quad2m (problem 3.2.1, negative)

Average Error: 33.8 → 6.6

Time: 6.4s

Precision: 64

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.48395925749684231185755183204764945523 \cdot 10^{145}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 1.027416821538068536725790738835144506948 \cdot 10^{-173}:\\ \;\;\;\;1 \cdot \frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 4.397833618396295559623484253976393644285 \cdot 10^{103}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r21204 = b_2;
        double r21205 = -r21204;
        double r21206 = r21204 * r21204;
        double r21207 = a;
        double r21208 = c;
        double r21209 = r21207 * r21208;
        double r21210 = r21206 - r21209;
        double r21211 = sqrt(r21210);
        double r21212 = r21205 - r21211;
        double r21213 = r21212 / r21207;
        return r21213;
}

↓

double f(double a, double b_2, double c) {
        double r21214 = b_2;
        double r21215 = -6.483959257496842e+145;
        bool r21216 = r21214 <= r21215;
        double r21217 = 1.0;
        double r21218 = -0.5;
        double r21219 = c;
        double r21220 = r21219 / r21214;
        double r21221 = r21218 * r21220;
        double r21222 = r21217 * r21221;
        double r21223 = 1.0274168215380685e-173;
        bool r21224 = r21214 <= r21223;
        double r21225 = r21214 * r21214;
        double r21226 = a;
        double r21227 = r21226 * r21219;
        double r21228 = r21225 - r21227;
        double r21229 = sqrt(r21228);
        double r21230 = r21229 - r21214;
        double r21231 = r21230 / r21219;
        double r21232 = r21217 / r21231;
        double r21233 = r21217 * r21232;
        double r21234 = 4.3978336183962956e+103;
        bool r21235 = r21214 <= r21234;
        double r21236 = -r21214;
        double r21237 = r21236 - r21229;
        double r21238 = r21237 / r21226;
        double r21239 = -2.0;
        double r21240 = r21214 / r21226;
        double r21241 = r21239 * r21240;
        double r21242 = r21235 ? r21238 : r21241;
        double r21243 = r21224 ? r21233 : r21242;
        double r21244 = r21216 ? r21222 : r21243;
        return r21244;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -6.483959257496842e+145

   1. Initial program 62.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-- 62.8

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 37.5

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 37.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 37.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

   8. Applied associate-/r* 37.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{1}}{a}}\]

   9. Simplified 37.3

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{a}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 50.7

       \[\leadsto \frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\]

   12. Applied *-un-lft-identity 50.7

       \[\leadsto \frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{\color{blue}{1 \cdot c}}}}{\sqrt{a} \cdot \sqrt{a}}\]

   13. Applied *-un-lft-identity 50.7

       \[\leadsto \frac{\frac{a}{\frac{\color{blue}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}{1 \cdot c}}}{\sqrt{a} \cdot \sqrt{a}}\]

   14. Applied times-frac 50.7

       \[\leadsto \frac{\frac{a}{\color{blue}{\frac{1}{1} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{\sqrt{a} \cdot \sqrt{a}}\]

   15. Applied add-sqr-sqrt 50.7

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{\frac{1}{1} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt{a} \cdot \sqrt{a}}\]

   16. Applied times-frac 50.7

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{a}}{\frac{1}{1}} \cdot \frac{\sqrt{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{\sqrt{a} \cdot \sqrt{a}}\]

   17. Applied times-frac 50.6

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{a}}{\frac{1}{1}}}{\sqrt{a}} \cdot \frac{\frac{\sqrt{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt{a}}}\]

   18. Simplified 50.6

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{\sqrt{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt{a}}\]

   19. Simplified 36.9

       \[\leadsto 1 \cdot \color{blue}{\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}\]

   20. Taylor expanded around -inf 1.6

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)}\]

   if -6.483959257496842e+145 < b_2 < 1.0274168215380685e-173

   1. Initial program 30.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-- 30.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 15.4

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 15.4

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 15.4

      \[\leadsto \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

   8. Applied associate-/r* 15.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{1}}{a}}\]

   9. Simplified 13.8

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{a}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 39.2

       \[\leadsto \frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\]

   12. Applied *-un-lft-identity 39.2

       \[\leadsto \frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{\color{blue}{1 \cdot c}}}}{\sqrt{a} \cdot \sqrt{a}}\]

   13. Applied *-un-lft-identity 39.2

       \[\leadsto \frac{\frac{a}{\frac{\color{blue}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}{1 \cdot c}}}{\sqrt{a} \cdot \sqrt{a}}\]

   14. Applied times-frac 39.2

       \[\leadsto \frac{\frac{a}{\color{blue}{\frac{1}{1} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{\sqrt{a} \cdot \sqrt{a}}\]

   15. Applied add-sqr-sqrt 39.1

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{\frac{1}{1} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt{a} \cdot \sqrt{a}}\]

   16. Applied times-frac 39.2

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{a}}{\frac{1}{1}} \cdot \frac{\sqrt{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{\sqrt{a} \cdot \sqrt{a}}\]

   17. Applied times-frac 37.7

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{a}}{\frac{1}{1}}}{\sqrt{a}} \cdot \frac{\frac{\sqrt{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt{a}}}\]

   18. Simplified 37.7

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{\sqrt{a}}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{\sqrt{a}}\]

   19. Simplified 9.7

       \[\leadsto 1 \cdot \color{blue}{\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}\]

   if 1.0274168215380685e-173 < b_2 < 4.3978336183962956e+103

   1. Initial program 6.4

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   if 4.3978336183962956e+103 < b_2

   1. Initial program 47.2

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-- 63.4

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 62.5

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 62.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

   6. Taylor expanded around 0 3.9

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.48395925749684231185755183204764945523 \cdot 10^{145}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 1.027416821538068536725790738835144506948 \cdot 10^{-173}:\\ \;\;\;\;1 \cdot \frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 4.397833618396295559623484253976393644285 \cdot 10^{103}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))