Average Error: 28.7 → 0.3
Time: 46.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-2 \cdot c}{b + \sqrt{(b \cdot b + \left(\left(a \cdot -4\right) \cdot c\right))_*}}\]
double f(double a, double b, double c) {
        double r2855299 = b;
        double r2855300 = -r2855299;
        double r2855301 = r2855299 * r2855299;
        double r2855302 = 4.0;
        double r2855303 = a;
        double r2855304 = r2855302 * r2855303;
        double r2855305 = c;
        double r2855306 = r2855304 * r2855305;
        double r2855307 = r2855301 - r2855306;
        double r2855308 = sqrt(r2855307);
        double r2855309 = r2855300 + r2855308;
        double r2855310 = 2.0;
        double r2855311 = r2855310 * r2855303;
        double r2855312 = r2855309 / r2855311;
        return r2855312;
}
↓
double f(double a, double b, double c) {
        double r2855313 = -2.0;
        double r2855314 = c;
        double r2855315 = r2855313 * r2855314;
        double r2855316 = b;
        double r2855317 = a;
        double r2855318 = -4.0;
        double r2855319 = r2855317 * r2855318;
        double r2855320 = r2855319 * r2855314;
        double r2855321 = fma(r2855316, r2855316, r2855320);
        double r2855322 = sqrt(r2855321);
        double r2855323 = r2855316 + r2855322;
        double r2855324 = r2855315 / r2855323;
        return r2855324;
}


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

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. Simplified28.7

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

  4. Applied *-un-lft-identity28.7

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

  5. Applied div-inv28.7

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

  6. Applied times-frac28.7

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

  7. Simplified28.6

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

  8. Simplified28.6

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

  10. Applied flip--28.8

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

  11. Applied associate-*l/28.8

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

  12. Simplified0.5

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

  13. Taylor expanded around -inf 0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019101 +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)))