Quadratic roots, narrow range

Average Error: 28.7 → 0.3

Time: 18.7s

Precision: 64

\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r41190 = b;
        double r41191 = -r41190;
        double r41192 = r41190 * r41190;
        double r41193 = 4.0;
        double r41194 = a;
        double r41195 = r41193 * r41194;
        double r41196 = c;
        double r41197 = r41195 * r41196;
        double r41198 = r41192 - r41197;
        double r41199 = sqrt(r41198);
        double r41200 = r41191 + r41199;
        double r41201 = 2.0;
        double r41202 = r41201 * r41194;
        double r41203 = r41200 / r41202;
        return r41203;
}

↓

double f(double a, double b, double c) {
        double r41204 = 1.0;
        double r41205 = 2.0;
        double r41206 = r41204 / r41205;
        double r41207 = 4.0;
        double r41208 = c;
        double r41209 = a;
        double r41210 = r41209 / r41209;
        double r41211 = r41208 / r41210;
        double r41212 = r41207 * r41211;
        double r41213 = b;
        double r41214 = -r41213;
        double r41215 = -r41207;
        double r41216 = r41208 * r41209;
        double r41217 = r41215 * r41216;
        double r41218 = fma(r41213, r41213, r41217);
        double r41219 = sqrt(r41218);
        double r41220 = r41214 - r41219;
        double r41221 = r41212 / r41220;
        double r41222 = r41206 * r41221;
        return r41222;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.7

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

3. Applied flip-+ 28.7

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

4. Simplified 0.5

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

5. Simplified 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied *-un-lft-identity 0.5

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

9. Applied times-frac 0.5

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

10. Applied times-frac 0.5

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

11. Simplified 0.5

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

12. Simplified 0.4

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

14. Applied *-un-lft-identity 0.4

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

15. Applied associate-*l* 0.4

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

16. Simplified 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))