Average Error: 33.4 → 9.7
Time: 37.8s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.91564606410173 \cdot 10^{+122}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.820248000579309 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

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

Derivation

  1. Split input into 3 regimes

  2. if b < -2.91564606410173e+122

    1. Initial program 50.0

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

    2. Initial simplification50.0

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

    4. Applied div-sub50.0

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

    5. Taylor expanded around -inf 2.6

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

    if -2.91564606410173e+122 < b < 5.820248000579309e-42

    1. Initial program 13.6

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

    2. Initial simplification13.6

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

    4. Applied div-sub13.6

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

    if 5.820248000579309e-42 < b

    1. Initial program 53.5

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

    2. Initial simplification53.5

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

    4. Applied div-sub54.4

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

    6. Applied add-sqr-sqrt54.4

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

    7. Applied sqrt-prod56.1

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

    8. Applied associate-/l*56.4

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

    9. Taylor expanded around inf 7.3

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

    10. Simplified7.3

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.91564606410173 \cdot 10^{+122}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.820248000579309 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 37.8s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))