Average Error: 24.1 → 7.5
Time: 1.5m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.999999999834416:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}}\right)}^{3} + {1.0}^{3}}{\left(1.0 \cdot 1.0 - \frac{1}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}} \cdot 1.0\right) + \frac{1}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}} \cdot \frac{1}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}}}}{2.0}\\ \end{array}\]

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) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) < -0.999999999834416

    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 32.6

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

    3. Simplified32.6

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

    if -0.999999999834416 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 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 *-un-lft-identity12.9

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    7. Applied clear-num0.1

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

    9. Applied flip3-+0.1

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.999999999834416:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}}\right)}^{3} + {1.0}^{3}}{\left(1.0 \cdot 1.0 - \frac{1}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}} \cdot 1.0\right) + \frac{1}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}} \cdot \frac{1}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta + \alpha\right)}}}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018230 
(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))