quadp (p42, positive)

Average Error: 33.1 → 13.3

Time: 30.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.2873109154899554 \cdot 10^{-271}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\frac{a \cdot -4}{a}}{\sqrt{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} + b}} \cdot \frac{c}{\sqrt{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} + b}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{b}{c} \cdot \frac{-1}{2}}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2112743 = b;
        double r2112744 = -r2112743;
        double r2112745 = r2112743 * r2112743;
        double r2112746 = 4.0;
        double r2112747 = a;
        double r2112748 = c;
        double r2112749 = r2112747 * r2112748;
        double r2112750 = r2112746 * r2112749;
        double r2112751 = r2112745 - r2112750;
        double r2112752 = sqrt(r2112751);
        double r2112753 = r2112744 + r2112752;
        double r2112754 = 2.0;
        double r2112755 = r2112754 * r2112747;
        double r2112756 = r2112753 / r2112755;
        return r2112756;
}

↓

double f(double a, double b, double c) {
        double r2112757 = b;
        double r2112758 = -1.2873109154899554e-271;
        bool r2112759 = r2112757 <= r2112758;
        double r2112760 = a;
        double r2112761 = c;
        double r2112762 = r2112760 * r2112761;
        double r2112763 = -4.0;
        double r2112764 = r2112757 * r2112757;
        double r2112765 = fma(r2112762, r2112763, r2112764);
        double r2112766 = sqrt(r2112765);
        double r2112767 = sqrt(r2112766);
        double r2112768 = -r2112757;
        double r2112769 = fma(r2112767, r2112767, r2112768);
        double r2112770 = r2112769 / r2112760;
        double r2112771 = 2.0;
        double r2112772 = r2112770 / r2112771;
        double r2112773 = 2.3648644896474148e+52;
        bool r2112774 = r2112757 <= r2112773;
        double r2112775 = 1.0;
        double r2112776 = r2112760 * r2112763;
        double r2112777 = r2112776 / r2112760;
        double r2112778 = r2112761 * r2112776;
        double r2112779 = fma(r2112757, r2112757, r2112778);
        double r2112780 = sqrt(r2112779);
        double r2112781 = r2112780 + r2112757;
        double r2112782 = sqrt(r2112781);
        double r2112783 = r2112777 / r2112782;
        double r2112784 = r2112761 / r2112782;
        double r2112785 = r2112783 * r2112784;
        double r2112786 = r2112775 / r2112785;
        double r2112787 = r2112775 / r2112786;
        double r2112788 = r2112787 / r2112771;
        double r2112789 = r2112757 / r2112761;
        double r2112790 = -0.5;
        double r2112791 = r2112789 * r2112790;
        double r2112792 = r2112775 / r2112791;
        double r2112793 = r2112792 / r2112771;
        double r2112794 = r2112774 ? r2112788 : r2112793;
        double r2112795 = r2112759 ? r2112772 : r2112794;
        return r2112795;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.1
Target	20.2
Herbie	13.3

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

Derivation

1. Split input into 3 regimes

2. if b < -1.2873109154899554e-271

   1. Initial program 20.9

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

   2. Simplified 20.9

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

   4. Applied add-sqr-sqrt 21.1

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

   5. Applied fma-neg 21.0

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

   if -1.2873109154899554e-271 < b < 2.3648644896474148e+52

   1. Initial program 28.4

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

   2. Simplified 28.4

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

   4. Applied flip-- 28.5

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

   5. Simplified 16.7

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

   7. Applied *-un-lft-identity 16.7

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

   8. Applied associate-/l* 16.8

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

   10. Applied *-un-lft-identity 16.8

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

   11. Applied associate-/l* 16.8

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

   12. Simplified 16.5

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

   14. Applied add-sqr-sqrt 16.7

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

   15. Applied *-un-lft-identity 16.7

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

   16. Applied times-frac 10.2

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

   17. Applied times-frac 10.3

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

   if 2.3648644896474148e+52 < b

   1. Initial program 56.4

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

   2. Simplified 56.4

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

   4. Applied flip-- 56.5

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

   5. Simplified 29.1

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

   7. Applied *-un-lft-identity 29.1

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

   8. Applied associate-/l* 29.2

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

   9. Taylor expanded around 0 4.5

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \frac{b}{c}}}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2873109154899554 \cdot 10^{-271}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\frac{a \cdot -4}{a}}{\sqrt{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} + b}} \cdot \frac{c}{\sqrt{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} + b}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{b}{c} \cdot \frac{-1}{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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)))