quadp (p42, positive)

Average Error: 33.3 → 10.3

Time: 16.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1261853 = b;
        double r1261854 = -r1261853;
        double r1261855 = r1261853 * r1261853;
        double r1261856 = 4.0;
        double r1261857 = a;
        double r1261858 = c;
        double r1261859 = r1261857 * r1261858;
        double r1261860 = r1261856 * r1261859;
        double r1261861 = r1261855 - r1261860;
        double r1261862 = sqrt(r1261861);
        double r1261863 = r1261854 + r1261862;
        double r1261864 = 2.0;
        double r1261865 = r1261864 * r1261857;
        double r1261866 = r1261863 / r1261865;
        return r1261866;
}

↓

double f(double a, double b, double c) {
        double r1261867 = b;
        double r1261868 = -3.263941314600607e+152;
        bool r1261869 = r1261867 <= r1261868;
        double r1261870 = c;
        double r1261871 = r1261870 / r1261867;
        double r1261872 = a;
        double r1261873 = r1261867 / r1261872;
        double r1261874 = r1261871 - r1261873;
        double r1261875 = 1.8378252714625124e-19;
        bool r1261876 = r1261867 <= r1261875;
        double r1261877 = r1261867 * r1261867;
        double r1261878 = 4.0;
        double r1261879 = r1261872 * r1261878;
        double r1261880 = r1261879 * r1261870;
        double r1261881 = r1261877 - r1261880;
        double r1261882 = sqrt(r1261881);
        double r1261883 = r1261882 - r1261867;
        double r1261884 = 0.5;
        double r1261885 = r1261883 * r1261884;
        double r1261886 = r1261885 / r1261872;
        double r1261887 = -r1261871;
        double r1261888 = r1261876 ? r1261886 : r1261887;
        double r1261889 = r1261869 ? r1261874 : r1261888;
        return r1261889;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.3
Target	20.3
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if b < -3.263941314600607e+152

   1. Initial program 60.1

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

   2. Simplified 60.1

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

   4. Applied div-inv 60.1

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

   5. Simplified 60.1

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

   6. Taylor expanded around -inf 2.3

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

   if -3.263941314600607e+152 < b < 1.8378252714625124e-19

   1. Initial program 14.2

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

   2. Simplified 14.1

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

   4. Applied div-inv 14.3

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

   5. Simplified 14.3

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

   7. Applied associate-*r/ 14.1

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

   if 1.8378252714625124e-19 < b

   1. Initial program 54.4

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

   2. Simplified 54.4

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

   4. Applied div-inv 54.4

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

   5. Simplified 54.4

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

   6. Taylor expanded around inf 7.0

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

   7. Simplified 7.0

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))