Average Error: 33.6 → 29.7
Time: 17.9s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 6.381659982219284 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{\frac{\sqrt{\sqrt[3]{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}} - b}{2}}}{\frac{a}{\sqrt{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 6.381659982219284e+143

    1. Initial program 27.0

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

    2. Simplified27.0

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

    3. Taylor expanded around -inf 27.0

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

    4. Simplified27.0

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

    6. Applied add-sqr-sqrt27.4

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

    7. Applied associate-/l*27.4

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

    9. Applied add-cube-cbrt27.8

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

    10. Applied sqrt-prod27.8

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

    if 6.381659982219284e+143 < b

    1. Initial program 61.7

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

    2. Simplified61.7

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

    3. Taylor expanded around -inf 61.7

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

    4. Simplified61.7

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

    5. Taylor expanded around 0 37.7

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification29.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6.381659982219284 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{\frac{\sqrt{\sqrt[3]{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}} - b}{2}}}{\frac{a}{\sqrt{\frac{\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))