Average Error: 3.4 → 1.4
Time: 7.9m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\frac{\left(1.0 + \beta\right) + (\alpha \cdot \beta + \alpha)_*}{\alpha + \left(2 + \beta\right)}}}{\sqrt{\alpha + \left(2 + \beta\right)}} \cdot \frac{\sqrt{\frac{\left(1.0 + \beta\right) + (\alpha \cdot \beta + \alpha)_*}{2 + \left(\beta + \alpha\right)}}}{\left(\left(2 + 1.0\right) + \left(\beta + \alpha\right)\right) \cdot \sqrt{2 + \left(\beta + \alpha\right)}} \le 0.08852945879169893:\\ \;\;\;\;\frac{\frac{\left(\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{1.0 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + 1.0\right) + \left(\beta + \alpha\right)\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (sqrt (/ (+ (fma alpha beta alpha) (+ beta 1.0)) (+ alpha (+ beta 2)))) (sqrt (+ alpha (+ beta 2)))) (/ (sqrt (/ (+ (+ beta 1.0) (fma alpha beta alpha)) (+ 2 (+ beta alpha)))) (* (+ (+ 1.0 2) (+ beta alpha)) (sqrt (+ 2 (+ beta alpha)))))) < 0.08852945879169893

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    4. Applied simplify0.1

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

    if 0.08852945879169893 < (* (/ (sqrt (/ (+ (fma alpha beta alpha) (+ beta 1.0)) (+ alpha (+ beta 2)))) (sqrt (+ alpha (+ beta 2)))) (/ (sqrt (/ (+ (+ beta 1.0) (fma alpha beta alpha)) (+ 2 (+ beta alpha)))) (* (+ (+ 1.0 2) (+ beta alpha)) (sqrt (+ 2 (+ beta alpha))))))

    1. Initial program 61.8

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

    2. Taylor expanded around inf 23.0

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

    3. Applied simplify23.0

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

  4. Applied simplify1.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\frac{\left(1.0 + \beta\right) + (\alpha \cdot \beta + \alpha)_*}{\alpha + \left(2 + \beta\right)}}}{\sqrt{\alpha + \left(2 + \beta\right)}} \cdot \frac{\sqrt{\frac{\left(1.0 + \beta\right) + (\alpha \cdot \beta + \alpha)_*}{2 + \left(\beta + \alpha\right)}}}{\left(\left(2 + 1.0\right) + \left(\beta + \alpha\right)\right) \cdot \sqrt{2 + \left(\beta + \alpha\right)}} \le 0.08852945879169893:\\ \;\;\;\;\frac{\frac{\left(\left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right) + 1.0\right) \cdot \frac{1}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{1.0 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + 1.0\right) + \left(\beta + \alpha\right)\right) \cdot \left(\left(2 + \alpha\right) + \beta\right)}\\ \end{array}}\]

Runtime

Time bar (total: 7.9m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +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)))