Average Error: 34.4 → 10.3
Time: 22.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\

\mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3174093 = b;
        double r3174094 = -r3174093;
        double r3174095 = r3174093 * r3174093;
        double r3174096 = 4.0;
        double r3174097 = a;
        double r3174098 = r3174096 * r3174097;
        double r3174099 = c;
        double r3174100 = r3174098 * r3174099;
        double r3174101 = r3174095 - r3174100;
        double r3174102 = sqrt(r3174101);
        double r3174103 = r3174094 + r3174102;
        double r3174104 = 2.0;
        double r3174105 = r3174104 * r3174097;
        double r3174106 = r3174103 / r3174105;
        return r3174106;
}
↓
double f(double a, double b, double c) {
        double r3174107 = b;
        double r3174108 = -1.7633154797394035e+89;
        bool r3174109 = r3174107 <= r3174108;
        double r3174110 = a;
        double r3174111 = r3174107 / r3174110;
        double r3174112 = -2.0;
        double r3174113 = 2.0;
        double r3174114 = c;
        double r3174115 = r3174114 / r3174107;
        double r3174116 = r3174113 * r3174115;
        double r3174117 = fma(r3174111, r3174112, r3174116);
        double r3174118 = r3174117 / r3174113;
        double r3174119 = 9.136492990928292e-23;
        bool r3174120 = r3174107 <= r3174119;
        double r3174121 = 1.0;
        double r3174122 = r3174121 / r3174110;
        double r3174123 = r3174107 * r3174107;
        double r3174124 = 4.0;
        double r3174125 = r3174124 * r3174114;
        double r3174126 = r3174110 * r3174125;
        double r3174127 = r3174123 - r3174126;
        double r3174128 = sqrt(r3174127);
        double r3174129 = r3174128 - r3174107;
        double r3174130 = r3174122 * r3174129;
        double r3174131 = r3174130 / r3174113;
        double r3174132 = -2.0;
        double r3174133 = r3174115 * r3174132;
        double r3174134 = r3174133 / r3174113;
        double r3174135 = r3174120 ? r3174131 : r3174134;
        double r3174136 = r3174109 ? r3174118 : r3174135;
        return r3174136;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around -inf 3.9

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

    4. Simplified3.9

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

    if -1.7633154797394035e+89 < b < 9.136492990928292e-23

    1. Initial program 15.0

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

    2. Simplified15.1

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

    4. Applied div-inv15.2

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

    if 9.136492990928292e-23 < b

    1. Initial program 55.5

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

    2. Simplified55.4

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

    3. Taylor expanded around inf 6.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))