The quadratic formula (r2)

Average Error: 33.3 → 6.6

Time: 3.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.5881437021072993 \cdot 10^{+120}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.7609432989173689 \cdot 10^{-218}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}\\ \mathbf{elif}\;b \le 7.506430474816716 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r23368045 = b;
        double r23368046 = -r23368045;
        double r23368047 = r23368045 * r23368045;
        double r23368048 = 4.0;
        double r23368049 = a;
        double r23368050 = c;
        double r23368051 = r23368049 * r23368050;
        double r23368052 = r23368048 * r23368051;
        double r23368053 = r23368047 - r23368052;
        double r23368054 = sqrt(r23368053);
        double r23368055 = r23368046 - r23368054;
        double r23368056 = 2.0;
        double r23368057 = r23368056 * r23368049;
        double r23368058 = r23368055 / r23368057;
        return r23368058;
}

↓

double f(double a, double b, double c) {
        double r23368059 = b;
        double r23368060 = -3.5881437021072993e+120;
        bool r23368061 = r23368059 <= r23368060;
        double r23368062 = c;
        double r23368063 = r23368062 / r23368059;
        double r23368064 = -r23368063;
        double r23368065 = 1.7609432989173689e-218;
        bool r23368066 = r23368059 <= r23368065;
        double r23368067 = 2.0;
        double r23368068 = r23368067 * r23368062;
        double r23368069 = -r23368059;
        double r23368070 = a;
        double r23368071 = r23368062 * r23368070;
        double r23368072 = -4.0;
        double r23368073 = r23368059 * r23368059;
        double r23368074 = fma(r23368071, r23368072, r23368073);
        double r23368075 = sqrt(r23368074);
        double r23368076 = r23368069 + r23368075;
        double r23368077 = r23368068 / r23368076;
        double r23368078 = 7.506430474816716e+75;
        bool r23368079 = r23368059 <= r23368078;
        double r23368080 = 0.5;
        double r23368081 = r23368080 / r23368070;
        double r23368082 = sqrt(r23368075);
        double r23368083 = r23368082 * r23368082;
        double r23368084 = r23368069 - r23368083;
        double r23368085 = r23368081 * r23368084;
        double r23368086 = r23368059 / r23368070;
        double r23368087 = r23368063 - r23368086;
        double r23368088 = r23368079 ? r23368085 : r23368087;
        double r23368089 = r23368066 ? r23368077 : r23368088;
        double r23368090 = r23368061 ? r23368064 : r23368089;
        return r23368090;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.3
Target	20.6
Herbie	6.6

\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -3.5881437021072993e+120

   1. Initial program 59.4

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

   2. Simplified 59.4

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

   4. Applied *-un-lft-identity 59.4

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

   5. Applied *-un-lft-identity 59.4

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

   6. Applied *-un-lft-identity 59.4

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

   7. Applied distribute-lft-out-- 59.4

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

   8. Applied times-frac 59.4

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

   9. Applied associate-/l* 59.4

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

   10. Simplified 59.4

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

   12. Applied div-inv 59.4

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

   13. Applied *-un-lft-identity 59.4

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

   14. Applied times-frac 59.4

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

   15. Applied *-un-lft-identity 59.4

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

   16. Applied times-frac 59.4

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

   17. Simplified 59.4

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

   18. Simplified 59.4

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

   19. Taylor expanded around -inf 1.9

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

   20. Simplified 1.9

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

   if -3.5881437021072993e+120 < b < 1.7609432989173689e-218

   1. Initial program 30.3

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

   2. Simplified 30.3

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

   4. Applied *-un-lft-identity 30.3

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

   5. Applied *-un-lft-identity 30.3

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

   6. Applied *-un-lft-identity 30.3

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

   7. Applied distribute-lft-out-- 30.3

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

   8. Applied times-frac 30.3

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

   9. Applied associate-/l* 30.3

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

   10. Simplified 30.3

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

   12. Applied div-inv 30.3

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

   13. Applied *-un-lft-identity 30.3

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

   14. Applied times-frac 30.3

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

   15. Applied *-un-lft-identity 30.3

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

   16. Applied times-frac 30.3

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

   17. Simplified 30.3

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

   18. Simplified 30.3

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

   20. Applied flip-- 30.5

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

   21. Applied associate-*l/ 30.5

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

   22. Simplified 14.4

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

   23. Taylor expanded around -inf 9.2

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

   if 1.7609432989173689e-218 < b < 7.506430474816716e+75

   1. Initial program 7.9

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

   2. Simplified 7.9

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

   4. Applied *-un-lft-identity 7.9

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

   5. Applied *-un-lft-identity 7.9

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

   6. Applied *-un-lft-identity 7.9

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

   7. Applied distribute-lft-out-- 7.9

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

   8. Applied times-frac 7.9

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

   9. Applied associate-/l* 8.0

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

   10. Simplified 8.0

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

   12. Applied div-inv 8.0

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

   13. Applied *-un-lft-identity 8.0

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

   14. Applied times-frac 8.1

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

   15. Applied *-un-lft-identity 8.1

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

   16. Applied times-frac 8.1

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

   17. Simplified 8.0

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

   18. Simplified 8.0

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

   20. Applied add-sqr-sqrt 8.0

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

   21. Applied sqrt-prod 8.2

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

   if 7.506430474816716e+75 < b

   1. Initial program 40.2

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

   2. Simplified 40.2

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

   4. Applied *-un-lft-identity 40.2

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

   5. Applied *-un-lft-identity 40.2

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

   6. Applied *-un-lft-identity 40.2

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

   7. Applied distribute-lft-out-- 40.2

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

   8. Applied times-frac 40.2

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

   9. Applied associate-/l* 40.3

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

   10. Simplified 40.3

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

   12. Applied div-inv 40.3

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

   13. Applied *-un-lft-identity 40.3

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

   14. Applied times-frac 40.3

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

   15. Applied *-un-lft-identity 40.3

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

   16. Applied times-frac 40.3

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

   17. Simplified 40.3

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

   18. Simplified 40.3

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

   19. Taylor expanded around inf 4.7

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.5881437021072993 \cdot 10^{+120}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.7609432989173689 \cdot 10^{-218}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}\\ \mathbf{elif}\;b \le 7.506430474816716 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 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)))