Average Error: 20.0 → 13.5
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \le -6.472477267158255 \cdot 10^{-260} \lor \neg \left(\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \le 1.976262583365 \cdot 10^{-322}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\sqrt[3]{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\right)\right)} \cdot {e}^{\left(\log \left(\sqrt[3]{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt[3]{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\right)\right)}}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (fma (* a 4) (- c) (* b b))))) 0) < -6.472477267158255e-260 or 1.976262583365e-322 < (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (fma (* a 4) (- c) (* b b))))) 0)

    1. Initial program 2.7

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

    2. Applied simplify2.7

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

    4. Applied add-exp-log2.7

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

    6. Applied pow12.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({\left(\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}^{1}\right)}}{a \cdot 2}\\ \end{array}\]

    7. Applied log-pow2.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{1 \cdot \log \left(\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)}}{a \cdot 2}\\ \end{array}\]

    8. Applied exp-prod2.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{1}\right)}^{\left(\log \left(\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b\right)\right)}}{a \cdot 2}\\ \end{array}\]

    9. Applied simplify2.7

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

    11. Applied add-cube-cbrt2.7

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

    12. Applied log-prod2.7

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

    13. Applied unpow-prod-up2.7

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

    if -6.472477267158255e-260 < (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (fma (* a 4) (- c) (* b b))))) 0) < 1.976262583365e-322

    1. Initial program 27.3

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

    2. Applied simplify27.2

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

    3. Taylor expanded around 0 18.1

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

  4. Applied simplify13.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \le -6.472477267158255 \cdot 10^{-260} \lor \neg \left(\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \le 1.976262583365 \cdot 10^{-322}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\sqrt[3]{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\right)\right)} \cdot {e}^{\left(\log \left(\sqrt[3]{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt[3]{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\right)\right)}}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018195 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))