Average Error: 33.4 → 9.9
Time: 2.4m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -2.1241737424767768 \cdot 10^{+144}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{if}\;-b \le -2.178398895180725 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 4\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}{a \cdot 2}\\ \mathbf{if}\;-b \le 5.282374269753992 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b + b}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.4
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- b) < -2.1241737424767768e+144

    1. Initial program 61.8

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

    2. Taylor expanded around inf 38.6

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

    3. Applied simplify1.7

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

    if -2.1241737424767768e+144 < (- b) < -2.178398895180725e-107

    1. Initial program 42.3

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

    3. Applied flip-+42.4

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

    4. Applied simplify15.1

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

    if -2.178398895180725e-107 < (- b) < 5.282374269753992e+37

    1. Initial program 12.7

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

    if 5.282374269753992e+37 < (- b)

    1. Initial program 34.7

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

    2. Taylor expanded around -inf 11.0

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

    3. Applied simplify6.3

      \[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b + b}{a \cdot 2}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify9.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -2.1241737424767768 \cdot 10^{+144}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{if}\;-b \le -2.178398895180725 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\left(a \cdot 4\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}{a \cdot 2}\\ \mathbf{if}\;-b \le 5.282374269753992 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b + b}{a \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(341049388 4193966283 3776730818 3836052170 128576249 3840315966)' 
(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)))