Average Error: 33.4 → 9.3
Time: 5.2m
Precision: 64
Internal Precision: 128
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.224659303475224 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -8.181768917333295 \cdot 10^{-83}:\\ \;\;\;\;\frac{c \cdot a}{a} \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \le 3.8865174268969883 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

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 < -3.224659303475224e-14

    1. Initial program 54.7

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

    2. Taylor expanded around inf 54.7

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified54.7

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

    4. Taylor expanded around -inf 6.1

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

    if -3.224659303475224e-14 < b_2 < -8.181768917333295e-83

    1. Initial program 37.8

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

    2. Taylor expanded around inf 37.8

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified37.8

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

    5. Applied *-un-lft-identity37.8

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

    6. Applied associate-/l*37.8

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

    8. Applied flip--37.9

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

    9. Applied associate-/r/38.0

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

    10. Applied add-cube-cbrt38.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{a}{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\]

    11. Applied times-frac38.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{a}{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{\sqrt[3]{1}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    12. Simplified18.2

      \[\leadsto \color{blue}{\frac{c \cdot a}{a}} \cdot \frac{\sqrt[3]{1}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    13. Simplified18.2

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

    if -8.181768917333295e-83 < b_2 < 3.8865174268969883e+99

    1. Initial program 12.7

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

    2. Taylor expanded around inf 12.7

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified12.7

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

    if 3.8865174268969883e+99 < b_2

    1. Initial program 44.9

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

    2. Taylor expanded around inf 44.9

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified44.9

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

    5. Applied *-un-lft-identity44.9

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

    6. Applied associate-/l*45.0

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

    7. Taylor expanded around 0 3.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.224659303475224 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -8.181768917333295 \cdot 10^{-83}:\\ \;\;\;\;\frac{c \cdot a}{a} \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \le 3.8865174268969883 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))