Average Error: 33.2 → 11.9
Time: 2.7m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.427092433976878 \cdot 10^{-11}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le 1.5848355210534097 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.5
Herbie11.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.427092433976878e-11

    1. Initial program 55.0

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

    2. Taylor expanded around -inf 45.5

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

    3. Applied simplify6.1

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

    if -6.427092433976878e-11 < b < 1.5848355210534097e-55

    1. Initial program 18.4

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

    3. Applied flip--22.8

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

    4. Applied simplify18.4

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

    5. Applied simplify18.4

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

    if 1.5848355210534097e-55 < b

    1. Initial program 26.3

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

    2. Taylor expanded around inf 13.3

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

    3. Applied simplify10.4

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

  4. Applied simplify11.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -6.427092433976878 \cdot 10^{-11}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le 1.5848355210534097 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018170 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))