The quadratic formula (r2)

Average Error: 34.7 → 10.5

Time: 7.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -6.77190793437936228 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le -1.466065355378786 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le -6.27327853977469935 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 1.57632464397167146 \cdot 10^{69}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r97856 = b;
        double r97857 = -r97856;
        double r97858 = r97856 * r97856;
        double r97859 = 4.0;
        double r97860 = a;
        double r97861 = c;
        double r97862 = r97860 * r97861;
        double r97863 = r97859 * r97862;
        double r97864 = r97858 - r97863;
        double r97865 = sqrt(r97864);
        double r97866 = r97857 - r97865;
        double r97867 = 2.0;
        double r97868 = r97867 * r97860;
        double r97869 = r97866 / r97868;
        return r97869;
}

↓

double f(double a, double b, double c) {
        double r97870 = b;
        double r97871 = -6.771907934379362e-77;
        bool r97872 = r97870 <= r97871;
        double r97873 = 1.0;
        double r97874 = 2.0;
        double r97875 = r97873 / r97874;
        double r97876 = -2.0;
        double r97877 = c;
        double r97878 = r97877 / r97870;
        double r97879 = r97876 * r97878;
        double r97880 = r97875 * r97879;
        double r97881 = -1.466065355378786e-87;
        bool r97882 = r97870 <= r97881;
        double r97883 = r97870 * r97870;
        double r97884 = 4.0;
        double r97885 = a;
        double r97886 = r97885 * r97877;
        double r97887 = r97884 * r97886;
        double r97888 = r97883 - r97887;
        double r97889 = -r97888;
        double r97890 = fma(r97870, r97870, r97889);
        double r97891 = r97875 * r97890;
        double r97892 = r97891 / r97885;
        double r97893 = -r97870;
        double r97894 = sqrt(r97888);
        double r97895 = r97893 + r97894;
        double r97896 = r97892 / r97895;
        double r97897 = -6.273278539774699e-139;
        bool r97898 = r97870 <= r97897;
        double r97899 = 1.5763246439716715e+69;
        bool r97900 = r97870 <= r97899;
        double r97901 = r97893 / r97885;
        double r97902 = r97894 / r97885;
        double r97903 = r97901 - r97902;
        double r97904 = r97875 * r97903;
        double r97905 = -1.0;
        double r97906 = r97870 / r97885;
        double r97907 = r97905 * r97906;
        double r97908 = r97900 ? r97904 : r97907;
        double r97909 = r97898 ? r97880 : r97908;
        double r97910 = r97882 ? r97896 : r97909;
        double r97911 = r97872 ? r97880 : r97910;
        return r97911;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.7
Target	21.3
Herbie	10.5

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

Derivation

1. Split input into 4 regimes

2. if b < -6.771907934379362e-77 or -1.466065355378786e-87 < b < -6.273278539774699e-139

   1. Initial program 51.1

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

   3. Applied clear-num 51.1

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

   5. Applied *-un-lft-identity 51.1

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

   6. Applied times-frac 51.1

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

   7. Applied add-cube-cbrt 51.1

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

   8. Applied times-frac 51.1

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

   9. Simplified 51.1

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

   10. Simplified 51.1

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

   11. Taylor expanded around -inf 11.6

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

   if -6.771907934379362e-77 < b < -1.466065355378786e-87

   1. Initial program 28.6

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

   3. Applied clear-num 28.7

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

   5. Applied flip-- 28.7

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

   6. Applied associate-/r/ 28.7

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

   7. Applied associate-/r* 28.7

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

   8. Simplified 28.8

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

   if -6.273278539774699e-139 < b < 1.5763246439716715e+69

   1. Initial program 11.6

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

   3. Applied clear-num 11.7

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

   5. Applied *-un-lft-identity 11.7

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

   6. Applied times-frac 11.7

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

   7. Applied add-cube-cbrt 11.7

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

   8. Applied times-frac 11.7

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

   9. Simplified 11.7

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

   10. Simplified 11.5

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

   12. Applied div-sub 11.5

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

   if 1.5763246439716715e+69 < b

   1. Initial program 42.5

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

   3. Applied clear-num 42.6

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

   4. Taylor expanded around 0 5.0

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.77190793437936228 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le -1.466065355378786 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le -6.27327853977469935 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 1.57632464397167146 \cdot 10^{69}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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