Average Error: 19.5 → 13.0
Time: 49.0s
Precision: 64
Internal Precision: 128
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.4267747929697456 \cdot 10^{+96}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{(\left(-\sqrt{b}\right) \cdot \left(\sqrt{b}\right) + \left(-\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right))_*}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \frac{1}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\right) \cdot 2\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b} \cdot 2\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 1.4267747929697456e+96

    1. Initial program 14.8

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

    2. Simplified14.7

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

    4. Applied add-sqr-sqrt14.8

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

    5. Applied distribute-lft-neg-in14.8

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

    6. Applied fma-neg14.8

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

    8. Applied div-inv14.8

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

    if 1.4267747929697456e+96 < b

    1. Initial program 44.4

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

    2. Simplified44.3

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

    3. Taylor expanded around 0 3.8

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.4267747929697456 \cdot 10^{+96}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{(\left(-\sqrt{b}\right) \cdot \left(\sqrt{b}\right) + \left(-\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right))_*}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \frac{1}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b}\right) \cdot 2\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019072 +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)))))))