Average Error: 32.7 → 8.8
Time: 3.9m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.1806211413086436 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(-b\right) + b}{a + a} - \frac{\frac{c}{b}}{1}\\ \mathbf{if}\;b \le -7.703564241260136 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{if}\;b \le 8.26868467476856 \cdot 10^{+105}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -3.1806211413086436e+62

    1. Initial program 56.2

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 41.0

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

    3. Applied simplify3.5

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

    if -3.1806211413086436e+62 < b < -7.703564241260136e-154

    1. Initial program 36.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip--36.9

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    4. Applied simplify15.8

      \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    if -7.703564241260136e-154 < b < 8.26868467476856e+105

    1. Initial program 10.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    if 8.26868467476856e+105 < b

    1. Initial program 45.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 10.6

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

    3. Applied simplify4.3

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

Runtime

Time bar (total: 3.9m)Debug log

herbie shell --seed '#(1743936871 1855164119 3668777427 1254258049 132811564 1366975197)' 
(FPCore (a b c)
  :name "NMSE p42, negative"
  :pre (and (>= (* b b) (* 4 (* a c))) (!= a 0))
  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))