quad2m (problem 3.2.1, negative)

Average Error: 33.6 → 8.5

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -31202666053494416225599488:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le -2.198298561806402727361079035927886117853 \cdot 10^{-291}:\\ \;\;\;\;1 \cdot \frac{\frac{1 \cdot \left(\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 5.379869857595589463811898652808656389365 \cdot 10^{128}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r15051 = b_2;
        double r15052 = -r15051;
        double r15053 = r15051 * r15051;
        double r15054 = a;
        double r15055 = c;
        double r15056 = r15054 * r15055;
        double r15057 = r15053 - r15056;
        double r15058 = sqrt(r15057);
        double r15059 = r15052 - r15058;
        double r15060 = r15059 / r15054;
        return r15060;
}

↓

double f(double a, double b_2, double c) {
        double r15061 = b_2;
        double r15062 = -3.1202666053494416e+25;
        bool r15063 = r15061 <= r15062;
        double r15064 = 1.0;
        double r15065 = -0.5;
        double r15066 = c;
        double r15067 = r15066 / r15061;
        double r15068 = r15065 * r15067;
        double r15069 = r15064 * r15068;
        double r15070 = -2.1982985618064027e-291;
        bool r15071 = r15061 <= r15070;
        double r15072 = 2.0;
        double r15073 = pow(r15061, r15072);
        double r15074 = r15073 - r15073;
        double r15075 = a;
        double r15076 = r15075 * r15066;
        double r15077 = r15074 + r15076;
        double r15078 = r15064 * r15077;
        double r15079 = r15078 / r15075;
        double r15080 = -r15061;
        double r15081 = r15061 * r15061;
        double r15082 = r15081 - r15076;
        double r15083 = sqrt(r15082);
        double r15084 = r15080 + r15083;
        double r15085 = r15079 / r15084;
        double r15086 = r15064 * r15085;
        double r15087 = 5.3798698575955895e+128;
        bool r15088 = r15061 <= r15087;
        double r15089 = r15080 - r15083;
        double r15090 = r15089 / r15075;
        double r15091 = r15064 * r15090;
        double r15092 = 0.5;
        double r15093 = r15092 * r15067;
        double r15094 = r15061 / r15075;
        double r15095 = r15072 * r15094;
        double r15096 = r15093 - r15095;
        double r15097 = r15064 * r15096;
        double r15098 = r15088 ? r15091 : r15097;
        double r15099 = r15071 ? r15086 : r15098;
        double r15100 = r15063 ? r15069 : r15099;
        return r15100;
}

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 < -3.1202666053494416e+25

   1. Initial program 56.4

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

   3. Applied clear-num 56.4

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

   5. Applied *-un-lft-identity 56.4

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

   6. Applied *-un-lft-identity 56.4

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

   7. Applied times-frac 56.4

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

   8. Applied add-cube-cbrt 56.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   9. Applied times-frac 56.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

   10. Simplified 56.4

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   11. Simplified 56.4

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

   12. Taylor expanded around -inf 4.8

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

   if -3.1202666053494416e+25 < b_2 < -2.1982985618064027e-291

   1. Initial program 27.7

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

   3. Applied clear-num 27.7

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

   5. Applied *-un-lft-identity 27.7

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

   6. Applied *-un-lft-identity 27.7

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

   7. Applied times-frac 27.7

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

   8. Applied add-cube-cbrt 27.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   9. Applied times-frac 27.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

   10. Simplified 27.7

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   11. Simplified 27.7

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

   13. Applied clear-num 27.7

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

   15. Applied flip-- 27.8

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

   16. Applied associate-/r/ 27.8

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

   17. Applied associate-/r* 27.8

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

   18. Simplified 16.2

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

   if -2.1982985618064027e-291 < b_2 < 5.3798698575955895e+128

   1. Initial program 8.7

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

   3. Applied clear-num 8.8

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

   5. Applied *-un-lft-identity 8.8

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

   6. Applied *-un-lft-identity 8.8

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

   7. Applied times-frac 8.8

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

   8. Applied add-cube-cbrt 8.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   9. Applied times-frac 8.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

   10. Simplified 8.8

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   11. Simplified 8.7

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

   if 5.3798698575955895e+128 < b_2

   1. Initial program 54.4

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

   3. Applied clear-num 54.4

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

   5. Applied *-un-lft-identity 54.4

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

   6. Applied *-un-lft-identity 54.4

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

   7. Applied times-frac 54.4

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

   8. Applied add-cube-cbrt 54.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   9. Applied times-frac 54.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

   10. Simplified 54.4

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   11. Simplified 54.4

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

   12. Taylor expanded around inf 3.1

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -31202666053494416225599488:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le -2.198298561806402727361079035927886117853 \cdot 10^{-291}:\\ \;\;\;\;1 \cdot \frac{\frac{1 \cdot \left(\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 5.379869857595589463811898652808656389365 \cdot 10^{128}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))