Average Error: 33.4 → 7.3
Time: 1.3m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b/2}{\frac{-1}{2}} \le -7.348936897450024 \cdot 10^{+109}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \mathbf{if}\;\frac{b/2}{\frac{-1}{2}} \le -5.163296647663301 \cdot 10^{-308}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b/2} \cdot \sqrt[3]{-b/2}\right) \cdot \left(\sqrt[3]{-b/2}\right) + \left(-\sqrt{b/2 \cdot b/2 - a \cdot c}\right))_*}{a}\\ \mathbf{if}\;\frac{b/2}{\frac{-1}{2}} \le 9.520254021401762 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{c}{1}}{\sqrt{b/2 \cdot b/2 - c \cdot a} - b/2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\frac{1}{2}\right) \cdot \frac{c}{b/2}}{1}\\ \end{array}\]

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 -1/2) < -7.348936897450024e+109

    1. Initial program 46.2

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

    2. Taylor expanded around inf 3.7

      \[\leadsto \color{blue}{-2 \cdot \frac{b/2}{a}}\]

    if -7.348936897450024e+109 < (/ b/2 -1/2) < -5.163296647663301e-308

    1. Initial program 9.2

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

    3. Applied add-cube-cbrt9.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b/2} \cdot \sqrt[3]{-b/2}\right) \cdot \sqrt[3]{-b/2}} - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]

    4. Applied fma-neg9.6

      \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{-b/2} \cdot \sqrt[3]{-b/2}\right) \cdot \left(\sqrt[3]{-b/2}\right) + \left(-\sqrt{b/2 \cdot b/2 - a \cdot c}\right))_*}}{a}\]

    if -5.163296647663301e-308 < (/ b/2 -1/2) < 9.520254021401762e+19

    1. Initial program 28.2

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

    3. Applied flip--28.3

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

    4. Applied simplify17.7

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

    5. Applied simplify17.7

      \[\leadsto \frac{\frac{c \cdot a}{\color{blue}{\sqrt{b/2 \cdot b/2 - a \cdot c} - b/2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity17.7

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

    8. Applied times-frac14.5

      \[\leadsto \frac{\color{blue}{\frac{c}{1} \cdot \frac{a}{\sqrt{b/2 \cdot b/2 - a \cdot c} - b/2}}}{a}\]

    9. Applied associate-/l*9.8

      \[\leadsto \color{blue}{\frac{\frac{c}{1}}{\frac{a}{\frac{a}{\sqrt{b/2 \cdot b/2 - a \cdot c} - b/2}}}}\]

    10. Applied simplify9.8

      \[\leadsto \frac{\frac{c}{1}}{\color{blue}{\sqrt{b/2 \cdot b/2 - c \cdot a} - b/2}}\]

    if 9.520254021401762e+19 < (/ b/2 -1/2)

    1. Initial program 55.6

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

    2. Taylor expanded around -inf 43.3

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

    3. Applied simplify4.9

      \[\leadsto \color{blue}{\frac{\left(-\frac{1}{2}\right) \cdot \frac{c}{b/2}}{1}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +o rules:numerics
(FPCore (a b/2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))