jeff quadratic root 1

Average Error: 19.6 → 13.8

Time: 22.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 1.5596668453223772 \cdot 10^{+21}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \cdot \sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \sqrt{\sqrt[3]{\left(\sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} - b}}{2}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r905209 = b;
        double r905210 = 0.0;
        bool r905211 = r905209 >= r905210;
        double r905212 = -r905209;
        double r905213 = r905209 * r905209;
        double r905214 = 4.0;
        double r905215 = a;
        double r905216 = r905214 * r905215;
        double r905217 = c;
        double r905218 = r905216 * r905217;
        double r905219 = r905213 - r905218;
        double r905220 = sqrt(r905219);
        double r905221 = r905212 - r905220;
        double r905222 = 2.0;
        double r905223 = r905222 * r905215;
        double r905224 = r905221 / r905223;
        double r905225 = r905222 * r905217;
        double r905226 = r905212 + r905220;
        double r905227 = r905225 / r905226;
        double r905228 = r905211 ? r905224 : r905227;
        return r905228;
}

↓

double f(double a, double b, double c) {
        double r905229 = b;
        double r905230 = 1.5596668453223772e+21;
        bool r905231 = r905229 <= r905230;
        double r905232 = 0.0;
        bool r905233 = r905229 >= r905232;
        double r905234 = -r905229;
        double r905235 = c;
        double r905236 = a;
        double r905237 = -4.0;
        double r905238 = r905236 * r905237;
        double r905239 = r905229 * r905229;
        double r905240 = fma(r905235, r905238, r905239);
        double r905241 = sqrt(r905240);
        double r905242 = r905234 - r905241;
        double r905243 = 2.0;
        double r905244 = r905243 * r905236;
        double r905245 = r905242 / r905244;
        double r905246 = r905241 - r905229;
        double r905247 = sqrt(r905246);
        double r905248 = cbrt(r905240);
        double r905249 = sqrt(r905248);
        double r905250 = r905248 * r905248;
        double r905251 = r905250 * r905248;
        double r905252 = cbrt(r905251);
        double r905253 = r905252 * r905248;
        double r905254 = sqrt(r905253);
        double r905255 = r905249 * r905254;
        double r905256 = r905255 - r905229;
        double r905257 = sqrt(r905256);
        double r905258 = r905247 * r905257;
        double r905259 = r905258 / r905243;
        double r905260 = r905235 / r905259;
        double r905261 = r905233 ? r905245 : r905260;
        double r905262 = r905236 / r905229;
        double r905263 = r905262 * r905235;
        double r905264 = r905263 - r905229;
        double r905265 = r905243 * r905264;
        double r905266 = r905265 / r905244;
        double r905267 = r905246 / r905243;
        double r905268 = r905235 / r905267;
        double r905269 = r905233 ? r905266 : r905268;
        double r905270 = r905231 ? r905261 : r905269;
        return r905270;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 1.5596668453223772e+21

   1. Initial program 15.9

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

   2. Simplified 15.9

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

   4. Applied add-sqr-sqrt 16.1

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

   6. Applied add-cube-cbrt 16.1

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

   7. Applied sqrt-prod 16.1

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

   9. Applied add-cbrt-cube 16.1

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

   if 1.5596668453223772e+21 < b

   1. Initial program 32.0

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

   2. Simplified 31.9

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

   3. Taylor expanded around inf 10.4

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

   4. Simplified 6.1

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

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.5596668453223772 \cdot 10^{+21}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b} \cdot \sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \sqrt{\sqrt[3]{\left(\sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} - b}}{2}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))