Average Error: 24.2 → 6.3
Time: 3.2m
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{\frac{{\left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)}^{3} + {1.0}^{3}}{\left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + \left(1.0 \cdot 1.0 - \left(\frac{\beta + \alpha}{e^{\log \left(\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot 1.0\right)}}{2.0} \le 1.1102230246251662 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{{\left({\left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)}^{3}\right)}^{3} + {\left({1.0}^{3}\right)}^{3}}{{\left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)}^{3} \cdot {\left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)}^{3} + \left({1.0}^{3} \cdot {1.0}^{3} - {\left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)}^{3} \cdot {1.0}^{3}\right)}}{\left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\alpha + \left(\left(\beta + 2.0\right) + i \cdot 2\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + \sqrt[3]{{\left(1.0 \cdot 1.0 - \frac{\alpha + \beta}{\sqrt{\left(2.0 + \alpha\right) + \left(2 \cdot i + \beta\right)}} \cdot \frac{\frac{\left(\beta - \alpha\right) \cdot 1.0}{2 \cdot i + \left(\alpha + \beta\right)}}{\sqrt{\left(2.0 + \alpha\right) + \left(2 \cdot i + \beta\right)}}\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 (* (/ (+ beta alpha) (sqrt (+ alpha (+ (+ beta 2.0) (* i 2))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 3) (pow 1.0 3)) (+ (* (* (/ (+ beta alpha) (sqrt (+ alpha (+ (+ beta 2.0) (* i 2))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) (* (/ (+ beta alpha) (sqrt (+ alpha (+ (+ beta 2.0) (* i 2))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))))) (- (* 1.0 1.0) (* (* (/ (+ beta alpha) (exp (log (sqrt (+ alpha (+ (+ beta 2.0) (* i 2))))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 1.0)))) 2.0) < 1.1102230246251662e-16

    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. Taylor expanded around inf 29.3

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

    3. Applied simplify29.3

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

    if 1.1102230246251662e-16 < (/ (/ (+ (pow (* (/ (+ beta alpha) (sqrt (+ alpha (+ (+ beta 2.0) (* i 2))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 3) (pow 1.0 3)) (+ (* (* (/ (+ beta alpha) (sqrt (+ alpha (+ (+ beta 2.0) (* i 2))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) (* (/ (+ beta alpha) (sqrt (+ alpha (+ (+ beta 2.0) (* i 2))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))))) (- (* 1.0 1.0) (* (* (/ (+ beta alpha) (exp (log (sqrt (+ alpha (+ (+ beta 2.0) (* i 2))))))) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 1.0)))) 2.0)

    1. Initial program 14.7

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

    3. Applied add-sqr-sqrt14.8

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

    4. Applied *-un-lft-identity14.8

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

    5. Applied times-frac0.6

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

    6. Applied times-frac0.6

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

    7. Applied simplify0.6

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

    9. Applied flip3-+0.7

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

    11. Applied flip3-+0.6

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

    13. Applied add-cbrt-cube0.6

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

    14. Applied simplify0.6

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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))