Average Error: 3.5 → 1.4
Time: 8.5m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\frac{\left(1.0 + \beta\right) + (\alpha \cdot \beta + \alpha)_*}{\left(\beta + \alpha\right) + 2}}}{\sqrt[3]{\left(\beta + \alpha\right) + 2} \cdot \left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right)} \cdot \frac{\sqrt{\frac{\left(\alpha + 1.0\right) + (\alpha \cdot \beta + \beta)_*}{\beta + \left(2 + \alpha\right)}}}{\sqrt[3]{\beta + \left(2 + \alpha\right)} \cdot \sqrt[3]{\beta + \left(2 + \alpha\right)}} \le +\infty:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 0.9

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

    if +inf.0 < (* (/ (sqrt (/ (+ (fma alpha beta beta) (+ alpha 1.0)) (+ beta (+ alpha 2)))) (* (cbrt (+ beta (+ alpha 2))) (cbrt (+ beta (+ alpha 2))))) (/ (sqrt (/ (+ (+ beta 1.0) (fma alpha beta alpha)) (+ 2 (+ beta alpha)))) (* (+ (+ 1.0 2) (+ beta alpha)) (cbrt (+ 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 11.6

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

    3. Applied simplify11.6

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

  4. Applied simplify1.4

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

Runtime

Time bar (total: 8.5m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +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)))