Average Error: 22.9 → 4.2
Time: 6.9m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\alpha + \left(\left(2.0 + i\right) + \left(\beta + i\right)\right)}} \cdot \left(\left(\sqrt[3]{\frac{\frac{\beta - \alpha}{\left(i + i\right) + \left(\alpha + \beta\right)}}{\sqrt{\alpha + \left(\left(2.0 + i\right) + \left(\beta + i\right)\right)}}} \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{\left(i + i\right) + \left(\alpha + \beta\right)}}{\sqrt{\alpha + \left(\left(2.0 + i\right) + \left(\beta + i\right)\right)}}}\right) \cdot \sqrt[3]{\frac{\frac{\beta - \alpha}{\left(i + i\right) + \left(\alpha + \beta\right)}}{\sqrt{\alpha + \left(\left(2.0 + i\right) + \left(\beta + i\right)\right)}}}\right) + 1.0\right)}^{3}}}{2.0} \le 0.12481298182409777:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \left(\left(\frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - \frac{\frac{4.0}{\alpha}}{\alpha}\right)\right) - \frac{\beta}{\alpha} \cdot \left(\frac{6.0}{\alpha} + \left(\frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\alpha + \left(\left(2.0 + i\right) + \left(\beta + i\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(i + i\right) + \left(\alpha + \beta\right)}}{\sqrt{\alpha + \left(\left(2.0 + i\right) + \left(\beta + i\right)\right)}} + 1.0\right)}^{3}}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (cbrt (pow (+ (* (/ (+ alpha beta) (sqrt (+ alpha (+ (+ 2.0 i) (+ beta i))))) (* (* (cbrt (/ (/ (- beta alpha) (+ (+ i i) (+ alpha beta))) (sqrt (+ alpha (+ (+ 2.0 i) (+ beta i)))))) (cbrt (/ (/ (- beta alpha) (+ (+ i i) (+ alpha beta))) (sqrt (+ alpha (+ (+ 2.0 i) (+ beta i))))))) (cbrt (/ (/ (- beta alpha) (+ (+ i i) (+ alpha beta))) (sqrt (+ alpha (+ (+ 2.0 i) (+ beta i)))))))) 1.0) 3)) 2.0) < 0.12481298182409777

    1. Initial program 60.8

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

    3. Applied add-sqr-sqrt60.8

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

    4. Applied *-un-lft-identity60.8

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

    5. Applied times-frac58.4

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

    6. Applied times-frac58.3

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

    7. Applied simplify58.3

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

    9. Applied add-log-exp58.3

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

    10. Taylor expanded around inf 23.8

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + 2.0 \cdot \frac{1}{\alpha}\right) - \left(6.0 \cdot \frac{\beta}{{\alpha}^{2}} + \left(2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}} + 4.0 \cdot \frac{1}{{\alpha}^{2}}\right)\right)}}{2.0}\]

    11. Applied simplify20.0

      \[\leadsto \color{blue}{\frac{\left(\frac{2.0}{\alpha} + \left(\left(\frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - \frac{\frac{4.0}{\alpha}}{\alpha}\right)\right) - \frac{\beta}{\alpha} \cdot \left(\frac{6.0}{\alpha} + \left(\frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right)\right)}{2.0}}\]

    if 0.12481298182409777 < (/ (cbrt (pow (+ (* (/ (+ alpha beta) (sqrt (+ alpha (+ (+ 2.0 i) (+ beta i))))) (* (* (cbrt (/ (/ (- beta alpha) (+ (+ i i) (+ alpha beta))) (sqrt (+ alpha (+ (+ 2.0 i) (+ beta i)))))) (cbrt (/ (/ (- beta alpha) (+ (+ i i) (+ alpha beta))) (sqrt (+ alpha (+ (+ 2.0 i) (+ beta i))))))) (cbrt (/ (/ (- beta alpha) (+ (+ i i) (+ alpha beta))) (sqrt (+ alpha (+ (+ 2.0 i) (+ beta i)))))))) 1.0) 3)) 2.0)

    1. Initial program 12.9

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

    3. Applied add-sqr-sqrt12.9

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

    4. Applied *-un-lft-identity12.9

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

    5. Applied times-frac0.0

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

    6. Applied times-frac0.0

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

    7. Applied simplify0.0

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

    9. Applied add-log-exp0.0

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

    11. Applied add-cbrt-cube0.0

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

    12. Applied simplify0.0

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

Runtime

Time bar (total: 6.9m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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))