Average Error: 23.6 → 4.2
Time: 2.5m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{\left(e^{\sqrt[3]{1}}\right)}^{\left(\sqrt[3]{{\left(\log \left((\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + (i \cdot 2 + 2.0)_*}\right) \cdot \left(\frac{\beta - \alpha}{\beta + (2 \cdot i + \alpha)_*}\right) + 1.0)_*\right)\right)}^{3}}\right)}}{2.0} \le 6.271856801363427 \cdot 10^{-13}:\\ \;\;\;\;\frac{(\left(-3.0\right) \cdot \left(\frac{\frac{2}{\alpha}}{\frac{\alpha}{\beta}}\right) + \left(\frac{2}{\alpha} \cdot \left(\beta + 1.0\right)\right))_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left((\left(\frac{1}{\sqrt{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}} \cdot \frac{\alpha + \beta}{\sqrt{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}}\right) \cdot \left(\frac{\beta - \alpha}{(i \cdot 2 + \left(\alpha + \beta\right))_*}\right) + 1.0)_*\right)}^{3}}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (pow (exp (cbrt 1)) (cbrt (pow (log (fma (/ (+ alpha beta) (+ (+ alpha beta) (fma i 2 2.0))) (/ (- beta alpha) (+ beta (fma 2 i alpha))) 1.0)) 3))) 2.0) < 6.271856801363427e-13

    1. Initial program 62.4

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

    2. Applied simplify60.4

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

    4. Applied add-cbrt-cube60.4

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

    5. Applied simplify60.4

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

    6. Taylor expanded around inf 44.2

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

    7. Applied simplify19.1

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

    if 6.271856801363427e-13 < (/ (pow (exp (cbrt 1)) (cbrt (pow (log (fma (/ (+ alpha beta) (+ (+ alpha beta) (fma i 2 2.0))) (/ (- beta alpha) (+ beta (fma 2 i alpha))) 1.0)) 3))) 2.0)

    1. Initial program 13.7

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

    2. Applied simplify0.3

      \[\leadsto \color{blue}{\frac{(\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}\right) + 1.0)_*}{2.0}}\]
    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-sqr-sqrt0.4

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

    8. Applied *-un-lft-identity0.4

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

    9. Applied times-frac0.4

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018166 +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))