Average Error: 23.4 → 6.2
Time: 5.8m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt[3]{\frac{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}}{\frac{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}{\beta + \alpha} \cdot \frac{\beta + \left(\left(2.0 + \alpha\right) + 2 \cdot i\right)}{\sqrt[3]{\beta - \alpha}}} + 1.0} \cdot \sqrt[3]{\frac{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}}{\frac{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}{\beta + \alpha} \cdot \frac{\beta + \left(\left(2.0 + \alpha\right) + 2 \cdot i\right)}{\sqrt[3]{\beta - \alpha}}} + 1.0}\right) \cdot \sqrt[3]{\left(\frac{\beta + \alpha}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\beta + \left(\left(2.0 + \alpha\right) + 2 \cdot i\right)}\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}\right) + 1.0}}{2.0} \le 3.552713678800501 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \frac{\beta + \alpha}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) + 2.0}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (* (cbrt (+ 1.0 (/ (* (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta)))) (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta))))) (* (/ (cbrt (+ (* i 2) (+ alpha beta))) (+ alpha beta)) (/ (+ beta (+ (* i 2) (+ 2.0 alpha))) (cbrt (- beta alpha))))))) (cbrt (+ 1.0 (/ (* (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta)))) (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta))))) (* (/ (cbrt (+ (* i 2) (+ alpha beta))) (+ alpha beta)) (/ (+ beta (+ (* i 2) (+ 2.0 alpha))) (cbrt (- beta alpha)))))))) (cbrt (+ 1.0 (* (* (/ (cbrt (- beta alpha)) (+ beta (+ (* i 2) (+ 2.0 alpha)))) (/ (+ alpha beta) (cbrt (+ (* i 2) (+ alpha beta))))) (* (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta)))) (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta))))))))) 2.0) < 3.552713678800501e-15

    1. Initial program 62.5

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

    2. 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 3.552713678800501e-15 < (/ (* (* (cbrt (+ 1.0 (/ (* (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta)))) (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta))))) (* (/ (cbrt (+ (* i 2) (+ alpha beta))) (+ alpha beta)) (/ (+ beta (+ (* i 2) (+ 2.0 alpha))) (cbrt (- beta alpha))))))) (cbrt (+ 1.0 (/ (* (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta)))) (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta))))) (* (/ (cbrt (+ (* i 2) (+ alpha beta))) (+ alpha beta)) (/ (+ beta (+ (* i 2) (+ 2.0 alpha))) (cbrt (- beta alpha)))))))) (cbrt (+ 1.0 (* (* (/ (cbrt (- beta alpha)) (+ beta (+ (* i 2) (+ 2.0 alpha)))) (/ (+ alpha beta) (cbrt (+ (* i 2) (+ alpha beta))))) (* (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta)))) (/ (cbrt (- beta alpha)) (cbrt (+ (* i 2) (+ alpha beta))))))))) 2.0)

    1. Initial program 13.5

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

    3. Applied *-un-lft-identity13.5

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

    4. Applied times-frac0.4

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

    5. Applied associate-/l*0.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}} + 1.0}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify6.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt[3]{\frac{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}}{\frac{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}{\beta + \alpha} \cdot \frac{\beta + \left(\left(2.0 + \alpha\right) + 2 \cdot i\right)}{\sqrt[3]{\beta - \alpha}}} + 1.0} \cdot \sqrt[3]{\frac{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}}{\frac{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}{\beta + \alpha} \cdot \frac{\beta + \left(\left(2.0 + \alpha\right) + 2 \cdot i\right)}{\sqrt[3]{\beta - \alpha}}} + 1.0}\right) \cdot \sqrt[3]{\left(\frac{\beta + \alpha}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\beta + \left(\left(2.0 + \alpha\right) + 2 \cdot i\right)}\right) \cdot \left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\beta + \alpha\right) + 2 \cdot i}}\right) + 1.0}}{2.0} \le 3.552713678800501 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0 + \frac{\beta + \alpha}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) + 2.0}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}}}{2.0}\\ \end{array}}\]

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(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))