quad2p (problem 3.2.1, positive)

Average Error: 34.2 → 7.2

Time: 16.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.063494556122608953103184521416208568921 \cdot 10^{74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le -4.058683644017214908087872893810993578525 \cdot 10^{-201}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} - b_2}}\\ \mathbf{elif}\;b_2 \le 9.585324748779202632561049799715830083757 \cdot 10^{76}:\\ \;\;\;\;\frac{-1}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} + b_2} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot c}{b_2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r29959 = b_2;
        double r29960 = -r29959;
        double r29961 = r29959 * r29959;
        double r29962 = a;
        double r29963 = c;
        double r29964 = r29962 * r29963;
        double r29965 = r29961 - r29964;
        double r29966 = sqrt(r29965);
        double r29967 = r29960 + r29966;
        double r29968 = r29967 / r29962;
        return r29968;
}

↓

double f(double a, double b_2, double c) {
        double r29969 = b_2;
        double r29970 = -2.063494556122609e+74;
        bool r29971 = r29969 <= r29970;
        double r29972 = c;
        double r29973 = r29972 / r29969;
        double r29974 = 0.5;
        double r29975 = a;
        double r29976 = r29969 / r29975;
        double r29977 = -2.0;
        double r29978 = r29976 * r29977;
        double r29979 = fma(r29973, r29974, r29978);
        double r29980 = -4.058683644017215e-201;
        bool r29981 = r29969 <= r29980;
        double r29982 = 1.0;
        double r29983 = -r29972;
        double r29984 = r29969 * r29969;
        double r29985 = fma(r29975, r29983, r29984);
        double r29986 = sqrt(r29985);
        double r29987 = r29986 - r29969;
        double r29988 = r29975 / r29987;
        double r29989 = r29982 / r29988;
        double r29990 = 9.585324748779203e+76;
        bool r29991 = r29969 <= r29990;
        double r29992 = -1.0;
        double r29993 = r29986 + r29969;
        double r29994 = r29992 / r29993;
        double r29995 = r29994 * r29972;
        double r29996 = -0.5;
        double r29997 = r29996 * r29972;
        double r29998 = r29997 / r29969;
        double r29999 = r29991 ? r29995 : r29998;
        double r30000 = r29981 ? r29989 : r29999;
        double r30001 = r29971 ? r29979 : r30000;
        return r30001;
}

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 < -2.063494556122609e+74

   1. Initial program 41.4

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

   2. Taylor expanded around -inf 5.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   3. Simplified 5.3

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

   if -2.063494556122609e+74 < b_2 < -4.058683644017215e-201

   1. Initial program 8.3

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

   3. Applied clear-num 8.4

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

   4. Simplified 8.4

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

   if -4.058683644017215e-201 < b_2 < 9.585324748779203e+76

   1. Initial program 28.3

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

   3. Applied flip-+ 28.4

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

   4. Simplified 16.6

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

   5. Simplified 16.6

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{-\left(b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}\right)}}}{a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 16.6

      \[\leadsto \frac{\frac{0 + a \cdot c}{-\left(b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}\right)}}{\color{blue}{1 \cdot a}}\]

   8. Applied neg-mul-1 16.6

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{-1 \cdot \left(b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}\right)}}}{1 \cdot a}\]

   9. Applied *-un-lft-identity 16.6

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + a \cdot c\right)}}{-1 \cdot \left(b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}\right)}}{1 \cdot a}\]

   10. Applied times-frac 16.6

       \[\leadsto \frac{\color{blue}{\frac{1}{-1} \cdot \frac{0 + a \cdot c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}}}{1 \cdot a}\]

   11. Applied times-frac 16.6

       \[\leadsto \color{blue}{\frac{\frac{1}{-1}}{1} \cdot \frac{\frac{0 + a \cdot c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}}{a}}\]

   12. Simplified 16.6

       \[\leadsto \color{blue}{-1} \cdot \frac{\frac{0 + a \cdot c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}}{a}\]

   13. Simplified 16.2

       \[\leadsto -1 \cdot \color{blue}{\frac{\frac{c \cdot a}{a}}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, {b_2}^{2}\right)}}}\]
   14. Using strategy rm

   15. Applied *-un-lft-identity 16.2

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

   16. Applied *-un-lft-identity 16.2

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

   17. Applied times-frac 10.2

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

   18. Applied times-frac 10.3

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

   19. Simplified 10.3

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

   20. Simplified 10.3

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

   if 9.585324748779203e+76 < b_2

   1. Initial program 58.5

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

   2. Taylor expanded around inf 3.4

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

   3. Simplified 3.4

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.063494556122608953103184521416208568921 \cdot 10^{74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le -4.058683644017214908087872893810993578525 \cdot 10^{-201}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} - b_2}}\\ \mathbf{elif}\;b_2 \le 9.585324748779202632561049799715830083757 \cdot 10^{76}:\\ \;\;\;\;\frac{-1}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} + b_2} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))