Average Error: 34.0 → 21.5
Time: 36.3s
Precision: 64
Internal Precision: 3136
\[\frac{\left(-b\right) + \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} = -\infty:\\ \;\;\;\;\frac{4 \cdot c}{2 \cdot a} \cdot \frac{a}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{elif}\;\frac{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} \le -1.3435509197748303 \cdot 10^{-236}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a}\\ \mathbf{elif}\;\frac{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} \le -0.0:\\ \;\;\;\;\frac{4 \cdot c}{\frac{\left(\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}{a}}\\ \mathbf{elif}\;\frac{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} \le 1.0862638443547306 \cdot 10^{+305}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus b

Bits error versus a

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes
  2. if (/ (+ (- b) (sqrt (- (pow b 2) (* (* 4 a) c)))) (* 2 a)) < -inf.0

    1. Initial program 62.1

      \[\frac{\left(-b\right) + \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm
    3. Applied flip-+63.3

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
    4. Applied associate-/l/63.3

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}\right)}}\]
    5. Simplified50.5

      \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}\right)}\]
    6. Using strategy rm
    7. Applied times-frac50.2

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

    if -inf.0 < (/ (+ (- b) (sqrt (- (pow b 2) (* (* 4 a) c)))) (* 2 a)) < -1.3435509197748303e-236 or -0.0 < (/ (+ (- b) (sqrt (- (pow b 2) (* (* 4 a) c)))) (* 2 a)) < 1.0862638443547306e+305

    1. Initial program 4.5

      \[\frac{\left(-b\right) + \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Taylor expanded around -inf 62.6

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{b}\right)\right)} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    3. Simplified4.5

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

    if -1.3435509197748303e-236 < (/ (+ (- b) (sqrt (- (pow b 2) (* (* 4 a) c)))) (* 2 a)) < -0.0

    1. Initial program 53.0

      \[\frac{\left(-b\right) + \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm
    3. Applied flip-+53.2

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
    4. Applied associate-/l/53.6

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}\right)}}\]
    5. Simplified15.3

      \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}\right)}\]
    6. Using strategy rm
    7. Applied associate-/l*5.8

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

    if 1.0862638443547306e+305 < (/ (+ (- b) (sqrt (- (pow b 2) (* (* 4 a) c)))) (* 2 a))

    1. Initial program 61.8

      \[\frac{\left(-b\right) + \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm
    3. Applied flip-+62.1

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
    4. Applied associate-/l/62.1

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}\right)}}\]
    5. Simplified51.4

      \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}\right)}\]
    6. Taylor expanded around -inf 51.3

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.
  4. Final simplification21.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} = -\infty:\\ \;\;\;\;\frac{4 \cdot c}{2 \cdot a} \cdot \frac{a}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{elif}\;\frac{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} \le -1.3435509197748303 \cdot 10^{-236}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a}\\ \mathbf{elif}\;\frac{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} \le -0.0:\\ \;\;\;\;\frac{4 \cdot c}{\frac{\left(\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(2 \cdot a\right)}{a}}\\ \mathbf{elif}\;\frac{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} \le 1.0862638443547306 \cdot 10^{+305}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 36.3s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (b a c)
  :name "(-b + sqrt(b^2 - 4*a*c)) / (2a)"
  (/ (+ (- b) (sqrt (- (pow b 2) (* (* 4 a) c)))) (* 2 a)))