Average Error: 4.0 → 1.4
Time: 9.9m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\left(2 + \alpha\right) + \beta}}}{\left(\left(2 + \beta\right) + \left(\alpha + 1.0\right)\right) \cdot \sqrt{\left(2 + \alpha\right) + \beta}} \cdot \left(\frac{\sqrt[3]{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\alpha + \left(2 + \beta\right)}}}{\sqrt{\alpha + \left(2 + \beta\right)}} \cdot \sqrt[3]{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\alpha + \left(2 + \beta\right)}}\right) \le +\infty:\\ \;\;\;\;\frac{\frac{\frac{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\sqrt{\left(2 + \alpha\right) + \beta}}}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \left(\alpha + 1.0\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (* (* (/ (cbrt (/ (+ (fma alpha beta beta) (+ alpha 1.0)) (+ alpha (+ beta 2)))) (sqrt (+ alpha (+ beta 2)))) (cbrt (/ (+ (fma alpha beta beta) (+ alpha 1.0)) (+ alpha (+ beta 2))))) (/ (cbrt (/ (+ (+ alpha 1.0) (fma alpha beta beta)) (+ (+ alpha 2) beta))) (* (+ (+ beta 2) (+ alpha 1.0)) (sqrt (+ (+ alpha 2) beta))))) < +inf.0

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied associate-/r*0.7

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

    5. Applied simplify0.7

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

    if +inf.0 < (* (* (/ (cbrt (/ (+ (fma alpha beta beta) (+ alpha 1.0)) (+ alpha (+ beta 2)))) (sqrt (+ alpha (+ beta 2)))) (cbrt (/ (+ (fma alpha beta beta) (+ alpha 1.0)) (+ alpha (+ beta 2))))) (/ (cbrt (/ (+ (+ alpha 1.0) (fma alpha beta beta)) (+ (+ alpha 2) beta))) (* (+ (+ beta 2) (+ alpha 1.0)) (sqrt (+ (+ alpha 2) beta)))))

    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 14.3

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

    3. Applied simplify14.3

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

  4. Applied simplify1.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\left(2 + \alpha\right) + \beta}}}{\left(\left(2 + \beta\right) + \left(\alpha + 1.0\right)\right) \cdot \sqrt{\left(2 + \alpha\right) + \beta}} \cdot \left(\frac{\sqrt[3]{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\alpha + \left(2 + \beta\right)}}}{\sqrt{\alpha + \left(2 + \beta\right)}} \cdot \sqrt[3]{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\alpha + \left(2 + \beta\right)}}\right) \le +\infty:\\ \;\;\;\;\frac{\frac{\frac{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\sqrt{\left(2 + \alpha\right) + \beta}}}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(2 + \beta\right) + \left(\alpha + 1.0\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 9.9m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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)))