Average Error: 3.6 → 1.3
Time: 6.3m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)} \le 1.7835984696848677 \cdot 10^{+308}:\\ \;\;\;\;\sqrt{\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}} \cdot \sqrt{\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)) < 1.7835984696848677e+308

    1. Initial program 0.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.2

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

    if 1.7835984696848677e+308 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

    1. Initial program 63.0

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Taylor expanded around inf 20.6

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

    3. Simplified20.6

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)} \le 1.7835984696848677 \cdot 10^{+308}:\\ \;\;\;\;\sqrt{\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}} \cdot \sqrt{\frac{\frac{\frac{\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]

Runtime

Time bar (total: 6.3m)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))