The quadratic formula (r1)

Average Error: 32.9 → 22.1

Time: 2.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 2.399747870429879 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)} - b}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(c \cdot -4\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)}\right) \cdot a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r21822708 = b;
        double r21822709 = -r21822708;
        double r21822710 = r21822708 * r21822708;
        double r21822711 = 4.0;
        double r21822712 = a;
        double r21822713 = r21822711 * r21822712;
        double r21822714 = c;
        double r21822715 = r21822713 * r21822714;
        double r21822716 = r21822710 - r21822715;
        double r21822717 = sqrt(r21822716);
        double r21822718 = r21822709 + r21822717;
        double r21822719 = 2.0;
        double r21822720 = r21822719 * r21822712;
        double r21822721 = r21822718 / r21822720;
        return r21822721;
}

↓

double f(double a, double b, double c) {
        double r21822722 = b;
        double r21822723 = 2.399747870429879e-86;
        bool r21822724 = r21822722 <= r21822723;
        double r21822725 = c;
        double r21822726 = -4.0;
        double r21822727 = r21822725 * r21822726;
        double r21822728 = a;
        double r21822729 = r21822722 * r21822722;
        double r21822730 = fma(r21822727, r21822728, r21822729);
        double r21822731 = sqrt(r21822730);
        double r21822732 = r21822731 - r21822722;
        double r21822733 = r21822732 / r21822728;
        double r21822734 = 0.5;
        double r21822735 = r21822733 * r21822734;
        double r21822736 = r21822727 * r21822728;
        double r21822737 = r21822722 + r21822731;
        double r21822738 = r21822737 * r21822728;
        double r21822739 = r21822736 / r21822738;
        double r21822740 = r21822734 * r21822739;
        double r21822741 = r21822724 ? r21822735 : r21822740;
        return r21822741;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	32.9
Target	20.3
Herbie	22.1

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

Derivation

1. Split input into 2 regimes

2. if b < 2.399747870429879e-86

   1. Initial program 20.0

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

   2. Simplified 20.0

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

   4. Applied *-un-lft-identity 20.0

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

   5. Applied *-un-lft-identity 20.0

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

   6. Applied *-un-lft-identity 20.0

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

   7. Applied distribute-lft-out-- 20.0

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

   8. Applied times-frac 20.0

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

   9. Applied associate-/l* 20.1

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

   10. Simplified 20.1

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

   12. Applied associate-/r/ 20.1

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

   13. Applied add-sqr-sqrt 20.1

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

   14. Applied times-frac 20.1

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

   15. Simplified 19.9

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

   16. Simplified 19.9

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

   if 2.399747870429879e-86 < b

   1. Initial program 52.1

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

   2. Simplified 52.2

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

   4. Applied *-un-lft-identity 52.2

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

   5. Applied *-un-lft-identity 52.2

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

   6. Applied *-un-lft-identity 52.2

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

   7. Applied distribute-lft-out-- 52.2

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

   8. Applied times-frac 52.2

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

   9. Applied associate-/l* 52.2

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

   10. Simplified 52.2

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

   12. Applied associate-/r/ 52.2

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

   13. Applied add-sqr-sqrt 52.2

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

   14. Applied times-frac 52.2

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

   15. Simplified 52.2

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

   16. Simplified 52.2

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

   18. Applied flip-- 52.2

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

   19. Applied associate-/l/ 53.4

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

   20. Simplified 25.2

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

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.399747870429879 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)} - b}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(c \cdot -4\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)}\right) \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +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)))