Average Error: 34.3 → 9.8
Time: 10.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.1113133556666892392847386011681009173 \cdot 10^{-81}:\\ \;\;\;\;-\frac{c \cdot \sqrt[3]{1}}{b} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\ \mathbf{elif}\;b \le 1.662466508012246493587212065137537026696 \cdot 10^{84}:\\ \;\;\;\;\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1\right) \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\\ \end{array}\]
\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 -5.1113133556666892392847386011681009173 \cdot 10^{-81}:\\
\;\;\;\;-\frac{c \cdot \sqrt[3]{1}}{b} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\

\mathbf{elif}\;b \le 1.662466508012246493587212065137537026696 \cdot 10^{84}:\\
\;\;\;\;\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)}\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r50144 = b;
        double r50145 = -r50144;
        double r50146 = r50144 * r50144;
        double r50147 = 4.0;
        double r50148 = a;
        double r50149 = c;
        double r50150 = r50148 * r50149;
        double r50151 = r50147 * r50150;
        double r50152 = r50146 - r50151;
        double r50153 = sqrt(r50152);
        double r50154 = r50145 - r50153;
        double r50155 = 2.0;
        double r50156 = r50155 * r50148;
        double r50157 = r50154 / r50156;
        return r50157;
}
↓
double f(double a, double b, double c) {
        double r50158 = b;
        double r50159 = -5.111313355666689e-81;
        bool r50160 = r50158 <= r50159;
        double r50161 = c;
        double r50162 = 1.0;
        double r50163 = cbrt(r50162);
        double r50164 = r50161 * r50163;
        double r50165 = r50164 / r50158;
        double r50166 = r50163 * r50163;
        double r50167 = r50165 * r50166;
        double r50168 = -r50167;
        double r50169 = 1.6624665080122465e+84;
        bool r50170 = r50158 <= r50169;
        double r50171 = -1.0;
        double r50172 = 2.0;
        double r50173 = a;
        double r50174 = r50172 * r50173;
        double r50175 = r50171 / r50174;
        double r50176 = -r50161;
        double r50177 = r50173 * r50176;
        double r50178 = 4.0;
        double r50179 = r50158 * r50158;
        double r50180 = fma(r50177, r50178, r50179);
        double r50181 = sqrt(r50180);
        double r50182 = r50158 + r50181;
        double r50183 = r50175 * r50182;
        double r50184 = -r50162;
        double r50185 = r50158 / r50173;
        double r50186 = r50161 / r50158;
        double r50187 = r50185 - r50186;
        double r50188 = r50184 * r50187;
        double r50189 = r50170 ? r50183 : r50188;
        double r50190 = r50160 ? r50168 : r50189;
        return r50190;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.3
Target21.1
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.111313355666689e-81

    1. Initial program 53.4

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

    2. Simplified53.4

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

    3. Taylor expanded around -inf 8.9

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

    4. Simplified9.6

      \[\leadsto -\color{blue}{\frac{1}{\frac{b}{c}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity9.6

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

    7. Applied *-un-lft-identity9.6

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

    8. Applied times-frac9.6

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

    9. Applied add-cube-cbrt9.6

      \[\leadsto -\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{b}{c}}\]

    10. Applied times-frac9.6

      \[\leadsto -\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{b}{c}}}\]

    11. Simplified9.6

      \[\leadsto -\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)} \cdot \frac{\sqrt[3]{1}}{\frac{b}{c}}\]

    12. Simplified8.9

      \[\leadsto -\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \color{blue}{\frac{\sqrt[3]{1} \cdot c}{b}}\]

    if -5.111313355666689e-81 < b < 1.6624665080122465e+84

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied div-inv13.3

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

    5. Simplified13.3

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

    if 1.6624665080122465e+84 < b

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around inf 3.5

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

    4. Simplified3.5

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.1113133556666892392847386011681009173 \cdot 10^{-81}:\\ \;\;\;\;-\frac{c \cdot \sqrt[3]{1}}{b} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\ \mathbf{elif}\;b \le 1.662466508012246493587212065137537026696 \cdot 10^{84}:\\ \;\;\;\;\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1\right) \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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

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