Average Error: 33.3 → 9.4
Time: 47.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.9485498422174874 \cdot 10^{+87}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.152686348507162 \cdot 10^{-27}:\\ \;\;\;\;\frac{c \cdot a}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\\ \mathbf{elif}\;b_2 \le -3.0036374972447675 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.355687789763098 \cdot 10^{+87}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.9485498422174874e+87 or -6.152686348507162e-27 < b_2 < -3.0036374972447675e-77

    1. Initial program 54.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 6.1

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -1.9485498422174874e+87 < b_2 < -6.152686348507162e-27

    1. Initial program 45.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--45.6

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Applied associate-/l/46.6

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}\]

    5. Simplified13.8

      \[\leadsto \frac{\color{blue}{a \cdot c}}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\]

    if -3.0036374972447675e-77 < b_2 < 9.355687789763098e+87

    1. Initial program 12.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-sub12.7

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity12.7

      \[\leadsto \frac{-b_2}{a} - \frac{\sqrt{\color{blue}{1 \cdot \left(b_2 \cdot b_2 - a \cdot c\right)}}}{a}\]

    6. Applied sqrt-prod12.7

      \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    7. Applied associate-/l*12.8

      \[\leadsto \frac{-b_2}{a} - \color{blue}{\frac{\sqrt{1}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

    8. Simplified12.8

      \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{1}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    if 9.355687789763098e+87 < b_2

    1. Initial program 41.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 4.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.9485498422174874 \cdot 10^{+87}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.152686348507162 \cdot 10^{-27}:\\ \;\;\;\;\frac{c \cdot a}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\\ \mathbf{elif}\;b_2 \le -3.0036374972447675 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.355687789763098 \cdot 10^{+87}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))