quadm (p42, negative)

Average Error: 35.8 → 7.2

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.1911797219824148 \cdot 10^{108}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.9472897126842423 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{4}}}{\frac{\left(2 \cdot a\right) \cdot \frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}}{c}}\\ \mathbf{elif}\;b \le 9.66145675560502778 \cdot 10^{79}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	35.8
Target	21.0
Herbie	7.2

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

Derivation

1. Split input into 4 regimes

2. if b < -2.1911797219824148e108

   1. Initial program 60.4

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

   2. Taylor expanded around -inf 2.8

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

   if -2.1911797219824148e108 < b < 1.9472897126842423e-302

   1. Initial program 29.3

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

   3. Applied flip-- 29.3

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

   4. Simplified 15.0

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

   5. Simplified 15.0

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]
   6. Using strategy rm

   7. Applied clear-num 15.2

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

   8. Simplified 15.2

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

   10. Applied add-sqr-sqrt 15.3

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

   11. Applied times-frac 15.3

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

   12. Applied *-un-lft-identity 15.3

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

   13. Applied times-frac 15.2

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

   14. Applied associate-/l* 15.1

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

   15. Simplified 10.3

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

   if 1.9472897126842423e-302 < b < 9.66145675560502778e79

   1. Initial program 9.1

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

   3. Applied div-inv 9.2

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

   if 9.66145675560502778e79 < b

   1. Initial program 44.1

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

   2. Taylor expanded around inf 3.9

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

   3. Simplified 3.9

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1911797219824148 \cdot 10^{108}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.9472897126842423 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{4}}}{\frac{\left(2 \cdot a\right) \cdot \frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}}{c}}\\ \mathbf{elif}\;b \le 9.66145675560502778 \cdot 10^{79}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(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)))