Average Error: 33.6 → 11.7
Time: 2.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3326732363290293 \cdot 10^{-08}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le -7.002005006180931 \cdot 10^{-73}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le -7.670780076949882 \cdot 10^{-135}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le 16578392417.099113:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.4
Herbie11.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.3326732363290293e-08 or -7.002005006180931e-73 < b < -7.670780076949882e-135

    1. Initial program 51.5

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

    2. Taylor expanded around -inf 46.7

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

    3. Applied simplify10.2

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

    if -1.3326732363290293e-08 < b < -7.002005006180931e-73

    1. Initial program 41.0

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

    if -7.670780076949882e-135 < b < 16578392417.099113

    1. Initial program 11.8

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

    if 16578392417.099113 < b

    1. Initial program 31.9

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

    2. Taylor expanded around inf 11.5

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

    3. Applied simplify7.3

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

  4. Applied simplify11.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.3326732363290293 \cdot 10^{-08}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le -7.002005006180931 \cdot 10^{-73}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le -7.670780076949882 \cdot 10^{-135}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le 16578392417.099113:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' 
(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)))