Average Error: 33.7 → 11.2
Time: 6.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.77413468628334187 \cdot 10^{-17}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.2919983862558445 \cdot 10^{30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, -\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.77413468628334187 \cdot 10^{-17}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.2919983862558445 \cdot 10^{30}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, -\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r74180 = b;
        double r74181 = -r74180;
        double r74182 = r74180 * r74180;
        double r74183 = 4.0;
        double r74184 = a;
        double r74185 = c;
        double r74186 = r74184 * r74185;
        double r74187 = r74183 * r74186;
        double r74188 = r74182 - r74187;
        double r74189 = sqrt(r74188);
        double r74190 = r74181 - r74189;
        double r74191 = 2.0;
        double r74192 = r74191 * r74184;
        double r74193 = r74190 / r74192;
        return r74193;
}
↓
double f(double a, double b, double c) {
        double r74194 = b;
        double r74195 = -3.774134686283342e-17;
        bool r74196 = r74194 <= r74195;
        double r74197 = -1.0;
        double r74198 = c;
        double r74199 = r74198 / r74194;
        double r74200 = r74197 * r74199;
        double r74201 = 1.2919983862558445e+30;
        bool r74202 = r74194 <= r74201;
        double r74203 = -r74194;
        double r74204 = cbrt(r74203);
        double r74205 = r74204 * r74204;
        double r74206 = r74194 * r74194;
        double r74207 = 4.0;
        double r74208 = a;
        double r74209 = r74208 * r74198;
        double r74210 = r74207 * r74209;
        double r74211 = r74206 - r74210;
        double r74212 = sqrt(r74211);
        double r74213 = -r74212;
        double r74214 = fma(r74205, r74204, r74213);
        double r74215 = 2.0;
        double r74216 = r74215 * r74208;
        double r74217 = r74214 / r74216;
        double r74218 = 1.0;
        double r74219 = r74194 / r74208;
        double r74220 = r74199 - r74219;
        double r74221 = r74218 * r74220;
        double r74222 = r74202 ? r74217 : r74221;
        double r74223 = r74196 ? r74200 : r74222;
        return r74223;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target20.5
Herbie11.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.774134686283342e-17

    1. Initial program 54.4

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

    2. Taylor expanded around -inf 6.7

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

    if -3.774134686283342e-17 < b < 1.2919983862558445e+30

    1. Initial program 16.8

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

    3. Applied add-cube-cbrt17.0

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

    4. Applied fma-neg17.0

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

    if 1.2919983862558445e+30 < b

    1. Initial program 34.2

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

    2. Taylor expanded around inf 6.6

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

    3. Simplified6.6

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.77413468628334187 \cdot 10^{-17}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.2919983862558445 \cdot 10^{30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, -\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))