Average Error: 33.5 → 9.9
Time: 1.9m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.5551074556147433 \cdot 10^{-63}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{if}\;b \le 3.4694830941587433 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{c}{b}\right) \cdot \left(a \cdot 2\right) + \left(-b\right))_* - b}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.4
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5551074556147433e-63

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 46.2

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

    3. Applied simplify8.5

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

    if -1.5551074556147433e-63 < b < 3.4694830941587433e+120

    1. Initial program 12.8

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

    3. Applied clear-num13.0

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    if 3.4694830941587433e+120 < b

    1. Initial program 49.0

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

    2. Taylor expanded around inf 10.6

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

    3. Applied simplify3.5

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

  4. Applied simplify9.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.5551074556147433 \cdot 10^{-63}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{if}\;b \le 3.4694830941587433 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{c}{b}\right) \cdot \left(a \cdot 2\right) + \left(-b\right))_* - b}{a \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018201 +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)))