Average Error: 34.0 → 9.1
Time: 22.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, a \cdot -4, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -2.900769547116861 \cdot 10^{+46}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\

\mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, a \cdot -4, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1706057 = b;
        double r1706058 = -r1706057;
        double r1706059 = r1706057 * r1706057;
        double r1706060 = 4.0;
        double r1706061 = a;
        double r1706062 = r1706060 * r1706061;
        double r1706063 = c;
        double r1706064 = r1706062 * r1706063;
        double r1706065 = r1706059 - r1706064;
        double r1706066 = sqrt(r1706065);
        double r1706067 = r1706058 + r1706066;
        double r1706068 = 2.0;
        double r1706069 = r1706068 * r1706061;
        double r1706070 = r1706067 / r1706069;
        return r1706070;
}
↓
double f(double a, double b, double c) {
        double r1706071 = b;
        double r1706072 = -2.900769547116861e+46;
        bool r1706073 = r1706071 <= r1706072;
        double r1706074 = c;
        double r1706075 = r1706074 / r1706071;
        double r1706076 = a;
        double r1706077 = r1706071 / r1706076;
        double r1706078 = r1706075 - r1706077;
        double r1706079 = 2.0;
        double r1706080 = r1706078 * r1706079;
        double r1706081 = r1706080 / r1706079;
        double r1706082 = 5.4213851798811764e-102;
        bool r1706083 = r1706071 <= r1706082;
        double r1706084 = 1.0;
        double r1706085 = r1706084 / r1706076;
        double r1706086 = -4.0;
        double r1706087 = r1706076 * r1706086;
        double r1706088 = r1706087 * r1706074;
        double r1706089 = fma(r1706071, r1706071, r1706088);
        double r1706090 = sqrt(r1706089);
        double r1706091 = r1706090 - r1706071;
        double r1706092 = r1706085 * r1706091;
        double r1706093 = r1706092 / r1706079;
        double r1706094 = 1.1597179970514171e+23;
        bool r1706095 = r1706071 <= r1706094;
        double r1706096 = cbrt(r1706076);
        double r1706097 = r1706096 * r1706096;
        double r1706098 = r1706084 / r1706097;
        double r1706099 = 0.0;
        double r1706100 = fma(r1706074, r1706087, r1706099);
        double r1706101 = r1706090 + r1706071;
        double r1706102 = r1706100 / r1706101;
        double r1706103 = r1706102 / r1706096;
        double r1706104 = r1706098 * r1706103;
        double r1706105 = r1706104 / r1706079;
        double r1706106 = -2.0;
        double r1706107 = r1706075 * r1706106;
        double r1706108 = r1706107 / r1706079;
        double r1706109 = r1706095 ? r1706105 : r1706108;
        double r1706110 = r1706083 ? r1706093 : r1706109;
        double r1706111 = r1706073 ? r1706081 : r1706110;
        return r1706111;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -2.900769547116861e+46

    1. Initial program 35.9

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

    2. Simplified35.9

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

    3. Taylor expanded around -inf 5.3

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

    4. Simplified5.3

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

    if -2.900769547116861e+46 < b < 5.4213851798811764e-102

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied div-inv12.9

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

    if 5.4213851798811764e-102 < b < 1.1597179970514171e+23

    1. Initial program 39.1

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

    2. Simplified39.1

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

    4. Applied add-cube-cbrt39.5

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

    5. Applied *-un-lft-identity39.5

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

    6. Applied times-frac39.5

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

    8. Applied flip--39.6

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

    9. Simplified17.8

      \[\leadsto \frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\]

    if 1.1597179970514171e+23 < b

    1. Initial program 55.6

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

    2. Simplified55.6

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

    3. Taylor expanded around inf 4.4

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, a \cdot -4, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))