Average Error: 20.1 → 13.7
Time: 29.4s
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5.523479422929744 \cdot 10^{+47}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{(\left(\left|\sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}\right|\right) \cdot \left(\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 5.523479422929744e+47

    1. Initial program 15.7

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

    2. Initial simplification15.7

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

    4. Applied add-cube-cbrt15.8

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

    5. Applied sqrt-prod15.8

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

    6. Applied fma-neg15.8

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

    7. Simplified15.8

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

    if 5.523479422929744e+47 < b

    1. Initial program 36.2

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

    2. Initial simplification36.2

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

    3. Taylor expanded around 0 6.1

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

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5.523479422929744 \cdot 10^{+47}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{(\left(\left|\sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}\right|\right) \cdot \left(\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Runtime

Time bar (total: 29.4s)Debug logProfile

herbie shell --seed 2018254 +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)))))))