Average Error: 23.7 → 6.2
Time: 5.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left(\frac{\frac{\alpha + \beta}{\sqrt{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}}}{\sqrt{1}} \cdot \frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}}}{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}} + 1.0\right)}^{3}}}{2.0} \le 1.1969591984239966 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(1.0 + \frac{\frac{\alpha + \beta}{\sqrt{\left(2 \cdot i + \left(2.0 + \alpha\right)\right) + \beta}}}{\frac{\sqrt{\left(2 \cdot i + \left(2.0 + \alpha\right)\right) + \beta}}{\frac{\beta - \alpha}{2 \cdot i + \left(\alpha + \beta\right)}}}\right)}^{3}}\right)}^{3}}}{2.0}\\ \end{array}\]

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 (/ (cbrt (pow (+ (* (/ (/ (+ alpha beta) (sqrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (sqrt 1)) (/ (/ (/ (- beta alpha) (+ (* i 2) (+ alpha beta))) (sqrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) 1.0) 3)) 2.0) < 1.1969591984239966e-16

    1. Initial program 62.7

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

    2. Taylor expanded around inf 27.5

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

    3. Applied simplify27.5

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

    if 1.1969591984239966e-16 < (/ (cbrt (pow (+ (* (/ (/ (+ alpha beta) (sqrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (sqrt 1)) (/ (/ (/ (- beta alpha) (+ (* i 2) (+ alpha beta))) (sqrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) 1.0) 3)) 2.0)

    1. Initial program 14.5

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

    3. Applied add-sqr-sqrt14.5

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

    4. Applied *-un-lft-identity14.5

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

    5. Applied times-frac1.1

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

    6. Applied times-frac1.1

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

    7. Applied simplify1.1

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

    9. Applied add-cbrt-cube1.1

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

    10. Applied simplify1.1

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

    12. Applied add-sqr-sqrt1.1

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

    13. Applied sqrt-prod1.2

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

    14. Applied add-sqr-sqrt1.2

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

    15. Applied sqrt-prod1.2

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

    16. Applied times-frac1.2

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

    17. Applied times-frac1.2

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

    18. Applied simplify1.2

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

    19. Applied simplify1.1

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

    21. Applied add-cbrt-cube1.1

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

    22. Applied simplify1.1

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

Runtime

Time bar (total: 5.7m)Debug logProfile

herbie shell --seed 2018207 
(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))