Quadratic roots, wide range

Average Error: 52.6 → 0.1

Time: 12.4s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r97 = b;
        double r98 = -r97;
        double r99 = r97 * r97;
        double r100 = 4.0;
        double r101 = a;
        double r102 = r100 * r101;
        double r103 = c;
        double r104 = r102 * r103;
        double r105 = r99 - r104;
        double r106 = sqrt(r105);
        double r107 = r98 + r106;
        double r108 = 2.0;
        double r109 = r108 * r101;
        double r110 = r107 / r109;
        return r110;
}

↓

double f(double a, double b, double c) {
        double r111 = 1.0;
        double r112 = 2.0;
        double r113 = r111 / r112;
        double r114 = 4.0;
        double r115 = c;
        double r116 = r114 * r115;
        double r117 = -1.0;
        double r118 = r116 / r117;
        double r119 = b;
        double r120 = r119 * r119;
        double r121 = a;
        double r122 = r114 * r121;
        double r123 = r122 * r115;
        double r124 = r120 - r123;
        double r125 = sqrt(r124);
        double r126 = r119 + r125;
        double r127 = r118 / r126;
        double r128 = r113 * r127;
        return r128;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

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

3. Applied flip-+ 52.6

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

4. Simplified 0.4

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

6. Applied add-sqr-sqrt 0.5

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

7. Applied distribute-rgt-neg-in 0.5

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

8. Applied fma-neg 0.4

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

10. Applied *-un-lft-identity 0.4

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

11. Applied *-un-lft-identity 0.4

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

12. Applied times-frac 0.4

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

13. Applied times-frac 0.4

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

14. Simplified 0.4

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

15. Simplified 0.2

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

17. Applied neg-mul-1 0.2

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

18. Applied associate-/r* 0.2

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

19. Simplified 0.1

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

20. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))