quadp (p42, positive)

Average Error: 33.6 → 9.0

Time: 21.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.3282248930815427 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{c}{\frac{a \cdot c}{b} \cdot 2} \cdot \frac{-4 \cdot a}{a}}{2}\\ \mathbf{elif}\;b \le -2.170460433232697 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 7.386841020175994 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{-4 \cdot a}{a} \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot a}{a} \cdot \frac{c}{b + b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2760056 = b;
        double r2760057 = -r2760056;
        double r2760058 = r2760056 * r2760056;
        double r2760059 = 4.0;
        double r2760060 = a;
        double r2760061 = c;
        double r2760062 = r2760060 * r2760061;
        double r2760063 = r2760059 * r2760062;
        double r2760064 = r2760058 - r2760063;
        double r2760065 = sqrt(r2760064);
        double r2760066 = r2760057 + r2760065;
        double r2760067 = 2.0;
        double r2760068 = r2760067 * r2760060;
        double r2760069 = r2760066 / r2760068;
        return r2760069;
}

↓

double f(double a, double b, double c) {
        double r2760070 = b;
        double r2760071 = -1.3282248930815427e+154;
        bool r2760072 = r2760070 <= r2760071;
        double r2760073 = c;
        double r2760074 = a;
        double r2760075 = r2760074 * r2760073;
        double r2760076 = r2760075 / r2760070;
        double r2760077 = 2.0;
        double r2760078 = r2760076 * r2760077;
        double r2760079 = r2760073 / r2760078;
        double r2760080 = -4.0;
        double r2760081 = r2760080 * r2760074;
        double r2760082 = r2760081 / r2760074;
        double r2760083 = r2760079 * r2760082;
        double r2760084 = r2760083 / r2760077;
        double r2760085 = -2.170460433232697e-296;
        bool r2760086 = r2760070 <= r2760085;
        double r2760087 = r2760070 * r2760070;
        double r2760088 = fma(r2760075, r2760080, r2760087);
        double r2760089 = sqrt(r2760088);
        double r2760090 = r2760089 - r2760070;
        double r2760091 = r2760090 / r2760074;
        double r2760092 = r2760091 / r2760077;
        double r2760093 = 7.386841020175994e+78;
        bool r2760094 = r2760070 <= r2760093;
        double r2760095 = fma(r2760073, r2760081, r2760087);
        double r2760096 = sqrt(r2760095);
        double r2760097 = r2760070 + r2760096;
        double r2760098 = r2760073 / r2760097;
        double r2760099 = r2760082 * r2760098;
        double r2760100 = r2760099 / r2760077;
        double r2760101 = r2760070 + r2760070;
        double r2760102 = r2760073 / r2760101;
        double r2760103 = r2760082 * r2760102;
        double r2760104 = r2760103 / r2760077;
        double r2760105 = r2760094 ? r2760100 : r2760104;
        double r2760106 = r2760086 ? r2760092 : r2760105;
        double r2760107 = r2760072 ? r2760084 : r2760106;
        return r2760107;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
Target	21.0
Herbie	9.0

\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -1.3282248930815427e+154

   1. Initial program 60.9

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

   2. Simplified 60.9

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

   4. Applied flip-- 62.3

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

   5. Simplified 62.5

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

   7. Applied *-un-lft-identity 62.5

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

   8. Applied *-un-lft-identity 62.5

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

   9. Applied *-un-lft-identity 62.5

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

   10. Applied times-frac 62.5

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

   11. Applied times-frac 62.5

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

   12. Simplified 62.5

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

   13. Simplified 62.4

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

   14. Taylor expanded around -inf 21.0

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

   if -1.3282248930815427e+154 < b < -2.170460433232697e-296

   1. Initial program 8.7

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

   2. Simplified 8.7

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

   if -2.170460433232697e-296 < b < 7.386841020175994e+78

   1. Initial program 30.4

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

   2. Simplified 30.4

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

   4. Applied flip-- 30.5

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

   5. Simplified 16.7

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

   7. Applied *-un-lft-identity 16.7

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

   8. Applied *-un-lft-identity 16.7

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

   9. Applied *-un-lft-identity 16.7

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

   10. Applied times-frac 16.7

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

   11. Applied times-frac 16.7

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

   12. Simplified 16.7

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

   13. Simplified 10.1

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

   if 7.386841020175994e+78 < b

   1. Initial program 57.6

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

   2. Simplified 57.6

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

   4. Applied flip-- 57.7

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

   5. Simplified 30.9

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

   7. Applied *-un-lft-identity 30.9

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

   8. Applied *-un-lft-identity 30.9

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

   9. Applied *-un-lft-identity 30.9

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

   10. Applied times-frac 30.9

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

   11. Applied times-frac 30.9

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

   12. Simplified 30.9

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

   13. Simplified 28.6

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

   14. Taylor expanded around 0 3.0

       \[\leadsto \frac{1 \cdot \left(\frac{c}{b + \color{blue}{b}} \cdot \frac{a \cdot -4}{a}\right)}{2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3282248930815427 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{c}{\frac{a \cdot c}{b} \cdot 2} \cdot \frac{-4 \cdot a}{a}}{2}\\ \mathbf{elif}\;b \le -2.170460433232697 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 7.386841020175994 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{-4 \cdot a}{a} \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot a}{a} \cdot \frac{c}{b + b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 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)))