quadm (p42, negative)

Average Error: 34.1 → 9.9

Time: 5.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.6006169463337958 \cdot 10^{-70}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \le 1.8526227140702638 \cdot 10^{107}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) - b \cdot 2}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.1
Target	20.7
Herbie	9.9

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -3.6006169463337958e-70

   1. Initial program 53.7

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

   2. Taylor expanded around -inf 8.8

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

   3. Simplified 8.8

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

   if -3.6006169463337958e-70 < b < 1.8526227140702638e107

   1. Initial program 12.5

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

   3. Applied div-inv 12.6

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

   4. Simplified 12.6

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

   if 1.8526227140702638e107 < b

   1. Initial program 49.7

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

   2. Taylor expanded around inf 10.3

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

   3. Simplified 4.3

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.6006169463337958 \cdot 10^{-70}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \mathbf{elif}\;b \le 1.8526227140702638 \cdot 10^{107}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) - b \cdot 2}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))