The quadratic formula (r1)

Average Error: 34.7 → 8.9

Time: 7.0s

Precision: 64

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -5.4369139762720996 \cdot 10^{56}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.70350838245532258 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.8597470564587674 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{1} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1} \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r80211 = b;
        double r80212 = -r80211;
        double r80213 = r80211 * r80211;
        double r80214 = 4.0;
        double r80215 = a;
        double r80216 = r80214 * r80215;
        double r80217 = c;
        double r80218 = r80216 * r80217;
        double r80219 = r80213 - r80218;
        double r80220 = sqrt(r80219);
        double r80221 = r80212 + r80220;
        double r80222 = 2.0;
        double r80223 = r80222 * r80215;
        double r80224 = r80221 / r80223;
        return r80224;
}

↓

double f(double a, double b, double c) {
        double r80225 = b;
        double r80226 = -5.4369139762720996e+56;
        bool r80227 = r80225 <= r80226;
        double r80228 = 1.0;
        double r80229 = c;
        double r80230 = r80229 / r80225;
        double r80231 = a;
        double r80232 = r80225 / r80231;
        double r80233 = r80230 - r80232;
        double r80234 = r80228 * r80233;
        double r80235 = -8.703508382455323e-221;
        bool r80236 = r80225 <= r80235;
        double r80237 = 4.0;
        double r80238 = 2.0;
        double r80239 = pow(r80225, r80238);
        double r80240 = r80239 - r80239;
        double r80241 = r80231 * r80229;
        double r80242 = r80237 * r80241;
        double r80243 = r80240 + r80242;
        double r80244 = r80243 / r80241;
        double r80245 = r80237 / r80244;
        double r80246 = -r80225;
        double r80247 = r80225 * r80225;
        double r80248 = r80237 * r80231;
        double r80249 = r80248 * r80229;
        double r80250 = r80247 - r80249;
        double r80251 = sqrt(r80250);
        double r80252 = r80246 + r80251;
        double r80253 = r80245 * r80252;
        double r80254 = 2.0;
        double r80255 = r80254 * r80231;
        double r80256 = r80253 / r80255;
        double r80257 = 1.8597470564587674e+138;
        bool r80258 = r80225 <= r80257;
        double r80259 = 1.0;
        double r80260 = r80259 / r80259;
        double r80261 = r80246 - r80251;
        double r80262 = r80259 / r80261;
        double r80263 = r80254 / r80237;
        double r80264 = r80263 * r80259;
        double r80265 = r80264 / r80229;
        double r80266 = r80262 / r80265;
        double r80267 = r80260 * r80266;
        double r80268 = -1.0;
        double r80269 = r80268 * r80230;
        double r80270 = r80260 * r80269;
        double r80271 = r80258 ? r80267 : r80270;
        double r80272 = r80236 ? r80256 : r80271;
        double r80273 = r80227 ? r80234 : r80272;
        return r80273;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.7
Target	21.9
Herbie	8.9

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

Derivation

1. Split input into 4 regimes

2. if b < -5.4369139762720996e+56

   1. Initial program 42.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Taylor expanded around -inf 5.1

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

   3. Simplified 5.1

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

   if -5.4369139762720996e+56 < b < -8.703508382455323e-221

   1. Initial program 8.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied flip-+ 35.2

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

   4. Simplified 35.3

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
   5. Using strategy rm

   6. Applied flip-- 35.3

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

   7. Applied associate-/r/ 35.4

      \[\leadsto \frac{\color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

   8. Simplified 16.9

      \[\leadsto \frac{\color{blue}{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\]

   if -8.703508382455323e-221 < b < 1.8597470564587674e+138

   1. Initial program 31.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied flip-+ 31.2

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

   4. Simplified 16.1

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 16.1

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

   7. Applied *-un-lft-identity 16.1

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

   8. Applied times-frac 16.1

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

   9. Applied associate-/l* 16.2

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

   10. Simplified 15.0

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
   11. Using strategy rm

   12. Applied div-inv 15.0

       \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

   13. Simplified 9.8

       \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}}\]

   if 1.8597470564587674e+138 < b

   1. Initial program 62.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied flip-+ 62.4

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

   4. Simplified 37.6

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 37.6

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

   7. Applied *-un-lft-identity 37.6

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

   8. Applied times-frac 37.6

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

   9. Applied associate-/l* 37.7

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

   10. Simplified 37.3

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
   11. Using strategy rm

   12. Applied div-inv 37.3

       \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

   13. Simplified 36.9

       \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}}\]

   14. Taylor expanded around inf 2.0

       \[\leadsto \frac{1}{1} \cdot \color{blue}{\left(-1 \cdot \frac{c}{b}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.4369139762720996 \cdot 10^{56}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.70350838245532258 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.8597470564587674 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{1} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1} \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))