Average Error: 33.5 → 9.2
Time: 1.0m
Precision: 64
Internal Precision: 3200
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.352627690078744 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{-c}{b}}{1}\\ \mathbf{if}\;b \le -2.0089291940149467 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}{a \cdot 2}\\ \mathbf{if}\;b \le -1.1884183040588929 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{-c}{b}}{1}\\ \mathbf{if}\;b \le 6.832382137151648 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot 2}{\frac{b}{c}} - \left(b + b\right)}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.9
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.352627690078744e+48 or -2.0089291940149467e-121 < b < -1.1884183040588929e-133

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 43.2

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

    3. Applied simplify4.8

      \[\leadsto \color{blue}{0 - \frac{\frac{c}{b}}{1}}\]

    if -4.352627690078744e+48 < b < -2.0089291940149467e-121

    1. Initial program 37.6

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

    3. Applied flip--37.7

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

    4. Applied simplify15.3

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

    5. Applied simplify15.3

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

    if -1.1884183040588929e-133 < b < 6.832382137151648e+104

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    if 6.832382137151648e+104 < b

    1. Initial program 46.2

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

    2. Taylor expanded around inf 9.4

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

    3. Applied simplify3.5

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

  4. Applied simplify9.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -4.352627690078744 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{-c}{b}}{1}\\ \mathbf{if}\;b \le -2.0089291940149467 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}{a \cdot 2}\\ \mathbf{if}\;b \le -1.1884183040588929 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{-c}{b}}{1}\\ \mathbf{if}\;b \le 6.832382137151648 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot 2}{\frac{b}{c}} - \left(b + b\right)}{a \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

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

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