Average Error: 3.8 → 1.4
Time: 6.6m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{\left(\beta + 1.0\right) + (\alpha \cdot \beta + \alpha)_*}{\left(\alpha + \beta\right) + 2}}}{\sqrt{\left(\alpha + \beta\right) + 2} \cdot \left(\left(\alpha + \beta\right) + \left(2 + 1.0\right)\right)} \cdot \frac{\sqrt[3]{\frac{(\beta \cdot \alpha + \beta)_* + \left(1.0 + \alpha\right)}{\alpha + \left(2 + \beta\right)}} \cdot \sqrt[3]{\frac{(\beta \cdot \alpha + \beta)_* + \left(1.0 + \alpha\right)}{\alpha + \left(2 + \beta\right)}}}{\sqrt{\alpha + \left(2 + \beta\right)}} \le +\infty:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{(\beta \cdot \alpha + \alpha)_* + \left(\beta + 1.0\right)}}{\left(\alpha + \beta\right) + 2} \cdot \sqrt{\left(\beta + 1.0\right) + (\alpha \cdot \beta + \alpha)_*}}{\left(\alpha + \beta\right) + 2}}{1.0 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(\alpha + \beta\right) + \left(2 + 1.0\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied add-sqr-sqrt0.6

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

    5. Applied times-frac0.6

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

    6. Applied simplify0.6

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

    7. Applied simplify0.6

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

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

    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 15.8

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

    3. Applied simplify15.8

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

  4. Applied simplify1.4

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

Runtime

Time bar (total: 6.6m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +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)))