Average Error: 23.3 → 9.0
Time: 3.8m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \le -5.095580114145825 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{if}\;\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \le -3.4088310934444843 \cdot 10^{+128}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1.0 + \frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + \left(i + i\right)}}}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + \left(i + i\right)\right)}} \cdot \frac{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{\sqrt{\left(i + i\right) + \left(\alpha + \beta\right)}}}{\sqrt{2.0 + \left(\left(i + i\right) + \left(\alpha + \beta\right)\right)}}\right)}^{3}}}{2.0}\\ \mathbf{if}\;\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \le -2.656761391033371 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1.0 + \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + \left(i + i\right)\right)}} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + i\right) + \left(i + \beta\right)}} \cdot \left(\alpha + \beta\right)}{\sqrt{2.0 + \left(\left(\alpha + i\right) + \left(i + \beta\right)\right)}}\right)}^{3}}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) < -5.095580114145825e+184 or -3.4088310934444843e+128 < (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) < -2.656761391033371e+20

    1. Initial program 59.0

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

    2. Taylor expanded around inf 36.0

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

    3. Applied simplify36.0

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

    if -5.095580114145825e+184 < (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) < -3.4088310934444843e+128

    1. Initial program 57.8

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

    3. Applied *-un-lft-identity57.8

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

    4. Applied *-un-lft-identity57.8

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

    5. Applied times-frac57.7

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

    6. Applied times-frac57.5

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

    7. Applied simplify57.5

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

    9. Applied add-cbrt-cube57.5

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

    10. Applied simplify57.5

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(1.0 + \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{2.0 + \left(\left(\alpha + \beta\right) + \left(i + i\right)\right)} \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}}{2.0}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt57.6

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

    13. Applied add-sqr-sqrt57.7

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

    14. Applied *-un-lft-identity57.7

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

    15. Applied times-frac57.6

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

    16. Applied times-frac57.7

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

    17. Applied associate-*l*57.7

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

    18. Applied simplify57.7

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

    if -2.656761391033371e+20 < (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i)))

    1. Initial program 12.5

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

    3. Applied *-un-lft-identity12.5

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

    4. Applied *-un-lft-identity12.5

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

    5. Applied times-frac0.3

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

    6. Applied times-frac0.3

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

    7. Applied simplify0.3

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

    9. Applied add-cbrt-cube0.3

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

    10. Applied simplify0.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(1.0 + \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{2.0 + \left(\left(\alpha + \beta\right) + \left(i + i\right)\right)} \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}}{2.0}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt0.3

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

    13. Applied add-cube-cbrt0.3

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

    14. Applied times-frac0.3

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

    15. Applied associate-*l*0.3

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

    16. Applied simplify0.3

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(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))