Average Error: 52.3 → 13.2
Time: 5.6m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 2.7149245189817217 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \frac{i}{4}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 2.7149245189817217e+127

    1. Initial program 38.7

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

    3. Applied times-frac15.8

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

    4. Applied associate-/l*15.7

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

    6. Applied add-sqr-sqrt15.7

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

    7. Applied associate-/l*15.8

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

    if 2.7149245189817217e+127 < 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. Using strategy rm

    3. Applied times-frac55.5

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

    4. Applied associate-/l*55.5

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

    6. Applied associate-/r/55.5

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

    7. Applied *-un-lft-identity55.5

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

    8. Applied times-frac55.4

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

    9. Applied times-frac55.5

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

    10. Taylor expanded around 0 11.4

      \[\leadsto \frac{\frac{i}{1}}{\color{blue}{4}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\alpha + \beta\right) + 2 \cdot i}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.7149245189817217 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1.0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \frac{i}{4}\\ \end{array}\]

Runtime

Time bar (total: 5.6m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes19.113.28.810.457.2%
herbie shell --seed 2018355 
(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)))