Average Error: 3.6 → 1.4
Time: 7.0m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}}{\sqrt{\left(\alpha + 2\right) + \left(1.0 + \beta\right)} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\frac{(\alpha \cdot \beta + 1.0)_* + \left(\beta + \alpha\right)}{\sqrt[3]{\alpha + \left(\beta + 2\right)}}}{\sqrt{\alpha + \left(\beta + 2\right)} \cdot \sqrt{\left(\alpha + 2\right) + \left(\beta + 1.0\right)}} \le +\infty:\\ \;\;\;\;\frac{\sqrt{\frac{(\alpha \cdot \beta + \alpha)_* + \left(\beta + 1.0\right)}{\alpha + \left(\beta + 2\right)}}}{\sqrt{\alpha + \left(\beta + 2\right)}} \cdot \frac{\sqrt{\frac{\left(\beta + 1.0\right) + (\alpha \cdot \beta + \alpha)_*}{2 + \left(\beta + \alpha\right)}}}{\left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right) \cdot \sqrt{2 + \left(\beta + \alpha\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + \alpha\right) + \beta\right) \cdot \left(\left(1.0 + 2\right) + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (/ (/ 1 (sqrt (+ (+ alpha beta) 2))) (cbrt (+ (+ alpha beta) 2))) (* (sqrt (+ (+ alpha 2) (+ 1.0 beta))) (cbrt (+ (+ alpha beta) 2)))) (/ (/ (+ (fma alpha beta 1.0) (+ beta alpha)) (cbrt (+ alpha (+ beta 2)))) (* (sqrt (+ alpha (+ beta 2))) (sqrt (+ (+ alpha 2) (+ beta 1.0)))))) < +inf.0

    1. Initial program 0.4

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

    3. Applied *-un-lft-identity0.4

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

    4. Applied add-sqr-sqrt0.9

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

    5. Applied add-sqr-sqrt0.5

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

    6. Applied times-frac0.5

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

    7. Applied times-frac0.5

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

    8. Applied simplify0.5

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

    9. Applied simplify0.5

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

    if +inf.0 < (* (/ (/ (/ 1 (sqrt (+ (+ alpha beta) 2))) (cbrt (+ (+ alpha beta) 2))) (* (sqrt (+ (+ alpha 2) (+ 1.0 beta))) (cbrt (+ (+ alpha beta) 2)))) (/ (/ (+ (fma alpha beta 1.0) (+ beta alpha)) (cbrt (+ alpha (+ beta 2)))) (* (sqrt (+ alpha (+ beta 2))) (sqrt (+ (+ alpha 2) (+ beta 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 18.4

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

    3. Applied simplify18.4

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

Runtime

Time bar (total: 7.0m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +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)))