Quadratic roots, narrow range

Average Error: 28.5 → 0.3

Time: 7.1s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r37288 = b;
        double r37289 = -r37288;
        double r37290 = r37288 * r37288;
        double r37291 = 4.0;
        double r37292 = a;
        double r37293 = r37291 * r37292;
        double r37294 = c;
        double r37295 = r37293 * r37294;
        double r37296 = r37290 - r37295;
        double r37297 = sqrt(r37296);
        double r37298 = r37289 + r37297;
        double r37299 = 2.0;
        double r37300 = r37299 * r37292;
        double r37301 = r37298 / r37300;
        return r37301;
}

↓

double f(double a, double b, double c) {
        double r37302 = 1.0;
        double r37303 = 2.0;
        double r37304 = r37302 / r37303;
        double r37305 = 4.0;
        double r37306 = a;
        double r37307 = c;
        double r37308 = r37306 * r37307;
        double r37309 = r37305 * r37308;
        double r37310 = r37309 / r37306;
        double r37311 = b;
        double r37312 = 0.0;
        double r37313 = r37312 - r37309;
        double r37314 = fma(r37311, r37311, r37313);
        double r37315 = sqrt(r37314);
        double r37316 = r37315 + r37311;
        double r37317 = -r37316;
        double r37318 = r37310 / r37317;
        double r37319 = r37304 * r37318;
        return r37319;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.5

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

3. Applied flip-+ 28.5

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

4. Simplified 0.5

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

6. Applied fma-neg 0.5

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

7. Simplified 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied sqrt-prod 0.5

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

12. Applied *-un-lft-identity 0.5

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

13. Applied *-un-lft-identity 0.5

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

14. Applied times-frac 0.5

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

15. Applied times-frac 0.5

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

16. Simplified 0.5

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :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 a) c)))) (* 2 a)))