jeff quadratic root 1

Average Error: 19.4 → 6.9

Time: 20.8s

Precision: 64

Math

\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.227893547082757383910869715431246603409 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, -b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)}}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}}\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r40145 = b;
        double r40146 = 0.0;
        bool r40147 = r40145 >= r40146;
        double r40148 = -r40145;
        double r40149 = r40145 * r40145;
        double r40150 = 4.0;
        double r40151 = a;
        double r40152 = r40150 * r40151;
        double r40153 = c;
        double r40154 = r40152 * r40153;
        double r40155 = r40149 - r40154;
        double r40156 = sqrt(r40155);
        double r40157 = r40148 - r40156;
        double r40158 = 2.0;
        double r40159 = r40158 * r40151;
        double r40160 = r40157 / r40159;
        double r40161 = r40158 * r40153;
        double r40162 = r40148 + r40156;
        double r40163 = r40161 / r40162;
        double r40164 = r40147 ? r40160 : r40163;
        return r40164;
}

↓

double f(double a, double b, double c) {
        double r40165 = b;
        double r40166 = -1.2278935470827574e+154;
        bool r40167 = r40165 <= r40166;
        double r40168 = 0.0;
        bool r40169 = r40165 >= r40168;
        double r40170 = -r40165;
        double r40171 = r40165 * r40165;
        double r40172 = 4.0;
        double r40173 = a;
        double r40174 = r40172 * r40173;
        double r40175 = c;
        double r40176 = r40174 * r40175;
        double r40177 = r40171 - r40176;
        double r40178 = sqrt(r40177);
        double r40179 = r40170 - r40178;
        double r40180 = 2.0;
        double r40181 = r40180 * r40173;
        double r40182 = r40179 / r40181;
        double r40183 = r40175 * r40180;
        double r40184 = r40165 / r40173;
        double r40185 = r40175 / r40184;
        double r40186 = fma(r40180, r40185, r40170);
        double r40187 = r40170 + r40186;
        double r40188 = r40183 / r40187;
        double r40189 = r40169 ? r40182 : r40188;
        double r40190 = 3.628799960716312e+50;
        bool r40191 = r40165 <= r40190;
        double r40192 = sqrt(r40170);
        double r40193 = -r40176;
        double r40194 = fma(r40165, r40165, r40193);
        double r40195 = cbrt(r40194);
        double r40196 = -r40175;
        double r40197 = fma(r40174, r40196, r40171);
        double r40198 = cbrt(r40197);
        double r40199 = r40198 * r40198;
        double r40200 = cbrt(r40199);
        double r40201 = cbrt(r40198);
        double r40202 = r40200 * r40201;
        double r40203 = r40195 * r40202;
        double r40204 = r40203 * r40195;
        double r40205 = sqrt(r40204);
        double r40206 = fma(r40192, r40192, r40205);
        double r40207 = r40183 / r40206;
        double r40208 = r40169 ? r40182 : r40207;
        double r40209 = -r40180;
        double r40210 = r40165 / r40175;
        double r40211 = r40173 / r40210;
        double r40212 = fma(r40209, r40211, r40165);
        double r40213 = r40170 - r40212;
        double r40214 = r40213 / r40181;
        double r40215 = fma(r40192, r40192, r40178);
        double r40216 = r40183 / r40215;
        double r40217 = r40169 ? r40214 : r40216;
        double r40218 = r40191 ? r40208 : r40217;
        double r40219 = r40167 ? r40189 : r40218;
        return r40219;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.2278935470827574e+154

   1. Initial program 37.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Taylor expanded around -inf 6.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}}\\ \end{array}\]

   3. Simplified 1.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, -b\right)}}\\ \end{array}\]

   if -1.2278935470827574e+154 < b < 3.628799960716312e+50

   1. Initial program 8.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\sqrt{-b} \cdot \sqrt{-b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   4. Applied fma-def 8.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\ \end{array}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}\\ \end{array}\]

   7. Simplified 8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}\\ \end{array}\]

   8. Simplified 8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}}\right)}\\ \end{array}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 8.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} \cdot \sqrt[3]{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}}\right)}\\ \end{array}\]

   11. Applied cbrt-prod 8.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}}\right)}\\ \end{array}\]

   12. Simplified 8.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}}\right)}\\ \end{array}\]

   13. Simplified 8.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)}}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(-c\right) \cdot \left(4 \cdot a\right)\right)}}\right)}\\ \end{array}\]

   if 3.628799960716312e+50 < b

   1. Initial program 38.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 38.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\sqrt{-b} \cdot \sqrt{-b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   4. Applied fma-def 38.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\ \end{array}\]

   5. Taylor expanded around inf 11.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \end{array}\]

   6. Simplified 6.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.227893547082757383910869715431246603409 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, -b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{\left(\sqrt[3]{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(4 \cdot a, -c, b \cdot b\right)}}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}}\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(\sqrt{-b}, \sqrt{-b}, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))