Average Error: 33.4 → 6.1
Time: 3.0m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0096397814194701 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\frac{c}{2} \cdot 4}{2}}{\frac{a}{\frac{b}{c}} - b}\\ \mathbf{if}\;b \le -8.812243082114111 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2}}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{if}\;b \le 2.9749766316821928 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.4
Target20.8
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.0096397814194701e+154

    1. Initial program 62.8

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

    3. Applied clear-num62.8

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

    5. Applied flip--62.9

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

    6. Applied associate-/r/62.9

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

    7. Applied simplify39.6

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

    8. Taylor expanded around -inf 7.6

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

    9. Applied simplify1.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{c}{2} \cdot 4}{2}}{\frac{a}{\frac{b}{c}} - b}}\]

    if -1.0096397814194701e+154 < b < -8.812243082114111e-292

    1. Initial program 34.1

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

    3. Applied clear-num34.1

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

    5. Applied flip--34.2

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

    6. Applied associate-/r/34.2

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

    7. Applied simplify8.2

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

    8. Taylor expanded around 0 8.2

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

    9. Applied simplify7.6

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

    if -8.812243082114111e-292 < b < 2.9749766316821928e+143

    1. Initial program 8.8

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

    3. Applied *-un-lft-identity8.8

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

    4. Applied times-frac8.7

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

    if 2.9749766316821928e+143 < b

    1. Initial program 56.7

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

    2. Taylor expanded around inf 9.6

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

    3. Applied simplify1.7

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

  4. Applied simplify6.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.0096397814194701 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\frac{c}{2} \cdot 4}{2}}{\frac{a}{\frac{b}{c}} - b}\\ \mathbf{if}\;b \le -8.812243082114111 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2}}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{if}\;b \le 2.9749766316821928 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2019053 
(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)))