quadp (p42, positive)

Average Error: 34.5 → 6.9

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.7868277875676773 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.459714357747724 \cdot 10^{-297}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.4805071658220791 \cdot 10^{102}:\\ \;\;\;\;\frac{4}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \frac{-c}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.5
Target	21.5
Herbie	6.9

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

Derivation

1. Split input into 4 regimes

2. if b < -4.7868277875676773e101

   1. Initial program 47.1

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

   2. Simplified 47.1

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

   3. Taylor expanded around -inf 4.1

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

   4. Simplified 4.1

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

   if -4.7868277875676773e101 < b < -2.459714357747724e-297

   1. Initial program 9.5

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

   2. Simplified 9.5

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

   if -2.459714357747724e-297 < b < 2.4805071658220791e102

   1. Initial program 32.0

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

   2. Simplified 32.0

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

   4. Applied flip-- 32.0

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

   5. Simplified 16.3

      \[\leadsto \frac{\frac{\color{blue}{0 - 4 \cdot \left(a \cdot c\right)}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2}\]

   6. Simplified 16.3

      \[\leadsto \frac{\frac{0 - 4 \cdot \left(a \cdot c\right)}{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2}\]
   7. Using strategy rm

   8. Applied sub0-neg 16.3

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

   9. Applied distribute-frac-neg 16.3

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

   10. Applied distribute-frac-neg 16.3

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

   11. Simplified 15.5

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

   13. Applied associate-/r* 15.5

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

   14. Simplified 9.1

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

   if 2.4805071658220791e102 < b

   1. Initial program 59.5

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

   2. Simplified 59.5

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

   3. Taylor expanded around inf 2.7

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.7868277875676773 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.459714357747724 \cdot 10^{-297}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.4805071658220791 \cdot 10^{102}:\\ \;\;\;\;\frac{4}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \frac{-c}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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