Average Error: 23.6 → 11.9
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.545313913952212 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right)}^{3} + {1.0}^{3}}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} + \frac{\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(1.0 \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) \cdot \left(1.0 \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right)}{1.0 \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} + 1.0 \cdot 1.0}}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.0344243922917673 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 8.434466939603523 \cdot 10^{+175}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta - \alpha}{\sqrt[3]{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \sqrt[3]{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta + \alpha}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if alpha < 8.545313913952212e+47

    1. Initial program 11.5

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

    3. Applied *-un-lft-identity11.5

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

    4. Applied times-frac1.2

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

    5. Simplified1.2

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

    7. Applied flip3-+1.2

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

    9. Applied flip--1.2

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

    if 8.545313913952212e+47 < alpha < 1.0344243922917673e+126 or 8.434466939603523e+175 < alpha

    1. Initial program 55.1

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

    2. Taylor expanded around -inf 40.1

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

    3. Simplified40.1

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

    if 1.0344243922917673e+126 < alpha < 8.434466939603523e+175

    1. Initial program 53.6

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

    3. Applied add-sqr-sqrt53.6

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

    4. Applied add-cube-cbrt53.6

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

    5. Applied times-frac37.4

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

    6. Applied times-frac37.3

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

    8. Applied add-cube-cbrt37.4

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

    9. Applied cbrt-prod37.2

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.545313913952212 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right)}^{3} + {1.0}^{3}}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} + \frac{\left(1.0 \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0\right) - \left(1.0 \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) \cdot \left(1.0 \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right)}{1.0 \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} + 1.0 \cdot 1.0}}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.0344243922917673 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 8.434466939603523 \cdot 10^{+175}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta - \alpha}{\sqrt[3]{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \sqrt[3]{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta + \alpha}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha} + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(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))