The quadratic formula (r1)

Average Error: 34.2 → 15.6

Time: 23.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 1.433410545098929447802382436964367105587 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 6.146061404567155775272975402192884048556 \cdot 10^{152}:\\ \;\;\;\;\frac{\left(\frac{c \cdot 4}{\sqrt{\sqrt{\mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)} + b}} \cdot \frac{a}{a}\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\sqrt[3]{b} \cdot \sqrt[3]{b}, \sqrt[3]{b}, \sqrt{\mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)}\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(c \cdot \left(-a\right), 4, 0\right)}{b + b}}{a}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r279771 = b;
        double r279772 = -r279771;
        double r279773 = r279771 * r279771;
        double r279774 = 4.0;
        double r279775 = a;
        double r279776 = r279774 * r279775;
        double r279777 = c;
        double r279778 = r279776 * r279777;
        double r279779 = r279773 - r279778;
        double r279780 = sqrt(r279779);
        double r279781 = r279772 + r279780;
        double r279782 = 2.0;
        double r279783 = r279782 * r279775;
        double r279784 = r279781 / r279783;
        return r279784;
}

↓

double f(double a, double b, double c) {
        double r279785 = b;
        double r279786 = 1.4334105450989294e-229;
        bool r279787 = r279785 <= r279786;
        double r279788 = a;
        double r279789 = -r279788;
        double r279790 = 4.0;
        double r279791 = r279789 * r279790;
        double r279792 = c;
        double r279793 = r279785 * r279785;
        double r279794 = fma(r279791, r279792, r279793);
        double r279795 = sqrt(r279794);
        double r279796 = r279795 / r279788;
        double r279797 = r279785 / r279788;
        double r279798 = r279796 - r279797;
        double r279799 = 2.0;
        double r279800 = r279798 / r279799;
        double r279801 = 6.146061404567156e+152;
        bool r279802 = r279785 <= r279801;
        double r279803 = r279792 * r279790;
        double r279804 = r279792 * r279789;
        double r279805 = fma(r279790, r279804, r279793);
        double r279806 = sqrt(r279805);
        double r279807 = r279806 + r279785;
        double r279808 = sqrt(r279807);
        double r279809 = r279803 / r279808;
        double r279810 = r279788 / r279788;
        double r279811 = r279809 * r279810;
        double r279812 = -1.0;
        double r279813 = cbrt(r279785);
        double r279814 = r279813 * r279813;
        double r279815 = fma(r279814, r279813, r279806);
        double r279816 = sqrt(r279815);
        double r279817 = r279812 / r279816;
        double r279818 = r279811 * r279817;
        double r279819 = r279818 / r279799;
        double r279820 = 0.0;
        double r279821 = fma(r279804, r279790, r279820);
        double r279822 = r279785 + r279785;
        double r279823 = r279821 / r279822;
        double r279824 = r279823 / r279788;
        double r279825 = r279824 / r279799;
        double r279826 = r279802 ? r279819 : r279825;
        double r279827 = r279787 ? r279800 : r279826;
        return r279827;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.0
Herbie	15.6

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

Derivation

1. Split input into 3 regimes

2. if b < 1.4334105450989294e-229

   1. Initial program 21.2

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

   2. Simplified 21.2

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

   4. Applied div-sub 21.2

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

   5. Simplified 21.2

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

   if 1.4334105450989294e-229 < b < 6.146061404567156e+152

   1. Initial program 37.7

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

   2. Simplified 37.7

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

   4. Applied flip-- 37.7

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

   5. Simplified 15.7

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

   6. Simplified 15.7

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

   8. Applied *-un-lft-identity 15.7

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

   9. Applied add-sqr-sqrt 15.9

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

   10. Applied *-un-lft-identity 15.9

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

   11. Applied times-frac 15.9

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

   12. Applied times-frac 14.9

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

   13. Simplified 14.9

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

   14. Simplified 7.5

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

   16. Applied add-cube-cbrt 7.6

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

   17. Applied fma-def 7.6

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

   if 6.146061404567156e+152 < b

   1. Initial program 63.8

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

   2. Simplified 63.8

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

   4. Applied flip-- 63.8

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

   5. Simplified 38.4

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

   6. Simplified 38.4

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

   7. Taylor expanded around 0 14.3

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\color{blue}{b} + b}}{a}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.433410545098929447802382436964367105587 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 6.146061404567155775272975402192884048556 \cdot 10^{152}:\\ \;\;\;\;\frac{\left(\frac{c \cdot 4}{\sqrt{\sqrt{\mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)} + b}} \cdot \frac{a}{a}\right) \cdot \frac{-1}{\sqrt{\mathsf{fma}\left(\sqrt[3]{b} \cdot \sqrt[3]{b}, \sqrt[3]{b}, \sqrt{\mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)}\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(c \cdot \left(-a\right), 4, 0\right)}{b + b}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))