jeff quadratic root 2

Average Error: 19.9 → 6.6

Time: 15.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{\frac{2}{\frac{\sqrt[3]{\frac{b}{c}} \cdot \sqrt[3]{\frac{b}{c}}}{a}}}{\sqrt[3]{\frac{b}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \le 2.620543139740264315993856298302188165155 \cdot 10^{84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{2 \cdot a}{\frac{b}{c}} + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{{b}^{6} - {\left(c \cdot \left(4 \cdot a\right)\right)}^{3}}}{\sqrt{\left(b \cdot b + c \cdot \left(4 \cdot a\right)\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right) + {b}^{4}}} - b}{2 \cdot a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r35141 = b;
        double r35142 = 0.0;
        bool r35143 = r35141 >= r35142;
        double r35144 = 2.0;
        double r35145 = c;
        double r35146 = r35144 * r35145;
        double r35147 = -r35141;
        double r35148 = r35141 * r35141;
        double r35149 = 4.0;
        double r35150 = a;
        double r35151 = r35149 * r35150;
        double r35152 = r35151 * r35145;
        double r35153 = r35148 - r35152;
        double r35154 = sqrt(r35153);
        double r35155 = r35147 - r35154;
        double r35156 = r35146 / r35155;
        double r35157 = r35147 + r35154;
        double r35158 = r35144 * r35150;
        double r35159 = r35157 / r35158;
        double r35160 = r35143 ? r35156 : r35159;
        return r35160;
}

↓

double f(double a, double b, double c) {
        double r35161 = b;
        double r35162 = -7.943482039519134e+75;
        bool r35163 = r35161 <= r35162;
        double r35164 = 0.0;
        bool r35165 = r35161 >= r35164;
        double r35166 = 2.0;
        double r35167 = c;
        double r35168 = r35166 * r35167;
        double r35169 = -2.0;
        double r35170 = r35161 * r35169;
        double r35171 = r35161 / r35167;
        double r35172 = cbrt(r35171);
        double r35173 = r35172 * r35172;
        double r35174 = a;
        double r35175 = r35173 / r35174;
        double r35176 = r35166 / r35175;
        double r35177 = r35176 / r35172;
        double r35178 = r35170 + r35177;
        double r35179 = r35168 / r35178;
        double r35180 = 1.0;
        double r35181 = r35167 / r35161;
        double r35182 = r35161 / r35174;
        double r35183 = r35181 - r35182;
        double r35184 = r35180 * r35183;
        double r35185 = r35165 ? r35179 : r35184;
        double r35186 = 2.6205431397402643e+84;
        bool r35187 = r35161 <= r35186;
        double r35188 = -r35161;
        double r35189 = r35161 * r35161;
        double r35190 = r35174 * r35167;
        double r35191 = 4.0;
        double r35192 = r35190 * r35191;
        double r35193 = r35189 - r35192;
        double r35194 = sqrt(r35193);
        double r35195 = sqrt(r35194);
        double r35196 = r35195 * r35195;
        double r35197 = r35188 - r35196;
        double r35198 = r35168 / r35197;
        double r35199 = r35194 - r35161;
        double r35200 = r35166 * r35174;
        double r35201 = r35199 / r35200;
        double r35202 = r35165 ? r35198 : r35201;
        double r35203 = r35200 / r35171;
        double r35204 = r35203 + r35170;
        double r35205 = r35168 / r35204;
        double r35206 = 6.0;
        double r35207 = pow(r35161, r35206);
        double r35208 = r35191 * r35174;
        double r35209 = r35167 * r35208;
        double r35210 = 3.0;
        double r35211 = pow(r35209, r35210);
        double r35212 = r35207 - r35211;
        double r35213 = sqrt(r35212);
        double r35214 = r35189 + r35209;
        double r35215 = r35214 * r35209;
        double r35216 = 4.0;
        double r35217 = pow(r35161, r35216);
        double r35218 = r35215 + r35217;
        double r35219 = sqrt(r35218);
        double r35220 = r35213 / r35219;
        double r35221 = r35220 - r35161;
        double r35222 = r35221 / r35200;
        double r35223 = r35165 ? r35205 : r35222;
        double r35224 = r35187 ? r35202 : r35223;
        double r35225 = r35163 ? r35185 : r35224;
        return r35225;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -7.943482039519134e+75

   1. Initial program 42.7

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

   2. Simplified 42.7

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

   3. Taylor expanded around inf 42.7

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

   4. Simplified 42.7

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

   5. Taylor expanded around -inf 9.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

   6. Simplified 4.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot 2}{\frac{b}{c}} - b \cdot 2}{2 \cdot a}\\ \end{array}\]

   7. Taylor expanded around 0 4.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}\\ \end{array}\]

   8. Simplified 4.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 4.2

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{2 \cdot a}{\color{blue}{\left(\sqrt[3]{\frac{b}{c}} \cdot \sqrt[3]{\frac{b}{c}}\right) \cdot \sqrt[3]{\frac{b}{c}}}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

   11. Applied associate-/r* 4.2

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \color{blue}{\frac{\frac{2 \cdot a}{\sqrt[3]{\frac{b}{c}} \cdot \sqrt[3]{\frac{b}{c}}}}{\sqrt[3]{\frac{b}{c}}}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

   12. Simplified 4.2

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{\color{blue}{\frac{2}{\frac{\sqrt[3]{\frac{b}{c}} \cdot \sqrt[3]{\frac{b}{c}}}{a}}}}{\sqrt[3]{\frac{b}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

   if -7.943482039519134e+75 < b < 2.6205431397402643e+84

   1. Initial program 9.0

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

   2. Simplified 9.0

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

   4. Applied add-sqr-sqrt 9.0

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

   5. Applied sqrt-prod 9.1

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

   6. Simplified 9.1

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

   7. Simplified 9.1

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

   if 2.6205431397402643e+84 < b

   1. Initial program 27.7

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

   2. Simplified 27.7

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

   3. Taylor expanded around inf 5.8

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

   4. Simplified 2.5

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

   6. Applied flip3-- 2.5

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

   7. Applied sqrt-div 2.5

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

   8. Simplified 2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(c \cdot a\right) \cdot 4\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)\right)}} - b}{2 \cdot a}\\ \end{array}\]

   9. Simplified 2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right) + {b}^{4}}} - b}{2 \cdot a}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2 + \frac{\frac{2}{\frac{\sqrt[3]{\frac{b}{c}} \cdot \sqrt[3]{\frac{b}{c}}}{a}}}{\sqrt[3]{\frac{b}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \le 2.620543139740264315993856298302188165155 \cdot 10^{84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{2 \cdot a}{\frac{b}{c}} + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{{b}^{6} - {\left(c \cdot \left(4 \cdot a\right)\right)}^{3}}}{\sqrt{\left(b \cdot b + c \cdot \left(4 \cdot a\right)\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right) + {b}^{4}}} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))