Average Error: 32.9 → 21.0
Time: 1.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 7.077732854953735 \cdot 10^{-263}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 4\right) \cdot \left(-c\right)}{2 \cdot a}}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 7.077732854953735e-263

    1. Initial program 19.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Applied simplify19.9

      \[\leadsto \color{blue}{\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt19.9

      \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}} - b}{2 \cdot a}\]

    5. Applied sqrt-prod20.1

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}} - b}{2 \cdot a}\]

    6. Applied fma-neg20.0

      \[\leadsto \frac{\color{blue}{(\left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}}{2 \cdot a}\]

    if 7.077732854953735e-263 < b

    1. Initial program 44.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Applied simplify45.0

      \[\leadsto \color{blue}{\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}}\]
    3. Using strategy rm

    4. Applied div-inv45.0

      \[\leadsto \color{blue}{\left(\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{2 \cdot a}}\]
    5. Using strategy rm

    6. Applied flip--45.1

      \[\leadsto \color{blue}{\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b \cdot b}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}} \cdot \frac{1}{2 \cdot a}\]

    7. Applied associate-*l/45.1

      \[\leadsto \color{blue}{\frac{\left(\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b \cdot b\right) \cdot \frac{1}{2 \cdot a}}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}\]

    8. Applied simplify21.9

      \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot 4\right) \cdot \left(-c\right)}{2 \cdot a}}}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))