Average Error: 23.4 → 6.6
Time: 4.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left((\left(\frac{\alpha + \beta}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\sqrt[3]{(i \cdot 2 + \left(\alpha + \beta\right))_*}}}{\sqrt[3]{(i \cdot 2 + \left(\alpha + \beta\right))_*} \cdot \left(\left(\sqrt[3]{\sqrt[3]{(i \cdot 2 + \left(\alpha + \beta\right))_*}} \cdot \sqrt[3]{\sqrt[3]{(i \cdot 2 + \left(\alpha + \beta\right))_*}}\right) \cdot \sqrt[3]{\sqrt[3]{(i \cdot 2 + \left(\alpha + \beta\right))_*}}\right)}\right) + 1.0)_*\right)}^{3}}}{2.0} \le 5.685556192513985 \cdot 10^{-16}:\\ \;\;\;\;(\left(\frac{4.0}{2.0}\right) \cdot \left(\frac{-1}{\alpha \cdot \alpha}\right) + \left(\frac{1}{\alpha}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt[3]{(\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}\right) + 1.0)_*} \cdot \sqrt[3]{(\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}\right) + 1.0)_*}\right) + \log \left(\sqrt[3]{(\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}\right) + 1.0)_*}\right)}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (cbrt (pow (fma (/ (+ alpha beta) (+ (fma i 2 beta) (+ 2.0 alpha))) (/ (/ (- beta alpha) (cbrt (fma i 2 (+ alpha beta)))) (* (cbrt (fma i 2 (+ alpha beta))) (* (* (cbrt (cbrt (fma i 2 (+ alpha beta)))) (cbrt (cbrt (fma i 2 (+ alpha beta))))) (cbrt (cbrt (fma i 2 (+ alpha beta))))))) 1.0) 3)) 2.0) < 5.685556192513985e-16

    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. Applied simplify60.8

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

    4. Applied add-exp-log60.8

      \[\leadsto \frac{\color{blue}{e^{\log \left((\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}\right) + 1.0)_*\right)}}}{2.0}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt60.9

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

    7. Applied *-un-lft-identity60.9

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

    8. Applied times-frac60.9

      \[\leadsto \frac{e^{\log \left((\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{(2 \cdot i + \left(\beta + \alpha\right))_*} \cdot \sqrt[3]{(2 \cdot i + \left(\beta + \alpha\right))_*}} \cdot \frac{\beta - \alpha}{\sqrt[3]{(2 \cdot i + \left(\beta + \alpha\right))_*}}\right)} + 1.0)_*\right)}}{2.0}\]
    9. Using strategy rm

    10. Applied add-cbrt-cube60.9

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

    11. Applied simplify60.9

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

    12. Taylor expanded around inf 62.5

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

    13. Applied simplify28.3

      \[\leadsto \color{blue}{(\left(\frac{4.0}{2.0}\right) \cdot \left(\frac{-1}{\alpha \cdot \alpha}\right) + \left(\frac{1}{\alpha}\right))_*}\]

    if 5.685556192513985e-16 < (/ (cbrt (pow (fma (/ (+ alpha beta) (+ (fma i 2 beta) (+ 2.0 alpha))) (/ (/ (- beta alpha) (cbrt (fma i 2 (+ alpha beta)))) (* (cbrt (fma i 2 (+ alpha beta))) (* (* (cbrt (cbrt (fma i 2 (+ alpha beta)))) (cbrt (cbrt (fma i 2 (+ alpha beta))))) (cbrt (cbrt (fma i 2 (+ alpha beta))))))) 1.0) 3)) 2.0)

    1. Initial program 13.6

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

    2. Applied simplify0.8

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

    4. Applied add-exp-log0.8

      \[\leadsto \frac{\color{blue}{e^{\log \left((\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}\right) + 1.0)_*\right)}}}{2.0}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt1.3

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

    7. Applied log-prod1.3

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

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(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))