Average Error: 33.3 → 9.7
Time: 1.4m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -5.578225228812023 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le 2.9211024028781706 \cdot 10^{+88}:\\ \;\;\;\;\left(\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b/2}{a}\right) - \left(\frac{b/2}{a} - \frac{c}{\frac{b/2}{\frac{1}{2}}}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b/2 < -5.578225228812023e-54

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 18.3

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

    3. Applied simplify7.5

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

    if -5.578225228812023e-54 < b/2 < 2.9211024028781706e+88

    1. Initial program 13.5

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

    3. Applied div-inv13.7

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

    if 2.9211024028781706e+88 < b/2

    1. Initial program 42.3

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

    3. Applied div-inv42.4

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

    4. Taylor expanded around inf 9.4

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

    5. Applied simplify3.6

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1063058331 1508344079 3191715834 2470104540 4213459606 1468189912)' 
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))