Average Error: 52.6 → 0.4
Time: 19.6s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{a \cdot 3}{\frac{b + \sqrt{\mathsf{fma}\left(\left(-c\right) \cdot a, 3, b \cdot b\right)}}{-c}}}{a \cdot 3}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{a \cdot 3}{\frac{b + \sqrt{\mathsf{fma}\left(\left(-c\right) \cdot a, 3, b \cdot b\right)}}{-c}}}{a \cdot 3}
double f(double a, double b, double c) {
        double r89266 = b;
        double r89267 = -r89266;
        double r89268 = r89266 * r89266;
        double r89269 = 3.0;
        double r89270 = a;
        double r89271 = r89269 * r89270;
        double r89272 = c;
        double r89273 = r89271 * r89272;
        double r89274 = r89268 - r89273;
        double r89275 = sqrt(r89274);
        double r89276 = r89267 + r89275;
        double r89277 = r89276 / r89271;
        return r89277;
}
↓
double f(double a, double b, double c) {
        double r89278 = a;
        double r89279 = 3.0;
        double r89280 = r89278 * r89279;
        double r89281 = b;
        double r89282 = c;
        double r89283 = -r89282;
        double r89284 = r89283 * r89278;
        double r89285 = r89281 * r89281;
        double r89286 = fma(r89284, r89279, r89285);
        double r89287 = sqrt(r89286);
        double r89288 = r89281 + r89287;
        double r89289 = r89288 / r89283;
        double r89290 = r89280 / r89289;
        double r89291 = r89290 / r89280;
        return r89291;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.6

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

  2. Simplified52.6

    \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(3 \cdot a, -c, b \cdot b\right)} - b}{3 \cdot a}}\]
  3. Using strategy rm

  4. Applied flip--52.6

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(3 \cdot a, -c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(3 \cdot a, -c, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(3 \cdot a, -c, b \cdot b\right)} + b}}}{3 \cdot a}\]

  5. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-c \cdot a, 3, 0\right)}}{\sqrt{\mathsf{fma}\left(3 \cdot a, -c, b \cdot b\right)} + b}}{3 \cdot a}\]

  6. Simplified0.5

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-c \cdot a, 3, 0\right)}{\color{blue}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}}}{3 \cdot a}\]
  7. Using strategy rm

  8. Applied clear-num0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}{\mathsf{fma}\left(-c \cdot a, 3, 0\right)}}}}{3 \cdot a}\]

  9. Simplified0.6

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{b + \sqrt{\mathsf{fma}\left(-a \cdot c, 3, b \cdot b\right)}}{-\left(a \cdot c\right) \cdot 3}}}}{3 \cdot a}\]

  10. Taylor expanded around 0 0.6

    \[\leadsto \frac{\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(-a \cdot c, 3, b \cdot b\right)}}{-\color{blue}{3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]

  11. Simplified0.4

    \[\leadsto \frac{\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(-a \cdot c, 3, b \cdot b\right)}}{-\color{blue}{\left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]
  12. Using strategy rm

  13. Applied distribute-rgt-neg-in0.4

    \[\leadsto \frac{\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(-a \cdot c, 3, b \cdot b\right)}}{\color{blue}{\left(3 \cdot a\right) \cdot \left(-c\right)}}}}{3 \cdot a}\]

  14. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{1}{\frac{\color{blue}{1 \cdot \left(b + \sqrt{\mathsf{fma}\left(-a \cdot c, 3, b \cdot b\right)}\right)}}{\left(3 \cdot a\right) \cdot \left(-c\right)}}}{3 \cdot a}\]

  15. Applied times-frac0.5

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{3 \cdot a} \cdot \frac{b + \sqrt{\mathsf{fma}\left(-a \cdot c, 3, b \cdot b\right)}}{-c}}}}{3 \cdot a}\]

  16. Applied associate-/r*0.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{1}{3 \cdot a}}}{\frac{b + \sqrt{\mathsf{fma}\left(-a \cdot c, 3, b \cdot b\right)}}{-c}}}}{3 \cdot a}\]

  17. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{a \cdot 3}}{\frac{b + \sqrt{\mathsf{fma}\left(-a \cdot c, 3, b \cdot b\right)}}{-c}}}{3 \cdot a}\]

  18. Final simplification0.4

    \[\leadsto \frac{\frac{a \cdot 3}{\frac{b + \sqrt{\mathsf{fma}\left(\left(-c\right) \cdot a, 3, b \cdot b\right)}}{-c}}}{a \cdot 3}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))