Average Error: 23.8 → 6.2
Time: 4.2m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt{\alpha + \beta} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\right) \cdot \sqrt{\alpha + \beta} + 1.0}{2.0} \le 1.6323084650749024 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\alpha + \beta}{\frac{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}}\right)}^{3} + {1.0}^{3}}{\left(1.0 \cdot 1.0 - 1.0 \cdot \frac{\alpha + \beta}{\frac{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}}\right) + \frac{\alpha + \beta}{\frac{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \frac{\alpha + \beta}{\frac{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}}}}{2.0}\\ \end{array}\]

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 (/ (+ (* (sqrt (+ beta alpha)) (* (sqrt (+ beta alpha)) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 1.0) 2.0) < 1.6323084650749024e-13

    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 29.1

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

    3. Simplified29.1

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

    if 1.6323084650749024e-13 < (/ (+ (* (sqrt (+ beta alpha)) (* (sqrt (+ beta alpha)) (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) 1.0) 2.0)

    1. Initial program 13.8

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

    3. Applied *-un-lft-identity13.8

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

    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 associate-/l*0.3

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

    6. Simplified0.3

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

    8. Applied flip3-+0.3

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt{\alpha + \beta} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\right) \cdot \sqrt{\alpha + \beta} + 1.0}{2.0} \le 1.6323084650749024 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\alpha + \beta}{\frac{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}}\right)}^{3} + {1.0}^{3}}{\left(1.0 \cdot 1.0 - 1.0 \cdot \frac{\alpha + \beta}{\frac{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}}\right) + \frac{\alpha + \beta}{\frac{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \frac{\alpha + \beta}{\frac{2.0 + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}}}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018216 
(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))