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed 2018336 +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)))