Average Error: 22.9 → 5.9
Time: 6.8m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\frac{\beta + \alpha}{\sqrt{i + \left(\left(2.0 + i\right) + \left(\beta + \alpha\right)\right)}} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt[3]{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt[3]{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt[3]{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0} \le 3.976882267942017 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1.0 + \left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\left|\sqrt[3]{\left(2.0 + i\right) + \left(\beta + \left(\alpha + i\right)\right)}\right|} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}}\right) \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* (* (/ (+ beta alpha) (sqrt (+ i (+ (+ 2.0 i) (+ beta alpha))))) (/ (* (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))) (* (cbrt (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))) (cbrt (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))))) (/ (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (cbrt (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))))) 1.0) 2.0) < 3.976882267942017e-09

    1. Initial program 62.1

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around inf 29.0

      \[\leadsto \frac{\color{blue}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Applied simplify29.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}}\]

    if 3.976882267942017e-09 < (/ (+ (* (* (/ (+ beta alpha) (sqrt (+ i (+ (+ 2.0 i) (+ beta alpha))))) (/ (* (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))))) (* (cbrt (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))) (cbrt (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))))) (/ (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2 i)))) (cbrt (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))))) 1.0) 2.0)

    1. Initial program 13.1

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt13.1

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity13.1

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    5. Applied times-frac0.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    6. Applied times-frac0.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    7. Applied simplify0.1

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{\sqrt{i + \left(\left(2.0 + i\right) + \left(\beta + \alpha\right)\right)}}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube0.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\frac{\beta + \alpha}{\sqrt{i + \left(\left(2.0 + i\right) + \left(\beta + \alpha\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{i + \left(\left(2.0 + i\right) + \left(\beta + \alpha\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{i + \left(\left(2.0 + i\right) + \left(\beta + \alpha\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}}{2.0}\]

    10. Applied simplify0.1

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(1.0 + \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}}{2.0}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt0.2

      \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)} \cdot \sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}\right) \cdot \sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}}} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}{2.0}\]

    13. Applied sqrt-prod0.2

      \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\color{blue}{\sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)} \cdot \sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}} \cdot \sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}}} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}{2.0}\]

    14. Applied *-un-lft-identity0.2

      \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \frac{\frac{\beta - \alpha}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + \left(i + i\right)\right)}}}{\sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)} \cdot \sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}} \cdot \sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}{2.0}\]

    15. Applied add-cube-cbrt0.2

      \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{1 \cdot \left(\left(\alpha + \beta\right) + \left(i + i\right)\right)}}{\sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)} \cdot \sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}} \cdot \sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}{2.0}\]

    16. Applied times-frac0.2

      \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + \left(i + i\right)}}}{\sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)} \cdot \sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}} \cdot \sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}{2.0}\]

    17. Applied times-frac0.2

      \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \color{blue}{\left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)} \cdot \sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}}\right)} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}{2.0}\]

    18. Applied simplify0.2

      \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \left(\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\left|\sqrt[3]{\left(2.0 + i\right) + \left(\beta + \left(\alpha + i\right)\right)}\right|}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\sqrt{\sqrt[3]{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}}\right) \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}{2.0}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 6.8m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))