Average Error: 33.3 → 6.8
Time: 41.6s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.318102466069036 \cdot 10^{+150}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -2.470005776589605 \cdot 10^{-265}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}}\\ \mathbf{elif}\;b \le 6.594734415149172 \cdot 10^{+81}:\\ \;\;\;\;-4 \cdot \frac{\frac{1}{2} \cdot c}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -4}{4 \cdot b}\\ \end{array}\]

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

Original33.3
Target20.7
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.318102466069036e+150

    1. Initial program 59.6

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

    2. Initial simplification59.6

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

    3. Taylor expanded around -inf 2.0

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

    4. Simplified2.0

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

    if -8.318102466069036e+150 < b < -2.470005776589605e-265

    1. Initial program 7.9

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

    2. Initial simplification7.9

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

    4. Applied clear-num8.1

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

    if -2.470005776589605e-265 < b < 6.594734415149172e+81

    1. Initial program 30.1

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

    2. Initial simplification30.1

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

    4. Applied flip--30.3

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

    5. Applied associate-/l/34.7

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

    6. Simplified20.9

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

    8. Applied associate-/l*16.5

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

    10. Applied *-un-lft-identity16.5

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

    11. Applied times-frac16.5

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

    12. Simplified16.5

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

    13. Simplified9.7

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

    if 6.594734415149172e+81 < b

    1. Initial program 58.0

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

    2. Initial simplification58.0

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

    4. Applied flip--58.1

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

    5. Applied associate-/l/58.4

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

    6. Simplified31.9

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

    8. Applied associate-/l*31.0

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

    9. Taylor expanded around 0 3.7

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.318102466069036 \cdot 10^{+150}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -2.470005776589605 \cdot 10^{-265}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}}\\ \mathbf{elif}\;b \le 6.594734415149172 \cdot 10^{+81}:\\ \;\;\;\;-4 \cdot \frac{\frac{1}{2} \cdot c}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -4}{4 \cdot b}\\ \end{array}\]

Runtime

Time bar (total: 41.6s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))