The quadratic formula (r1)

Average Error: 33.3 → 10.0

Time: 23.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.82289647433212 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.289226058156428 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1459853 = b;
        double r1459854 = -r1459853;
        double r1459855 = r1459853 * r1459853;
        double r1459856 = 4.0;
        double r1459857 = a;
        double r1459858 = r1459856 * r1459857;
        double r1459859 = c;
        double r1459860 = r1459858 * r1459859;
        double r1459861 = r1459855 - r1459860;
        double r1459862 = sqrt(r1459861);
        double r1459863 = r1459854 + r1459862;
        double r1459864 = 2.0;
        double r1459865 = r1459864 * r1459857;
        double r1459866 = r1459863 / r1459865;
        return r1459866;
}

↓

double f(double a, double b, double c) {
        double r1459867 = b;
        double r1459868 = -4.82289647433212e+153;
        bool r1459869 = r1459867 <= r1459868;
        double r1459870 = c;
        double r1459871 = r1459870 / r1459867;
        double r1459872 = a;
        double r1459873 = r1459867 / r1459872;
        double r1459874 = r1459871 - r1459873;
        double r1459875 = 2.0;
        double r1459876 = r1459874 * r1459875;
        double r1459877 = r1459876 / r1459875;
        double r1459878 = 3.289226058156428e-70;
        bool r1459879 = r1459867 <= r1459878;
        double r1459880 = -4.0;
        double r1459881 = r1459872 * r1459880;
        double r1459882 = r1459881 * r1459870;
        double r1459883 = fma(r1459867, r1459867, r1459882);
        double r1459884 = sqrt(r1459883);
        double r1459885 = r1459884 - r1459867;
        double r1459886 = r1459885 / r1459872;
        double r1459887 = r1459886 / r1459875;
        double r1459888 = -2.0;
        double r1459889 = r1459888 * r1459871;
        double r1459890 = r1459889 / r1459875;
        double r1459891 = r1459879 ? r1459887 : r1459890;
        double r1459892 = r1459869 ? r1459877 : r1459891;
        return r1459892;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.3
Target	20.7
Herbie	10.0

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

Derivation

1. Split input into 3 regimes

2. if b < -4.82289647433212e+153

   1. Initial program 60.9

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

   2. Simplified 60.9

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

   4. Applied div-inv 60.9

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

   6. Applied un-div-inv 60.9

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

   7. Taylor expanded around -inf 2.3

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

   8. Simplified 2.3

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

   if -4.82289647433212e+153 < b < 3.289226058156428e-70

   1. Initial program 12.4

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

   2. Simplified 12.4

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

   4. Applied div-inv 12.6

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

   6. Applied un-div-inv 12.4

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

   if 3.289226058156428e-70 < b

   1. Initial program 52.2

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

   2. Simplified 52.2

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

   4. Applied div-inv 52.2

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

   5. Taylor expanded around inf 9.1

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.82289647433212 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.289226058156428 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))