Average Error: 24.1 → 6.2
Time: 1.7m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{\left({\left(e^{\log \left((\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{(i \cdot 2 + \alpha)_* + \beta}\right) + 1.0)_*\right)}\right)}^{3}\right)}^{\frac{1}{3}}}{2.0} \le 6.8557558835442065 \cdot 10^{-09}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(2.0 - \frac{4.0}{\alpha}\right) + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right))_*}{2.0}\\ \mathbf{if}\;\frac{{\left({\left(e^{\log \left((\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{(i \cdot 2 + \alpha)_* + \beta}\right) + 1.0)_*\right)}\right)}^{3}\right)}^{\frac{1}{3}}}{2.0} \le +\infty:\\ \;\;\;\;\frac{{\left({\left(e^{\log \left((\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{(i \cdot 2 + \alpha)_* + \beta}\right) + 1.0)_*\right)}\right)}^{3}\right)}^{\frac{1}{3}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(e^{\log \left((\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + (2 \cdot i + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{(i \cdot 2 + \alpha)_* + \beta}\right) + 1.0)_*\right)}\right)}^{3}\right)}^{\frac{1}{3}}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (pow (pow (exp (log (fma (/ (+ alpha beta) (+ (+ alpha beta) (fma 2 i 2.0))) (/ (- beta alpha) (+ (fma i 2 alpha) beta)) 1.0))) 3) 1/3) 2.0) < 6.8557558835442065e-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. Applied simplify60.1

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

    4. Applied add-cbrt-cube60.1

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

    5. Applied simplify60.1

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

    6. Taylor expanded around inf 50.0

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

    7. Applied simplify28.9

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

    if 6.8557558835442065e-09 < (/ (pow (pow (exp (log (fma (/ (+ alpha beta) (+ (+ alpha beta) (fma 2 i 2.0))) (/ (- beta alpha) (+ (fma i 2 alpha) beta)) 1.0))) 3) 1/3) 2.0)

    1. Initial program 14.0

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

    2. Applied simplify0.1

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

    4. Applied add-cbrt-cube0.2

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

    5. Applied simplify0.2

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

    7. Applied pow1/30.1

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

    9. Applied add-exp-log0.1

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

Runtime

Time bar (total: 1.7m)Debug logProfile

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