Average Error: 34.3 → 12.9
Time: 6.9s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.4009313388987055 \cdot 10^{134}:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -4.22044913050568174 \cdot 10^{-254}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.5579717161169453 \cdot 10^{143}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{a \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -3.4009313388987055e134

    1. Initial program 57.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around -inf 10.7

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a}\]

    if -3.4009313388987055e134 < b < -4.22044913050568174e-254

    1. Initial program 7.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt8.1

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

    if -4.22044913050568174e-254 < b < 2.5579717161169453e143

    1. Initial program 32.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+32.9

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

    4. Simplified16.1

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
    5. Using strategy rm

    6. Applied associate-/r*16.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}}\]

    7. Simplified16.1

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

    9. Applied clear-num16.2

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{3 \cdot \left(a \cdot c\right)}{3 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}}\]

    10. Simplified16.1

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{a \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

    if 2.5579717161169453e143 < b

    1. Initial program 62.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+62.9

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

    4. Simplified37.6

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt37.6

      \[\leadsto \frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}{3 \cdot a}\]

    7. Taylor expanded around inf 15.3

      \[\leadsto \frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.4009313388987055 \cdot 10^{134}:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -4.22044913050568174 \cdot 10^{-254}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.5579717161169453 \cdot 10^{143}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{a \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))