Quadratic roots, narrow range

Average Error: 28.8 → 0.3

Time: 48.4s

Precision: 64

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r3033386 = b;
        double r3033387 = -r3033386;
        double r3033388 = r3033386 * r3033386;
        double r3033389 = 4.0;
        double r3033390 = a;
        double r3033391 = r3033389 * r3033390;
        double r3033392 = c;
        double r3033393 = r3033391 * r3033392;
        double r3033394 = r3033388 - r3033393;
        double r3033395 = sqrt(r3033394);
        double r3033396 = r3033387 + r3033395;
        double r3033397 = 2.0;
        double r3033398 = r3033397 * r3033390;
        double r3033399 = r3033396 / r3033398;
        return r3033399;
}

↓

double f(double a, double b, double c) {
        double r3033400 = -2.0;
        double r3033401 = c;
        double r3033402 = r3033400 * r3033401;
        double r3033403 = b;
        double r3033404 = a;
        double r3033405 = -4.0;
        double r3033406 = r3033404 * r3033405;
        double r3033407 = r3033406 * r3033401;
        double r3033408 = fma(r3033403, r3033403, r3033407);
        double r3033409 = sqrt(r3033408);
        double r3033410 = sqrt(r3033409);
        double r3033411 = fma(r3033410, r3033410, r3033403);
        double r3033412 = r3033402 / r3033411;
        return r3033412;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.8

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

2. Simplified 28.8

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

4. Applied *-un-lft-identity 28.8

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

5. Applied div-inv 28.8

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

6. Applied times-frac 28.8

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

7. Simplified 28.8

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

8. Simplified 28.8

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

10. Applied flip-- 28.9

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

11. Applied associate-*l/ 28.9

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

12. Simplified 0.4

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

13. Taylor expanded around 0 0.3

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

15. Applied add-sqr-sqrt 0.4

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

16. Applied fma-def 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019112 +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 a) c)))) (* 2 a)))