Average Error: 33.6 → 6.6
Time: 1.6m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0508280330364059 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b + b}{2 \cdot a}\\ \mathbf{if}\;b \le 2.6599776360757905 \cdot 10^{-277}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le 2.4872041728097723 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot \frac{c}{1}}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.9
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.0508280330364059e+79

    1. Initial program 41.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around -inf 10.0

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a}\]

    3. Applied simplify4.5

      \[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b + b}{2 \cdot a}}\]

    if -1.0508280330364059e+79 < b < 2.6599776360757905e-277

    1. Initial program 9.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    if 2.6599776360757905e-277 < b < 2.4872041728097723e+127

    1. Initial program 34.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+34.3

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    4. Applied simplify16.0

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity16.0

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    7. Applied times-frac16.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}}\]

    8. Applied simplify7.9

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{c}{1} \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\]

    if 2.4872041728097723e+127 < b

    1. Initial program 61.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around inf 39.6

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a}\]

    3. Applied simplify2.1

      \[\leadsto \color{blue}{\left(-1\right) \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify6.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.0508280330364059 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b + b}{2 \cdot a}\\ \mathbf{if}\;b \le 2.6599776360757905 \cdot 10^{-277}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le 2.4872041728097723 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot \frac{c}{1}}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))