The quadratic formula (r1)

Average Error: 33.5 → 11.9

Time: 42.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.3725796156555912 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.207624111695675 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 4.664677641347216 \cdot 10^{-111}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.922674299151799 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a} \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3916704 = b;
        double r3916705 = -r3916704;
        double r3916706 = r3916704 * r3916704;
        double r3916707 = 4.0;
        double r3916708 = a;
        double r3916709 = r3916707 * r3916708;
        double r3916710 = c;
        double r3916711 = r3916709 * r3916710;
        double r3916712 = r3916706 - r3916711;
        double r3916713 = sqrt(r3916712);
        double r3916714 = r3916705 + r3916713;
        double r3916715 = 2.0;
        double r3916716 = r3916715 * r3916708;
        double r3916717 = r3916714 / r3916716;
        return r3916717;
}

↓

double f(double a, double b, double c) {
        double r3916718 = b;
        double r3916719 = -1.3725796156555912e+127;
        bool r3916720 = r3916718 <= r3916719;
        double r3916721 = c;
        double r3916722 = r3916721 / r3916718;
        double r3916723 = a;
        double r3916724 = r3916718 / r3916723;
        double r3916725 = r3916722 - r3916724;
        double r3916726 = 2.0;
        double r3916727 = r3916725 * r3916726;
        double r3916728 = r3916727 / r3916726;
        double r3916729 = 3.207624111695675e-187;
        bool r3916730 = r3916718 <= r3916729;
        double r3916731 = -4.0;
        double r3916732 = r3916723 * r3916731;
        double r3916733 = r3916732 * r3916721;
        double r3916734 = fma(r3916718, r3916718, r3916733);
        double r3916735 = sqrt(r3916734);
        double r3916736 = r3916735 / r3916723;
        double r3916737 = r3916736 - r3916724;
        double r3916738 = r3916737 / r3916726;
        double r3916739 = 4.664677641347216e-111;
        bool r3916740 = r3916718 <= r3916739;
        double r3916741 = -2.0;
        double r3916742 = r3916741 * r3916722;
        double r3916743 = r3916742 / r3916726;
        double r3916744 = 1.922674299151799e-16;
        bool r3916745 = r3916718 <= r3916744;
        double r3916746 = r3916718 * r3916718;
        double r3916747 = fma(r3916721, r3916732, r3916746);
        double r3916748 = r3916747 - r3916746;
        double r3916749 = r3916748 / r3916723;
        double r3916750 = 1.0;
        double r3916751 = sqrt(r3916747);
        double r3916752 = r3916718 + r3916751;
        double r3916753 = r3916750 / r3916752;
        double r3916754 = r3916749 * r3916753;
        double r3916755 = r3916754 / r3916726;
        double r3916756 = r3916745 ? r3916755 : r3916743;
        double r3916757 = r3916740 ? r3916743 : r3916756;
        double r3916758 = r3916730 ? r3916738 : r3916757;
        double r3916759 = r3916720 ? r3916728 : r3916758;
        return r3916759;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.5
Target	21.2
Herbie	11.9

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

Derivation

1. Split input into 4 regimes

2. if b < -1.3725796156555912e+127

   1. Initial program 51.4

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

   2. Simplified 51.4

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

   3. Taylor expanded around -inf 2.3

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

   4. Simplified 2.3

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

   if -1.3725796156555912e+127 < b < 3.207624111695675e-187

   1. Initial program 10.4

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

   2. Simplified 10.4

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

   4. Applied div-sub 10.4

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

   if 3.207624111695675e-187 < b < 4.664677641347216e-111 or 1.922674299151799e-16 < b

   1. Initial program 50.1

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

   2. Simplified 50.1

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

   4. Applied div-sub 50.8

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

   5. Taylor expanded around inf 12.0

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

   if 4.664677641347216e-111 < b < 1.922674299151799e-16

   1. Initial program 36.3

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

   2. Simplified 36.3

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

   4. Applied *-un-lft-identity 36.3

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

   5. Applied associate-/l* 36.4

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

   7. Applied flip-- 36.4

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

   8. Applied associate-/r/ 36.5

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

   9. Applied add-cube-cbrt 36.5

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

   10. Applied times-frac 36.5

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

   11. Simplified 36.4

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

   12. Simplified 36.4

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

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3725796156555912 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.207624111695675 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 4.664677641347216 \cdot 10^{-111}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.922674299151799 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a} \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +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)))