Average Error: 23.4 → 4.0
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left((\left(\frac{\alpha + \beta}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\sqrt[3]{\beta - \alpha}}{(i \cdot 2 + \left(\alpha + \beta\right))_*}\right) + 1.0)_*\right)}^{3}}}{2.0} \le 7.575589886591226 \cdot 10^{-15}:\\ \;\;\;\;\frac{(\left(-3.0\right) \cdot \left(\frac{\frac{2}{\alpha}}{\frac{\alpha}{\beta}}\right) + \left(\frac{2}{\alpha} \cdot \left(\beta + 1.0\right)\right))_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{e}^{\left(\log \left({\left((\left(\frac{\alpha + \beta}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\beta - \alpha}{(i \cdot 2 + \left(\alpha + \beta\right))_*}\right) + 1.0)_*\right)}^{3}\right)\right)}}}{2.0}\\ \end{array}\]

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))) (* (* (cbrt (- beta alpha)) (cbrt (- beta alpha))) (/ (cbrt (- beta alpha)) (fma i 2 (+ alpha beta)))) 1.0) 3)) 2.0) < 7.575589886591226e-15

    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 simplify60.6

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

    4. Applied add-cbrt-cube60.6

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

    5. Applied simplify60.6

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

    6. Taylor expanded around inf 44.5

      \[\leadsto \frac{\color{blue}{\left(e^{\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)} + 1.0 \cdot \frac{e^{\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)}}{\beta}\right) - 3.0 \cdot \frac{e^{\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)}}{\alpha}}}{2.0}\]

    7. Applied simplify18.6

      \[\leadsto \color{blue}{\frac{(\left(-3.0\right) \cdot \left(\frac{\frac{2}{\alpha}}{\frac{\alpha}{\beta}}\right) + \left(\frac{2}{\alpha} \cdot \left(\beta + 1.0\right)\right))_*}{2.0}}\]

    if 7.575589886591226e-15 < (/ (cbrt (pow (fma (/ (+ alpha beta) (+ (fma i 2 beta) (+ 2.0 alpha))) (* (* (cbrt (- beta alpha)) (cbrt (- beta alpha))) (/ (cbrt (- beta alpha)) (fma i 2 (+ alpha beta)))) 1.0) 3)) 2.0)

    1. Initial program 13.8

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

    2. Applied simplify0.4

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

    4. Applied add-cbrt-cube0.4

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

    5. Applied simplify0.4

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

    7. Applied add-exp-log0.6

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

    9. Applied *-un-lft-identity0.6

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

    10. Applied exp-prod0.4

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

    11. Applied simplify0.4

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018214 +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))