Average Error: 3.6 → 2.1
Time: 6.7m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)} \le 0.0:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}\]

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) (+ (+ 1.0 2) (+ beta alpha)))) < 0.0

    1. Initial program 13.9

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

    2. Taylor expanded around inf 14.1

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

    3. Applied simplify7.9

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

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

    1. Initial program 0.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify2.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)} \le 0.0:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(\beta + \alpha\right) + \left(1.0 + 2\right)\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}}\]

Runtime

Time bar (total: 6.7m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +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)))