Average Error: 18.9 → 7.4
Time: 1.4m
Precision: 64
Internal Precision: 384
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.717288898696744 \cdot 10^{+88}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b + b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \le 1.437759751871048 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot a\right) \cdot \frac{c}{b} - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -5.717288898696744e+88

    1. Initial program 41.4

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

    2. Taylor expanded around -inf 9.0

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

    3. Applied simplify4.5

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

    if -5.717288898696744e+88 < b < 1.437759751871048e+105

    1. Initial program 8.4

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

    3. Applied add-exp-log10.1

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

    if 1.437759751871048e+105 < b

    1. Initial program 30.5

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

    2. Taylor expanded around inf 6.4

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

    3. Applied simplify2.5

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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))))