quadm (p42, negative)

Average Error: 33.9 → 8.8

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.860355455153336511492756422997945623034 \cdot 10^{55}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -4.593198226330854148819304728272718450887 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-1\right) + 1\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.368471338029900067384853691743532212305 \cdot 10^{94}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r101724 = b;
        double r101725 = -r101724;
        double r101726 = r101724 * r101724;
        double r101727 = 4.0;
        double r101728 = a;
        double r101729 = c;
        double r101730 = r101728 * r101729;
        double r101731 = r101727 * r101730;
        double r101732 = r101726 - r101731;
        double r101733 = sqrt(r101732);
        double r101734 = r101725 - r101733;
        double r101735 = 2.0;
        double r101736 = r101735 * r101728;
        double r101737 = r101734 / r101736;
        return r101737;
}

↓

double f(double a, double b, double c) {
        double r101738 = b;
        double r101739 = -3.8603554551533365e+55;
        bool r101740 = r101738 <= r101739;
        double r101741 = -1.0;
        double r101742 = c;
        double r101743 = r101742 / r101738;
        double r101744 = r101741 * r101743;
        double r101745 = -4.593198226330854e-145;
        bool r101746 = r101738 <= r101745;
        double r101747 = r101738 * r101738;
        double r101748 = 2.0;
        double r101749 = pow(r101738, r101748);
        double r101750 = r101747 - r101749;
        double r101751 = 4.0;
        double r101752 = a;
        double r101753 = r101752 * r101742;
        double r101754 = r101751 * r101753;
        double r101755 = r101750 + r101754;
        double r101756 = r101747 - r101754;
        double r101757 = sqrt(r101756);
        double r101758 = r101757 - r101738;
        double r101759 = r101755 / r101758;
        double r101760 = 1.0;
        double r101761 = -r101760;
        double r101762 = r101761 + r101760;
        double r101763 = r101757 * r101762;
        double r101764 = r101759 + r101763;
        double r101765 = 2.0;
        double r101766 = r101765 * r101752;
        double r101767 = r101764 / r101766;
        double r101768 = 2.3684713380299e+94;
        bool r101769 = r101738 <= r101768;
        double r101770 = -r101738;
        double r101771 = r101770 - r101757;
        double r101772 = r101771 / r101766;
        double r101773 = 1.0;
        double r101774 = r101738 / r101752;
        double r101775 = r101743 - r101774;
        double r101776 = r101773 * r101775;
        double r101777 = r101769 ? r101772 : r101776;
        double r101778 = r101746 ? r101767 : r101777;
        double r101779 = r101740 ? r101744 : r101778;
        return r101779;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.9
Target	21.0
Herbie	8.8

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -3.8603554551533365e+55

   1. Initial program 57.7

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

   2. Taylor expanded around -inf 3.8

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

   if -3.8603554551533365e+55 < b < -4.593198226330854e-145

   1. Initial program 36.9

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

   3. Applied add-cube-cbrt 38.3

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

   4. Applied add-cube-cbrt 37.3

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

   5. Applied distribute-lft-neg-in 37.3

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

   6. Applied prod-diff 37.3

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

   7. Simplified 38.1

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

   8. Simplified 37.9

      \[\leadsto \frac{\left(\left(-\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b} - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-1\right) + 1\right)}}{2 \cdot a}\]
   9. Using strategy rm

   10. Applied flip-- 37.9

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

   11. Simplified 17.4

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

   12. Simplified 17.2

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

   if -4.593198226330854e-145 < b < 2.3684713380299e+94

   1. Initial program 11.1

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

   if 2.3684713380299e+94 < b

   1. Initial program 44.8

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

   2. Taylor expanded around inf 3.8

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

   3. Simplified 3.8

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.860355455153336511492756422997945623034 \cdot 10^{55}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -4.593198226330854148819304728272718450887 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-1\right) + 1\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.368471338029900067384853691743532212305 \cdot 10^{94}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))