Average Error: 23.9 → 6.7
Time: 2.6m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta + \alpha}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \le 1.605930327664669 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \alpha}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* (/ (+ beta alpha) (* (cbrt (+ (+ beta alpha) (* 2 i))) (cbrt (+ (+ beta alpha) (* 2 i))))) (/ (/ (- beta alpha) (cbrt (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0) 2.0) < 1.605930327664669e-12

    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. Taylor expanded around inf 31.1

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

    3. Applied simplify31.1

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

    if 1.605930327664669e-12 < (/ (+ (* (/ (+ beta alpha) (* (cbrt (+ (+ beta alpha) (* 2 i))) (cbrt (+ (+ beta alpha) (* 2 i))))) (/ (/ (- beta alpha) (cbrt (+ (+ 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 add-cube-cbrt14.2

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

    5. Applied times-frac0.5

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

    6. Applied times-frac0.5

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

    7. Applied simplify0.5

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018198 
(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))