Average Error: 33.2 → 10.8
Time: 18.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\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 -2.2415082771065304 \cdot 10^{-131}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1257204 = b;
        double r1257205 = -r1257204;
        double r1257206 = r1257204 * r1257204;
        double r1257207 = 4.0;
        double r1257208 = a;
        double r1257209 = c;
        double r1257210 = r1257208 * r1257209;
        double r1257211 = r1257207 * r1257210;
        double r1257212 = r1257206 - r1257211;
        double r1257213 = sqrt(r1257212);
        double r1257214 = r1257205 - r1257213;
        double r1257215 = 2.0;
        double r1257216 = r1257215 * r1257208;
        double r1257217 = r1257214 / r1257216;
        return r1257217;
}
↓
double f(double a, double b, double c) {
        double r1257218 = b;
        double r1257219 = -2.2415082771065304e-131;
        bool r1257220 = r1257218 <= r1257219;
        double r1257221 = -2.0;
        double r1257222 = c;
        double r1257223 = r1257222 / r1257218;
        double r1257224 = r1257221 * r1257223;
        double r1257225 = 2.0;
        double r1257226 = r1257224 / r1257225;
        double r1257227 = 2.559678284282607e+69;
        bool r1257228 = r1257218 <= r1257227;
        double r1257229 = 1.0;
        double r1257230 = a;
        double r1257231 = r1257229 / r1257230;
        double r1257232 = -r1257218;
        double r1257233 = -4.0;
        double r1257234 = r1257230 * r1257233;
        double r1257235 = r1257218 * r1257218;
        double r1257236 = fma(r1257234, r1257222, r1257235);
        double r1257237 = sqrt(r1257236);
        double r1257238 = r1257232 - r1257237;
        double r1257239 = r1257231 * r1257238;
        double r1257240 = r1257239 / r1257225;
        double r1257241 = r1257218 / r1257230;
        double r1257242 = r1257223 - r1257241;
        double r1257243 = r1257242 * r1257225;
        double r1257244 = r1257243 / r1257225;
        double r1257245 = r1257228 ? r1257240 : r1257244;
        double r1257246 = r1257220 ? r1257226 : r1257245;
        return r1257246;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target19.9
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.2415082771065304e-131

    1. Initial program 49.6

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

    2. Simplified49.7

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

    3. Taylor expanded around -inf 12.4

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

    if -2.2415082771065304e-131 < b < 2.559678284282607e+69

    1. Initial program 11.4

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

    2. Simplified11.4

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

    4. Applied div-inv11.5

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

    if 2.559678284282607e+69 < b

    1. Initial program 38.9

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

    2. Simplified38.9

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

    4. Applied div-inv39.0

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

    5. Taylor expanded around inf 4.8

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

    6. Simplified4.8

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 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)))