Average Error: 3.6 → 1.4
Time: 8.0m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\frac{(\alpha \cdot \beta + 1.0)_* + \left(\alpha + \beta\right)}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}}}{\sqrt{\left(2 + 1.0\right) + \left(\alpha + \beta\right)}} \cdot \frac{\sqrt{\frac{(\alpha \cdot \beta + 1.0)_* + \left(\alpha + \beta\right)}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}}}{\sqrt{\left(2 + 1.0\right) + \left(\alpha + \beta\right)}} \le 0.08347471864511563:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + 1.0\right) + \left(\alpha + \beta\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (sqrt (/ (+ (+ beta alpha) (fma alpha beta 1.0)) (* (+ beta (+ alpha 2)) (+ beta (+ alpha 2))))) (sqrt (+ (+ 1.0 2) (+ beta alpha)))) (/ (sqrt (/ (+ (+ beta alpha) (fma alpha beta 1.0)) (* (+ beta (+ alpha 2)) (+ beta (+ alpha 2))))) (sqrt (+ (+ 1.0 2) (+ beta alpha))))) < 0.08347471864511563

    1. Initial program 0.1

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

    if 0.08347471864511563 < (* (/ (sqrt (/ (+ (+ beta alpha) (fma alpha beta 1.0)) (* (+ beta (+ alpha 2)) (+ beta (+ alpha 2))))) (sqrt (+ (+ 1.0 2) (+ beta alpha)))) (/ (sqrt (/ (+ (+ beta alpha) (fma alpha beta 1.0)) (* (+ beta (+ alpha 2)) (+ beta (+ alpha 2))))) (sqrt (+ (+ 1.0 2) (+ beta alpha)))))

    1. Initial program 60.9

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

    2. Taylor expanded around inf 22.7

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

    3. Applied simplify22.7

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

  4. Applied simplify1.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\frac{(\alpha \cdot \beta + 1.0)_* + \left(\alpha + \beta\right)}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}}}{\sqrt{\left(2 + 1.0\right) + \left(\alpha + \beta\right)}} \cdot \frac{\sqrt{\frac{(\alpha \cdot \beta + 1.0)_* + \left(\alpha + \beta\right)}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}}}{\sqrt{\left(2 + 1.0\right) + \left(\alpha + \beta\right)}} \le 0.08347471864511563:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha}\right) \cdot \left(\frac{2.0}{\alpha} - 1.0\right) + 1)_*}{\left(\left(2 + 1.0\right) + \left(\alpha + \beta\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 8.0m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +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)))