jeff quadratic root 2

Average Error: 18.9 → 8.3

Time: 23.4s

Precision: 64

Math

\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -5.748232111683622 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 5.966643100467746 \cdot 10^{+48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}} \cdot \left|\sqrt[3]{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(\frac{a}{\frac{b}{c}}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r954229 = b;
        double r954230 = 0.0;
        bool r954231 = r954229 >= r954230;
        double r954232 = 2.0;
        double r954233 = c;
        double r954234 = r954232 * r954233;
        double r954235 = -r954229;
        double r954236 = r954229 * r954229;
        double r954237 = 4.0;
        double r954238 = a;
        double r954239 = r954237 * r954238;
        double r954240 = r954239 * r954233;
        double r954241 = r954236 - r954240;
        double r954242 = sqrt(r954241);
        double r954243 = r954235 - r954242;
        double r954244 = r954234 / r954243;
        double r954245 = r954235 + r954242;
        double r954246 = r954232 * r954238;
        double r954247 = r954245 / r954246;
        double r954248 = r954231 ? r954244 : r954247;
        return r954248;
}

↓

double f(double a, double b, double c) {
        double r954249 = b;
        double r954250 = -5.748232111683622e+41;
        bool r954251 = r954249 <= r954250;
        double r954252 = 0.0;
        bool r954253 = r954249 >= r954252;
        double r954254 = 2.0;
        double r954255 = c;
        double r954256 = r954254 * r954255;
        double r954257 = -r954249;
        double r954258 = r954249 * r954249;
        double r954259 = 4.0;
        double r954260 = a;
        double r954261 = r954259 * r954260;
        double r954262 = r954261 * r954255;
        double r954263 = r954258 - r954262;
        double r954264 = sqrt(r954263);
        double r954265 = r954257 - r954264;
        double r954266 = r954256 / r954265;
        double r954267 = r954260 * r954255;
        double r954268 = r954267 / r954249;
        double r954269 = r954268 - r954249;
        double r954270 = r954254 * r954269;
        double r954271 = r954254 * r954260;
        double r954272 = r954270 / r954271;
        double r954273 = r954253 ? r954266 : r954272;
        double r954274 = 5.966643100467746e+48;
        bool r954275 = r954249 <= r954274;
        double r954276 = -4.0;
        double r954277 = fma(r954276, r954267, r954258);
        double r954278 = sqrt(r954277);
        double r954279 = cbrt(r954278);
        double r954280 = r954279 * r954279;
        double r954281 = sqrt(r954280);
        double r954282 = cbrt(r954277);
        double r954283 = fabs(r954282);
        double r954284 = r954281 * r954283;
        double r954285 = r954257 - r954284;
        double r954286 = r954256 / r954285;
        double r954287 = r954264 + r954257;
        double r954288 = r954287 / r954271;
        double r954289 = r954253 ? r954286 : r954288;
        double r954290 = r954249 / r954255;
        double r954291 = r954260 / r954290;
        double r954292 = -2.0;
        double r954293 = fma(r954291, r954292, r954249);
        double r954294 = r954257 - r954293;
        double r954295 = r954256 / r954294;
        double r954296 = cbrt(r954263);
        double r954297 = r954296 * r954296;
        double r954298 = sqrt(r954297);
        double r954299 = sqrt(r954296);
        double r954300 = r954298 * r954299;
        double r954301 = r954257 + r954300;
        double r954302 = r954301 / r954271;
        double r954303 = r954253 ? r954295 : r954302;
        double r954304 = r954275 ? r954289 : r954303;
        double r954305 = r954251 ? r954273 : r954304;
        return r954305;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -5.748232111683622e+41

   1. Initial program 34.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   2. Taylor expanded around -inf 11.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

   3. Simplified 11.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

   if -5.748232111683622e+41 < b < 5.966643100467746e+48

   1. Initial program 9.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
   2. Using strategy rm

   3. Applied add-exp-log 11.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 11.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   6. Applied sqrt-prod 11.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \color{blue}{\left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   7. Applied log-prod 11.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\color{blue}{\log \left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) + \log \left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   8. Applied exp-sum 11.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{e^{\log \left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} \cdot e^{\log \left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   9. Simplified 10.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left|\sqrt[3]{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right|} \cdot e^{\log \left(\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   10. Simplified 9.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right| \cdot \color{blue}{\sqrt{\sqrt[3]{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
   11. Using strategy rm

   12. Applied add-sqr-sqrt 9.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right| \cdot \sqrt{\sqrt[3]{\color{blue}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   13. Applied cbrt-prod 9.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right| \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

   if 5.966643100467746e+48 < b

   1. Initial program 25.0

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 25.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

   4. Applied sqrt-prod 25.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

   5. Taylor expanded around inf 6.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

   6. Simplified 3.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, -2, b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.748232111683622 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 5.966643100467746 \cdot 10^{+48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}} \cdot \left|\sqrt[3]{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(\frac{a}{\frac{b}{c}}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))