quadm (p42, negative)

Average Error: 34.5 → 6.7

Time: 4.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.61268387266151013 \cdot 10^{141}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.5402182456312607 \cdot 10^{-243}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 2.8568501197790958 \cdot 10^{109}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r92148 = b;
        double r92149 = -r92148;
        double r92150 = r92148 * r92148;
        double r92151 = 4.0;
        double r92152 = a;
        double r92153 = c;
        double r92154 = r92152 * r92153;
        double r92155 = r92151 * r92154;
        double r92156 = r92150 - r92155;
        double r92157 = sqrt(r92156);
        double r92158 = r92149 - r92157;
        double r92159 = 2.0;
        double r92160 = r92159 * r92152;
        double r92161 = r92158 / r92160;
        return r92161;
}

↓

double f(double a, double b, double c) {
        double r92162 = b;
        double r92163 = -2.61268387266151e+141;
        bool r92164 = r92162 <= r92163;
        double r92165 = -1.0;
        double r92166 = c;
        double r92167 = r92166 / r92162;
        double r92168 = r92165 * r92167;
        double r92169 = 2.5402182456312607e-243;
        bool r92170 = r92162 <= r92169;
        double r92171 = 1.0;
        double r92172 = 2.0;
        double r92173 = r92172 * r92166;
        double r92174 = r92162 * r92162;
        double r92175 = 4.0;
        double r92176 = a;
        double r92177 = r92176 * r92166;
        double r92178 = r92175 * r92177;
        double r92179 = r92174 - r92178;
        double r92180 = sqrt(r92179);
        double r92181 = r92180 - r92162;
        double r92182 = r92173 / r92181;
        double r92183 = r92171 * r92182;
        double r92184 = 2.8568501197790958e+109;
        bool r92185 = r92162 <= r92184;
        double r92186 = -r92162;
        double r92187 = r92186 - r92180;
        double r92188 = r92172 * r92176;
        double r92189 = r92171 / r92188;
        double r92190 = r92187 * r92189;
        double r92191 = r92162 / r92176;
        double r92192 = r92165 * r92191;
        double r92193 = r92185 ? r92190 : r92192;
        double r92194 = r92170 ? r92183 : r92193;
        double r92195 = r92164 ? r92168 : r92194;
        return r92195;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.5
Target	20.6
Herbie	6.7

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

Derivation

1. Split input into 4 regimes

2. if b < -2.61268387266151e+141

   1. Initial program 62.7

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

   2. Taylor expanded around -inf 1.9

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

   if -2.61268387266151e+141 < b < 2.5402182456312607e-243

   1. Initial program 32.8

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

   3. Applied div-inv 32.8

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

   5. Applied flip-- 32.9

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

   6. Simplified 15.9

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

   7. Simplified 15.9

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

   9. Applied *-un-lft-identity 15.9

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

   10. Applied *-un-lft-identity 15.9

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

   11. Applied times-frac 15.9

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

   12. Applied associate-*l* 15.9

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

   13. Simplified 14.5

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

   14. Taylor expanded around 0 8.6

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

   if 2.5402182456312607e-243 < b < 2.8568501197790958e+109

   1. Initial program 8.7

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

   3. Applied div-inv 8.9

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

   if 2.8568501197790958e+109 < b

   1. Initial program 49.1

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

   3. Applied div-inv 49.1

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

   5. Applied flip-- 63.2

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

   6. Simplified 62.2

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

   7. Simplified 62.2

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

   8. Taylor expanded around 0 3.6

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.61268387266151013 \cdot 10^{141}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.5402182456312607 \cdot 10^{-243}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 2.8568501197790958 \cdot 10^{109}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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