Average Error: 34.0 → 8.7
Time: 1.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -3.615631846772334 \cdot 10^{+78}:\\ \;\;\;\;\frac{c}{\frac{b/2}{\frac{-1}{2}}}\\ \mathbf{if}\;b/2 \le -6.177983723551376 \cdot 10^{-248}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b/2 \cdot b/2 - a \cdot c} - b/2}}{a}\\ \mathbf{if}\;b/2 \le 9.149481453724965 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{1}{2} - \frac{b/2}{a} \cdot 2\\ \end{array}\]

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 < -3.615631846772334e+78

    1. Initial program 58.1

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

    2. Taylor expanded around -inf 14.5

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

    3. Applied simplify3.1

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

    if -3.615631846772334e+78 < b/2 < -6.177983723551376e-248

    1. Initial program 33.3

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

    3. Applied flip--33.4

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

    4. Applied simplify16.1

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

    5. Applied simplify16.1

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

    if -6.177983723551376e-248 < b/2 < 9.149481453724965e+131

    1. Initial program 9.5

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

    3. Applied clear-num9.6

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

    if 9.149481453724965e+131 < b/2

    1. Initial program 53.9

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

    2. Taylor expanded around inf 10.4

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

    3. Applied simplify2.9

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' +o rules:numerics
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))