Average Error: 34.9 → 10.1
Time: 20.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\
\;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\end{array}
double f(double a, double b, double c) {
        double r4469195 = b;
        double r4469196 = -r4469195;
        double r4469197 = r4469195 * r4469195;
        double r4469198 = 4.0;
        double r4469199 = a;
        double r4469200 = r4469198 * r4469199;
        double r4469201 = c;
        double r4469202 = r4469200 * r4469201;
        double r4469203 = r4469197 - r4469202;
        double r4469204 = sqrt(r4469203);
        double r4469205 = r4469196 + r4469204;
        double r4469206 = 2.0;
        double r4469207 = r4469206 * r4469199;
        double r4469208 = r4469205 / r4469207;
        return r4469208;
}
↓
double f(double a, double b, double c) {
        double r4469209 = b;
        double r4469210 = -3.6803290429888884e+148;
        bool r4469211 = r4469209 <= r4469210;
        double r4469212 = c;
        double r4469213 = r4469212 / r4469209;
        double r4469214 = a;
        double r4469215 = r4469209 / r4469214;
        double r4469216 = r4469213 - r4469215;
        double r4469217 = 1.0;
        double r4469218 = r4469216 * r4469217;
        double r4469219 = 4.6129908231112306e-104;
        bool r4469220 = r4469209 <= r4469219;
        double r4469221 = -r4469212;
        double r4469222 = 4.0;
        double r4469223 = r4469222 * r4469214;
        double r4469224 = r4469221 * r4469223;
        double r4469225 = fma(r4469209, r4469209, r4469224);
        double r4469226 = sqrt(r4469225);
        double r4469227 = r4469226 - r4469209;
        double r4469228 = 1.0;
        double r4469229 = 2.0;
        double r4469230 = r4469214 * r4469229;
        double r4469231 = r4469228 / r4469230;
        double r4469232 = r4469227 * r4469231;
        double r4469233 = -1.0;
        double r4469234 = r4469213 * r4469233;
        double r4469235 = r4469220 ? r4469232 : r4469234;
        double r4469236 = r4469211 ? r4469218 : r4469235;
        return r4469236;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.9
Target21.3
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.6803290429888884e+148

    1. Initial program 62.1

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

    2. Taylor expanded around -inf 2.3

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

    3. Simplified2.3

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

    if -3.6803290429888884e+148 < b < 4.6129908231112306e-104

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    4. Simplified12.3

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

    6. Applied fma-neg12.3

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

    8. Applied associate-/r/12.3

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

    if 4.6129908231112306e-104 < b

    1. Initial program 52.7

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

    2. Taylor expanded around inf 9.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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)))