Average Error: 52.7 → 12.9
Time: 3.1m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 5.236931143686764 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{\left(i + \alpha\right) + \beta}{\frac{\left(i + i\right) + \left(\alpha + \beta\right)}{i}} \cdot \frac{\left(\alpha + \beta\right) \cdot i + \left(i \cdot i + \beta \cdot \alpha\right)}{\left(i + i\right) + \left(\alpha + \beta\right)}}{\left(\left(i + i\right) + \left(\alpha + \beta\right)\right) \cdot \left(\left(i + i\right) + \left(\alpha + \beta\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\frac{0.25}{i}}{i}} \cdot \frac{1}{16}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 5.236931143686764e+131

    1. Initial program 39.8

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

    2. Taylor expanded around 0 39.8

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

    3. Applied simplify15.1

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

    if 5.236931143686764e+131 < 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. Taylor expanded around 0 62.1

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

    3. Applied simplify57.1

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

    5. Applied add-exp-log57.1

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

    6. Taylor expanded around inf 11.2

      \[\leadsto e^{\color{blue}{\log \frac{1}{16} + 0.25 \cdot \frac{1}{{i}^{2}}}}\]

    7. Applied simplify11.2

      \[\leadsto \color{blue}{e^{\frac{\frac{0.25}{i}}{i}} \cdot \frac{1}{16}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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)))