Average Error: 22.8 → 6.1
Time: 4.4m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\alpha + \beta}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \frac{\beta - \alpha}{\sqrt[3]{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \le 3.827131753869617 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{\alpha}}{2.0} \cdot \left(\left(\frac{4.0}{\alpha} + 2.0\right) + \frac{8.0}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}} \cdot \sqrt[3]{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right) \cdot \sqrt[3]{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\\ \end{array}\]

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 (/ (+ (/ (* (/ (+ alpha beta) (* (cbrt (+ (+ alpha beta) (* 2 i))) (cbrt (+ (+ alpha beta) (* 2 i))))) (/ (- beta alpha) (* (cbrt (sqrt (+ (+ alpha beta) (* 2 i)))) (cbrt (sqrt (+ (+ alpha beta) (* 2 i))))))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0) < 3.827131753869617e-12

    1. Initial program 62.3

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

    2. Taylor expanded around inf 29.4

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

    3. Applied simplify29.4

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

    if 3.827131753869617e-12 < (/ (+ (/ (* (/ (+ alpha beta) (* (cbrt (+ (+ alpha beta) (* 2 i))) (cbrt (+ (+ alpha beta) (* 2 i))))) (/ (- beta alpha) (* (cbrt (sqrt (+ (+ alpha beta) (* 2 i)))) (cbrt (sqrt (+ (+ alpha beta) (* 2 i))))))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0)

    1. Initial program 13.2

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

    3. Applied associate-/l*0.3

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

    5. Applied add-cube-cbrt0.5

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

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed 2018195 
(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))