Average Error: 33.3 → 10.9
Time: 41.1s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.7380529006372414 \cdot 10^{+159}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\left(\left(\frac{a \cdot c}{b} \cdot 2 - b\right) + \left(-b\right)\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{elif}\;b \le -1.807148991901003 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{a \cdot 2}}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}\\ \mathbf{elif}\;b \le 4.45363456258042 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

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

Original33.3
Target21.0
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.7380529006372414e+159

    1. Initial program 62.9

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

    2. Initial simplification62.9

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

    4. Applied flip--62.9

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

    5. Applied associate-/l/62.9

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

    6. Simplified38.1

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

    7. Taylor expanded around -inf 14.4

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

    if -2.7380529006372414e+159 < b < -1.807148991901003e-135

    1. Initial program 40.6

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

    2. Initial simplification40.6

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

    4. Applied flip--40.7

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

    5. Applied associate-/l/43.2

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

    6. Simplified17.9

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

    8. Applied associate-/r*13.5

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

    9. Simplified13.5

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

    if -1.807148991901003e-135 < b < 4.45363456258042e+153

    1. Initial program 10.4

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

    2. Initial simplification10.4

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

    if 4.45363456258042e+153 < b

    1. Initial program 60.6

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

    2. Initial simplification60.6

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

    3. Taylor expanded around inf 2.4

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

    4. Simplified2.4

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.7380529006372414 \cdot 10^{+159}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\left(\left(\frac{a \cdot c}{b} \cdot 2 - b\right) + \left(-b\right)\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{elif}\;b \le -1.807148991901003 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{a \cdot 2}}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}\\ \mathbf{elif}\;b \le 4.45363456258042 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Runtime

Time bar (total: 41.1s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))