jeff quadratic root 2

Average Error: 19.9 → 6.6

Time: 16.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r35100 = b;
        double r35101 = 0.0;
        bool r35102 = r35100 >= r35101;
        double r35103 = 2.0;
        double r35104 = c;
        double r35105 = r35103 * r35104;
        double r35106 = -r35100;
        double r35107 = r35100 * r35100;
        double r35108 = 4.0;
        double r35109 = a;
        double r35110 = r35108 * r35109;
        double r35111 = r35110 * r35104;
        double r35112 = r35107 - r35111;
        double r35113 = sqrt(r35112);
        double r35114 = r35106 - r35113;
        double r35115 = r35105 / r35114;
        double r35116 = r35106 + r35113;
        double r35117 = r35103 * r35109;
        double r35118 = r35116 / r35117;
        double r35119 = r35102 ? r35115 : r35118;
        return r35119;
}

↓

double f(double a, double b, double c) {
        double r35120 = b;
        double r35121 = -7.943482039519134e+75;
        bool r35122 = r35120 <= r35121;
        double r35123 = 0.0;
        bool r35124 = r35120 >= r35123;
        double r35125 = 2.0;
        double r35126 = c;
        double r35127 = r35125 * r35126;
        double r35128 = -2.0;
        double r35129 = r35120 * r35128;
        double r35130 = a;
        double r35131 = r35120 / r35126;
        double r35132 = r35130 / r35131;
        double r35133 = r35132 * r35125;
        double r35134 = cbrt(r35133);
        double r35135 = r35125 * r35130;
        double r35136 = r35135 / r35131;
        double r35137 = cbrt(r35136);
        double r35138 = cbrt(r35137);
        double r35139 = r35138 * r35138;
        double r35140 = r35138 * r35139;
        double r35141 = r35140 * r35134;
        double r35142 = r35134 * r35141;
        double r35143 = r35129 + r35142;
        double r35144 = r35127 / r35143;
        double r35145 = 2.0;
        double r35146 = r35120 * r35145;
        double r35147 = r35136 - r35146;
        double r35148 = r35147 / r35135;
        double r35149 = r35124 ? r35144 : r35148;
        double r35150 = 2.6205431397402643e+84;
        bool r35151 = r35120 <= r35150;
        double r35152 = -r35120;
        double r35153 = r35120 * r35120;
        double r35154 = 4.0;
        double r35155 = r35154 * r35130;
        double r35156 = r35155 * r35126;
        double r35157 = r35153 - r35156;
        double r35158 = sqrt(r35157);
        double r35159 = sqrt(r35158);
        double r35160 = r35159 * r35159;
        double r35161 = r35152 - r35160;
        double r35162 = r35127 / r35161;
        double r35163 = r35152 + r35158;
        double r35164 = r35163 / r35135;
        double r35165 = r35124 ? r35162 : r35164;
        double r35166 = r35136 + r35129;
        double r35167 = r35127 / r35166;
        double r35168 = log(r35136);
        double r35169 = exp(r35168);
        double r35170 = r35169 - r35146;
        double r35171 = r35170 / r35135;
        double r35172 = r35124 ? r35167 : r35171;
        double r35173 = r35151 ? r35165 : r35172;
        double r35174 = r35122 ? r35149 : r35173;
        return r35174;
}

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. Taylor expanded around inf 42.7

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

   3. Simplified 42.7

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

   4. Taylor expanded around -inf 9.5

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

   5. Simplified 4.3

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

   7. Applied add-cube-cbrt 4.3

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

   8. Simplified 4.3

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

   9. Simplified 4.3

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

   11. Applied add-cube-cbrt 4.3

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

   12. Simplified 4.3

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

   13. Simplified 4.3

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

   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. Using strategy rm

   3. Applied add-sqr-sqrt 9.0

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

   4. Applied sqrt-prod 9.1

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

   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. Taylor expanded around inf 5.8

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

   3. Simplified 2.5

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

   4. Taylor expanded around -inf 2.5

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

   5. Simplified 2.5

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

   7. Applied add-exp-log 2.5

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

   8. Applied add-exp-log 2.5

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

   9. Applied div-exp 2.5

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

   10. Applied add-exp-log 2.5

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

   11. Applied add-exp-log 2.5

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

   12. Applied prod-exp 2.5

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

   13. Applied div-exp 2.5

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

   14. Simplified 2.5

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

4. Final simplification 6.6

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

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))))