Average Error: 23.7 → 6.0
Time: 4.0m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt[3]{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}\right) \cdot \sqrt[3]{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}}{2.0} \le 5.551115123125783 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot \left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\beta + \left(\left(2.0 + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\beta + \left(\left(2.0 + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (* (cbrt (+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)) (cbrt (+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0))) (cbrt (+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0))) 2.0) < 5.551115123125783e-17

    1. Initial program 62.6

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

    2. Taylor expanded around inf 28.2

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

    3. Applied simplify28.2

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

    if 5.551115123125783e-17 < (/ (* (* (cbrt (+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0)) (cbrt (+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0))) (cbrt (+ (* (+ beta alpha) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0))) 2.0)

    1. Initial program 14.1

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

    3. Applied *-un-lft-identity14.1

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

    4. Applied *-un-lft-identity14.1

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

    5. Applied times-frac0.5

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

    6. Applied times-frac0.5

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

    7. Applied simplify0.5

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

    9. Applied add-cube-cbrt0.7

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

    10. Applied *-un-lft-identity0.7

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    11. Applied add-cube-cbrt0.5

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    12. Applied times-frac0.5

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    13. Applied times-frac0.5

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + 1.0}{2.0}\]

    14. Applied simplify0.5

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

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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))