Average Error: 3.3 → 1.1
Time: 7.4m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{\left(\beta \cdot \alpha + \left(\beta + 1.0\right)\right) + \alpha}{\left(\beta + \alpha\right) + 2}}}{\sqrt{\left(\beta + \alpha\right) + 2} \cdot \left(\left(\beta + 1.0\right) + \left(\alpha + 2\right)\right)} \cdot \frac{\sqrt[3]{\frac{\left(\beta + 1.0\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\beta + 2\right) + \alpha}}}{\frac{\sqrt{\left(\beta + 2\right) + \alpha}}{\sqrt[3]{\frac{\left(\beta + 1.0\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \le +\infty:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0.25 + 0.5}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 0.5

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

    if +inf.0 < (* (/ (cbrt (/ (+ (+ alpha (* beta alpha)) (+ beta 1.0)) (+ (+ beta 2) alpha))) (/ (sqrt (+ (+ beta 2) alpha)) (cbrt (/ (+ (+ alpha (* beta alpha)) (+ beta 1.0)) (+ (+ beta 2) alpha))))) (/ (cbrt (/ (+ alpha (+ (+ beta 1.0) (* beta alpha))) (+ 2 (+ beta alpha)))) (* (+ (+ beta 1.0) (+ alpha 2)) (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 0 58.0

      \[\leadsto \frac{\frac{\color{blue}{0.5 + \left(0.25 \cdot \beta + 0.25 \cdot \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    3. Applied simplify14.0

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(\alpha + \beta\right) + 0.5}{\left(\left(2 + 1.0\right) + \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify1.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{\left(\beta \cdot \alpha + \left(\beta + 1.0\right)\right) + \alpha}{\left(\beta + \alpha\right) + 2}}}{\sqrt{\left(\beta + \alpha\right) + 2} \cdot \left(\left(\beta + 1.0\right) + \left(\alpha + 2\right)\right)} \cdot \frac{\sqrt[3]{\frac{\left(\beta + 1.0\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\beta + 2\right) + \alpha}}}{\frac{\sqrt{\left(\beta + 2\right) + \alpha}}{\sqrt[3]{\frac{\left(\beta + 1.0\right) + \left(\alpha + \beta \cdot \alpha\right)}{\left(\beta + 2\right) + \alpha}}}} \le +\infty:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot 0.25 + 0.5}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + \left(2 + 1.0\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 7.4m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(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)))