quadp (p42, positive)

Average Error: 34.2 → 6.8

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.78597839720700856 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{elif}\;b \le 3.3471968526963819 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{\frac{\frac{2}{4}}{c} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)} \cdot \frac{\frac{1}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r94954 = b;
        double r94955 = -r94954;
        double r94956 = r94954 * r94954;
        double r94957 = 4.0;
        double r94958 = a;
        double r94959 = c;
        double r94960 = r94958 * r94959;
        double r94961 = r94957 * r94960;
        double r94962 = r94956 - r94961;
        double r94963 = sqrt(r94962);
        double r94964 = r94955 + r94963;
        double r94965 = 2.0;
        double r94966 = r94965 * r94958;
        double r94967 = r94964 / r94966;
        return r94967;
}

↓

double f(double a, double b, double c) {
        double r94968 = b;
        double r94969 = -5.238946631357967e+127;
        bool r94970 = r94968 <= r94969;
        double r94971 = 1.0;
        double r94972 = c;
        double r94973 = r94972 / r94968;
        double r94974 = a;
        double r94975 = r94968 / r94974;
        double r94976 = r94973 - r94975;
        double r94977 = r94971 * r94976;
        double r94978 = 1.7859783972070086e-284;
        bool r94979 = r94968 <= r94978;
        double r94980 = 1.0;
        double r94981 = 2.0;
        double r94982 = r94980 / r94981;
        double r94983 = -r94968;
        double r94984 = r94968 * r94968;
        double r94985 = 4.0;
        double r94986 = r94974 * r94972;
        double r94987 = r94985 * r94986;
        double r94988 = r94984 - r94987;
        double r94989 = sqrt(r94988);
        double r94990 = r94983 + r94989;
        double r94991 = r94990 / r94974;
        double r94992 = r94982 * r94991;
        double r94993 = 3.347196852696382e+84;
        bool r94994 = r94968 <= r94993;
        double r94995 = r94981 / r94985;
        double r94996 = r94995 / r94972;
        double r94997 = cbrt(r94980);
        double r94998 = r94997 * r94997;
        double r94999 = r94996 * r94998;
        double r95000 = r94980 / r94999;
        double r95001 = r94980 / r94997;
        double r95002 = r94983 - r94989;
        double r95003 = r95001 / r95002;
        double r95004 = r95000 * r95003;
        double r95005 = -1.0;
        double r95006 = r95005 * r94973;
        double r95007 = r94994 ? r95004 : r95006;
        double r95008 = r94979 ? r94992 : r95007;
        double r95009 = r94970 ? r94977 : r95008;
        return r95009;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.6
Herbie	6.8

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

Derivation

1. Split input into 4 regimes

2. if b < -5.238946631357967e+127

   1. Initial program 54.2

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

   2. Taylor expanded around -inf 3.3

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

   3. Simplified 3.3

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

   if -5.238946631357967e+127 < b < 1.7859783972070086e-284

   1. Initial program 8.6

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

   3. Applied *-un-lft-identity 8.6

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

   4. Applied times-frac 8.6

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

   if 1.7859783972070086e-284 < b < 3.347196852696382e+84

   1. Initial program 33.2

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

   3. Applied flip-+ 33.2

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

   4. Simplified 16.5

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

   6. Applied *-un-lft-identity 16.5

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]

   7. Applied *-un-lft-identity 16.5

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]

   8. Applied times-frac 16.5

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

   9. Applied associate-/l* 16.7

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}\]

   10. Simplified 16.1

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

   12. Applied associate-/l* 16.2

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

   13. Simplified 10.1

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

   15. Applied add-cube-cbrt 10.1

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

   16. Applied add-sqr-sqrt 10.1

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

   17. Applied times-frac 10.1

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

   18. Applied times-frac 9.9

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

   19. Simplified 9.9

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

   20. Simplified 9.9

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

   if 3.347196852696382e+84 < b

   1. Initial program 58.6

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

   2. Taylor expanded around inf 2.9

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.78597839720700856 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{elif}\;b \le 3.3471968526963819 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{\frac{\frac{2}{4}}{c} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)} \cdot \frac{\frac{1}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))