Average Error: 23.5 → 5.9
Time: 5.5m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left(\frac{\frac{\alpha + \beta}{\sqrt[3]{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}} \cdot \sqrt[3]{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}}}{\sqrt{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}} \cdot \frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt[3]{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}}}{\sqrt{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}} + 1.0\right)}^{3}}}{2.0} \le 2.836508805614813 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\\ \end{array}\]

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) (* (cbrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2))))) (cbrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2))))))) (sqrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (/ (/ (/ (- beta alpha) (+ (* i 2) (+ alpha beta))) (cbrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (sqrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2))))))) 1.0) 3)) 2.0) < 2.836508805614813e-12

    1. Initial program 62.4

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

    2. Taylor expanded around inf 30.2

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

    3. Applied simplify30.2

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

    if 2.836508805614813e-12 < (/ (cbrt (pow (+ (* (/ (/ (+ alpha beta) (* (cbrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2))))) (cbrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2))))))) (sqrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (/ (/ (/ (- beta alpha) (+ (* i 2) (+ alpha beta))) (cbrt (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (sqrt (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 *-un-lft-identity14.5

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

    4. Applied times-frac0.3

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

    5. Applied simplify0.3

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

Runtime

Time bar (total: 5.5m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(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))