Average Error: 33.2 → 7.0
Time: 49.4s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.325623142188407 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -9.494305176106116 \cdot 10^{-225}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.7760913692530284 \cdot 10^{+28}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -1.325623142188407e+151

    1. Initial program 59.5

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

    3. Applied div-inv59.5

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

    4. Taylor expanded around -inf 2.1

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

    if -1.325623142188407e+151 < b < -9.494305176106116e-225

    1. Initial program 7.1

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

    if -9.494305176106116e-225 < b < 1.7760913692530284e+28

    1. Initial program 25.9

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

    3. Applied div-inv25.9

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

    5. Applied flip-+26.1

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

    6. Applied associate-*l/26.2

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

    7. Simplified16.7

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

    8. Taylor expanded around inf 11.1

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

    if 1.7760913692530284e+28 < b

    1. Initial program 55.7

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

    3. Applied div-inv55.7

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

    4. Taylor expanded around inf 4.9

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

    5. Simplified4.9

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.325623142188407 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -9.494305176106116 \cdot 10^{-225}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.7760913692530284 \cdot 10^{+28}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))