quadp (p42, positive)

Average Error: 33.6 → 6.9

Time: 20.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le -9.252735565156383 \cdot 10^{-297}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 5.6843076821754435 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} + b} \cdot \left(-4 \cdot c\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3178771 = b;
        double r3178772 = -r3178771;
        double r3178773 = r3178771 * r3178771;
        double r3178774 = 4.0;
        double r3178775 = a;
        double r3178776 = c;
        double r3178777 = r3178775 * r3178776;
        double r3178778 = r3178774 * r3178777;
        double r3178779 = r3178773 - r3178778;
        double r3178780 = sqrt(r3178779);
        double r3178781 = r3178772 + r3178780;
        double r3178782 = 2.0;
        double r3178783 = r3178782 * r3178775;
        double r3178784 = r3178781 / r3178783;
        return r3178784;
}

↓

double f(double a, double b, double c) {
        double r3178785 = b;
        double r3178786 = -2.1144981103869975e+131;
        bool r3178787 = r3178785 <= r3178786;
        double r3178788 = c;
        double r3178789 = r3178788 / r3178785;
        double r3178790 = a;
        double r3178791 = r3178785 / r3178790;
        double r3178792 = r3178789 - r3178791;
        double r3178793 = 2.0;
        double r3178794 = r3178792 * r3178793;
        double r3178795 = r3178794 / r3178793;
        double r3178796 = -9.252735565156383e-297;
        bool r3178797 = r3178785 <= r3178796;
        double r3178798 = 1.0;
        double r3178799 = -4.0;
        double r3178800 = r3178790 * r3178788;
        double r3178801 = r3178785 * r3178785;
        double r3178802 = fma(r3178799, r3178800, r3178801);
        double r3178803 = sqrt(r3178802);
        double r3178804 = r3178803 - r3178785;
        double r3178805 = r3178790 / r3178804;
        double r3178806 = r3178798 / r3178805;
        double r3178807 = r3178806 / r3178793;
        double r3178808 = 5.6843076821754435e+85;
        bool r3178809 = r3178785 <= r3178808;
        double r3178810 = r3178803 + r3178785;
        double r3178811 = r3178798 / r3178810;
        double r3178812 = r3178799 * r3178788;
        double r3178813 = r3178811 * r3178812;
        double r3178814 = r3178813 / r3178793;
        double r3178815 = -2.0;
        double r3178816 = r3178789 * r3178815;
        double r3178817 = r3178816 / r3178793;
        double r3178818 = r3178809 ? r3178814 : r3178817;
        double r3178819 = r3178797 ? r3178807 : r3178818;
        double r3178820 = r3178787 ? r3178795 : r3178819;
        return r3178820;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
Target	21.0
Herbie	6.9

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

Derivation

1. Split input into 4 regimes

2. if b < -2.1144981103869975e+131

   1. Initial program 53.8

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

   2. Simplified 53.7

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

   4. Applied *-un-lft-identity 53.7

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

   5. Applied associate-/r* 53.7

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

   6. Simplified 53.7

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

   7. Taylor expanded around -inf 2.6

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

   8. Simplified 2.6

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

   if -2.1144981103869975e+131 < b < -9.252735565156383e-297

   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

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

   4. Applied clear-num 8.9

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

   if -9.252735565156383e-297 < b < 5.6843076821754435e+85

   1. Initial program 30.7

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

   2. Simplified 30.7

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

   4. Applied clear-num 30.7

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

   6. Applied flip-- 30.9

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

   7. Applied associate-/r/ 30.9

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

   8. Applied *-un-lft-identity 30.9

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

   9. Applied times-frac 30.9

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

   10. Simplified 16.2

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

   11. Taylor expanded around 0 10.0

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

   if 5.6843076821754435e+85 < b

   1. Initial program 58.0

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

   2. Simplified 58.0

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

   4. Applied *-un-lft-identity 58.0

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

   5. Applied associate-/r* 58.0

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

   6. Simplified 58.0

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

   7. Taylor expanded around inf 2.6

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le -9.252735565156383 \cdot 10^{-297}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 5.6843076821754435 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} + b} \cdot \left(-4 \cdot c\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

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)))