jeff quadratic root 2

Average Error: 19.6 → 6.7

Time: 56.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.82289647433212 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 4.866803345215698 \cdot 10^{+83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-\sqrt{b}, \sqrt{b}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}{\sqrt{\mathsf{fma}\left(b \cdot b, b \cdot b, \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)\right)}} + \left(-b\right)}{a \cdot 2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r926761 = b;
        double r926762 = 0.0;
        bool r926763 = r926761 >= r926762;
        double r926764 = 2.0;
        double r926765 = c;
        double r926766 = r926764 * r926765;
        double r926767 = -r926761;
        double r926768 = r926761 * r926761;
        double r926769 = 4.0;
        double r926770 = a;
        double r926771 = r926769 * r926770;
        double r926772 = r926771 * r926765;
        double r926773 = r926768 - r926772;
        double r926774 = sqrt(r926773);
        double r926775 = r926767 - r926774;
        double r926776 = r926766 / r926775;
        double r926777 = r926767 + r926774;
        double r926778 = r926764 * r926770;
        double r926779 = r926777 / r926778;
        double r926780 = r926763 ? r926776 : r926779;
        return r926780;
}

↓

double f(double a, double b, double c) {
        double r926781 = b;
        double r926782 = -4.82289647433212e+153;
        bool r926783 = r926781 <= r926782;
        double r926784 = 0.0;
        bool r926785 = r926781 >= r926784;
        double r926786 = 2.0;
        double r926787 = c;
        double r926788 = r926786 * r926787;
        double r926789 = a;
        double r926790 = r926789 / r926781;
        double r926791 = r926790 * r926787;
        double r926792 = r926791 - r926781;
        double r926793 = r926786 * r926792;
        double r926794 = r926788 / r926793;
        double r926795 = r926781 / r926787;
        double r926796 = r926789 / r926795;
        double r926797 = r926796 - r926781;
        double r926798 = r926797 * r926786;
        double r926799 = r926789 * r926786;
        double r926800 = r926798 / r926799;
        double r926801 = r926785 ? r926794 : r926800;
        double r926802 = 4.866803345215698e+83;
        bool r926803 = r926781 <= r926802;
        double r926804 = sqrt(r926781);
        double r926805 = -r926804;
        double r926806 = r926781 * r926781;
        double r926807 = 4.0;
        double r926808 = r926807 * r926789;
        double r926809 = r926808 * r926787;
        double r926810 = r926806 - r926809;
        double r926811 = cbrt(r926810);
        double r926812 = sqrt(r926811);
        double r926813 = r926811 * r926811;
        double r926814 = sqrt(r926813);
        double r926815 = -r926814;
        double r926816 = r926812 * r926815;
        double r926817 = fma(r926805, r926804, r926816);
        double r926818 = -r926812;
        double r926819 = r926812 * r926814;
        double r926820 = fma(r926818, r926814, r926819);
        double r926821 = r926817 + r926820;
        double r926822 = r926788 / r926821;
        double r926823 = sqrt(r926810);
        double r926824 = -r926781;
        double r926825 = r926823 + r926824;
        double r926826 = r926825 / r926799;
        double r926827 = r926785 ? r926822 : r926826;
        double r926828 = r926781 * r926806;
        double r926829 = r926828 * r926828;
        double r926830 = r926809 * r926809;
        double r926831 = r926809 * r926830;
        double r926832 = r926829 - r926831;
        double r926833 = sqrt(r926832);
        double r926834 = r926806 + r926809;
        double r926835 = r926809 * r926834;
        double r926836 = fma(r926806, r926806, r926835);
        double r926837 = sqrt(r926836);
        double r926838 = r926833 / r926837;
        double r926839 = r926838 + r926824;
        double r926840 = r926839 / r926799;
        double r926841 = r926785 ? r926794 : r926840;
        double r926842 = r926803 ? r926827 : r926841;
        double r926843 = r926783 ? r926801 : r926842;
        return r926843;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -4.82289647433212e+153

   1. Initial program 60.9

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

   2. Taylor expanded around inf 60.9

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

   3. Simplified 60.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 60.9

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

   6. Applied sqrt-prod 60.9

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

   7. Taylor expanded around -inf 11.4

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

   8. Simplified 2.5

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

   if -4.82289647433212e+153 < b < 4.866803345215698e+83

   1. Initial program 8.7

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

   3. Applied add-cube-cbrt 8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   4. Applied sqrt-prod 8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   5. Applied add-sqr-sqrt 8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) - \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   6. Applied distribute-lft-neg-in 8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-\sqrt{b}\right) \cdot \sqrt{b}} - \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   7. Applied prod-diff 8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(-\sqrt{b}, \sqrt{b}, -\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) + \mathsf{fma}\left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   if 4.866803345215698e+83 < b

   1. Initial program 28.8

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

   2. Taylor expanded around inf 6.5

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

   3. Simplified 2.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
   4. Using strategy rm

   5. Applied flip3-- 2.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a}\\ \end{array}\]

   6. Applied sqrt-div 2.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a}\\ \end{array}\]

   7. Simplified 2.9

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

   8. Simplified 2.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \frac{\sqrt{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}{\sqrt{\mathsf{fma}\left(b \cdot b, b \cdot b, \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)\right)}}}{2 \cdot a}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.82289647433212 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 4.866803345215698 \cdot 10^{+83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-\sqrt{b}, \sqrt{b}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}{\sqrt{\mathsf{fma}\left(b \cdot b, b \cdot b, \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)\right)}} + \left(-b\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))