Average Error: 52.7 → 11.1
Time: 6.6m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{i}{4 \cdot \left(\beta + \alpha\right) + i \cdot 8} \cdot \frac{\left(\beta + \alpha\right) + i}{\left(i + \beta\right) + \left(i + \alpha\right)} \le 6.595186430306066 \cdot 10^{-147}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{i}{4 \cdot \left(\beta + \alpha\right) + i \cdot 8} \cdot \frac{\left(\beta + \alpha\right) + i}{\left(i + \beta\right) + \left(i + \alpha\right)}\right)}^{3}}\\ \mathbf{if}\;\frac{i}{4 \cdot \left(\beta + \alpha\right) + i \cdot 8} \cdot \frac{\left(\beta + \alpha\right) + i}{\left(i + \beta\right) + \left(i + \alpha\right)} \le 0.061562236797838885:\\ \;\;\;\;\frac{i}{\frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\left(\alpha + i\right) + \beta}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{4 \cdot \left(\alpha + \beta\right) + 8 \cdot i}}{\frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if (* (/ i (+ (* 4 (+ beta alpha)) (* i 8))) (/ (+ (+ beta alpha) i) (+ (+ i beta) (+ i alpha)))) < 6.595186430306066e-147

    1. Initial program 62.9

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

    3. Applied times-frac39.2

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

    4. Applied associate-/l*39.2

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

    5. Taylor expanded around 0 61.4

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{4 \cdot \beta + \left(4 \cdot \alpha + 8 \cdot i\right)}}\]

    6. Applied simplify59.8

      \[\leadsto \color{blue}{\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube15.9

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}} \cdot \frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}\right) \cdot \frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}}}\]

    9. Applied simplify15.9

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{i}{4 \cdot \left(\beta + \alpha\right) + i \cdot 8} \cdot \frac{\left(\beta + \alpha\right) + i}{\left(i + \beta\right) + \left(i + \alpha\right)}\right)}^{3}}}\]

    if 6.595186430306066e-147 < (* (/ i (+ (* 4 (+ beta alpha)) (* i 8))) (/ (+ (+ beta alpha) i) (+ (+ i beta) (+ i alpha)))) < 0.061562236797838885

    1. Initial program 53.2

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

    3. Applied *-un-lft-identity53.2

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

    4. Applied times-frac42.6

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

    5. Applied times-frac42.6

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

    6. Applied simplify42.6

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

    if 0.061562236797838885 < (* (/ i (+ (* 4 (+ beta alpha)) (* i 8))) (/ (+ (+ beta alpha) i) (+ (+ i beta) (+ i alpha))))

    1. Initial program 51.3

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

    3. Applied times-frac37.5

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

    4. Applied associate-/l*37.5

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

    5. Taylor expanded around 0 38.0

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{4 \cdot \beta + \left(4 \cdot \alpha + 8 \cdot i\right)}}\]

    6. Applied simplify1.0

      \[\leadsto \color{blue}{\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}}\]
    7. Using strategy rm

    8. Applied associate-/r*0.9

      \[\leadsto \color{blue}{\frac{\frac{i}{4 \cdot \left(\alpha + \beta\right) + 8 \cdot i}}{\frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 6.6m)Debug logProfile

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