Average Error: 3.5 → 1.9
Time: 4.5m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(\beta + 2\right) + \left(1.0 + \alpha\right)\right)} \le 0.0:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(\beta + 2\right) + \left(1.0 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{(\beta \cdot \alpha + \beta)_* + \left(\alpha + 1.0\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\beta + 2\right) + \left(\alpha + 1.0\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (fma (/ 1 alpha) (- (/ 2.0 alpha) 1.0) 1) (* (+ (+ 2 alpha) beta) (+ (+ beta 2) (+ 1.0 alpha)))) < 0.0

    1. Initial program 13.8

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

    2. Taylor expanded around inf 13.8

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

    3. Applied simplify7.1

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

    if 0.0 < (/ (fma (/ 1 alpha) (- (/ 2.0 alpha) 1.0) 1) (* (+ (+ 2 alpha) beta) (+ (+ beta 2) (+ 1.0 alpha))))

    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. Applied simplify0.2

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

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))