Average Error: 19.5 → 13.7
Time: 27.8s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 7.145891006680855 \cdot 10^{-06}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b} \cdot 2\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}} \cdot \sqrt[3]{\frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}\right) \cdot \sqrt[3]{\frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}\right) \cdot 2\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 7.145891006680855e-06

    1. Initial program 15.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified15.6

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt15.8

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

    5. Applied sqrt-prod15.8

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

    if 7.145891006680855e-06 < b

    1. Initial program 30.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified30.9

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \end{array}}\]

    3. Taylor expanded around inf 11.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

    4. Simplified7.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

    5. Taylor expanded around 0 7.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} - \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \end{array}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt7.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(\left(\sqrt[3]{\frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}} \cdot \sqrt[3]{\frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}}\right) \cdot \sqrt[3]{\frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}}\right)}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 7.145891006680855 \cdot 10^{-06}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b} \cdot 2\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}} \cdot \sqrt[3]{\frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}\right) \cdot \sqrt[3]{\frac{c}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}}\right) \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))