quadp (p42, positive)

Average Error: 33.8 → 10.3

Time: 15.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.91962817906715367126033645969528543778 \cdot 10^{153}:\\ \;\;\;\;\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;b \le 2.071930020515770918527743961403466592109 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r77164 = b;
        double r77165 = -r77164;
        double r77166 = r77164 * r77164;
        double r77167 = 4.0;
        double r77168 = a;
        double r77169 = c;
        double r77170 = r77168 * r77169;
        double r77171 = r77167 * r77170;
        double r77172 = r77166 - r77171;
        double r77173 = sqrt(r77172);
        double r77174 = r77165 + r77173;
        double r77175 = 2.0;
        double r77176 = r77175 * r77168;
        double r77177 = r77174 / r77176;
        return r77177;
}

↓

double f(double a, double b, double c) {
        double r77178 = b;
        double r77179 = -4.919628179067154e+153;
        bool r77180 = r77178 <= r77179;
        double r77181 = 2.0;
        double r77182 = c;
        double r77183 = r77182 / r77178;
        double r77184 = r77181 * r77183;
        double r77185 = 2.0;
        double r77186 = a;
        double r77187 = r77178 / r77186;
        double r77188 = r77185 * r77187;
        double r77189 = r77184 - r77188;
        double r77190 = 1.0;
        double r77191 = r77190 / r77181;
        double r77192 = r77189 * r77191;
        double r77193 = 2.071930020515771e-74;
        bool r77194 = r77178 <= r77193;
        double r77195 = r77178 * r77178;
        double r77196 = 4.0;
        double r77197 = r77186 * r77182;
        double r77198 = r77196 * r77197;
        double r77199 = r77195 - r77198;
        double r77200 = sqrt(r77199);
        double r77201 = r77200 - r77178;
        double r77202 = r77201 / r77186;
        double r77203 = r77191 * r77202;
        double r77204 = -2.0;
        double r77205 = r77204 * r77183;
        double r77206 = r77191 * r77205;
        double r77207 = r77194 ? r77203 : r77206;
        double r77208 = r77180 ? r77192 : r77207;
        return r77208;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	20.4
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if b < -4.919628179067154e+153

   1. Initial program 63.8

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

   2. Simplified 63.8

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

   4. Applied clear-num 63.8

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

   6. Applied *-un-lft-identity 63.8

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

   7. Applied add-cube-cbrt 63.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   8. Applied times-frac 63.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]

   9. Simplified 63.8

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   10. Simplified 63.8

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

   12. Applied add-sqr-sqrt 63.8

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

   13. Applied times-frac 63.8

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

   14. Applied associate-*l* 63.8

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

   15. Simplified 63.8

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

   16. Taylor expanded around -inf 2.1

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

   if -4.919628179067154e+153 < b < 2.071930020515771e-74

   1. Initial program 12.8

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

   2. Simplified 12.8

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

   4. Applied clear-num 13.0

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

   6. Applied *-un-lft-identity 13.0

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

   7. Applied add-cube-cbrt 13.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   8. Applied times-frac 13.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]

   9. Simplified 13.0

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   10. Simplified 13.0

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

   12. Applied add-sqr-sqrt 13.0

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

   13. Applied times-frac 13.0

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

   14. Applied associate-*l* 13.0

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

   15. Simplified 12.8

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

   if 2.071930020515771e-74 < b

   1. Initial program 53.1

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

   2. Simplified 53.1

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

   4. Applied clear-num 53.2

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

   6. Applied *-un-lft-identity 53.2

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

   7. Applied add-cube-cbrt 53.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   8. Applied times-frac 53.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]

   9. Simplified 53.2

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   10. Simplified 53.1

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

   12. Applied add-sqr-sqrt 53.1

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

   13. Applied times-frac 53.1

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

   14. Applied associate-*l* 53.1

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

   15. Simplified 53.1

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

   16. Taylor expanded around inf 9.2

       \[\leadsto 1 \cdot \left(\frac{\sqrt{1}}{2} \cdot \color{blue}{\left(-2 \cdot \frac{c}{b}\right)}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.91962817906715367126033645969528543778 \cdot 10^{153}:\\ \;\;\;\;\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;b \le 2.071930020515770918527743961403466592109 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))