Average Error: 33.2 → 10.3
Time: 29.4s
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -4.005537006288397 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le 1.026219862232508 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b/2} + \frac{\left(-b/2\right) - b/2}{a}\\ \end{array}\]

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 < -4.005537006288397e-85

    1. Initial program 51.7

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

    2. Taylor expanded around -inf 20.9

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

    3. Applied simplify10.3

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

    if -4.005537006288397e-85 < b/2 < 1.026219862232508e+81

    1. Initial program 12.4

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

    3. Applied clear-num12.6

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

    if 1.026219862232508e+81 < b/2

    1. Initial program 40.7

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

    3. Applied clear-num40.8

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

    4. Taylor expanded around inf 11.1

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

    5. Applied simplify4.6

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

Runtime

Time bar (total: 29.4s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(FPCore (a b/2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))