Average Error: 52.6 → 0.3
Time: 16.7s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\frac{\sqrt{\mathsf{fma}\left(3 \cdot \left(-a\right), c, {b}^{2}\right)} + b}{c} \cdot \frac{a \cdot 3}{3 \cdot \left(-a\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{1}{\frac{\sqrt{\mathsf{fma}\left(3 \cdot \left(-a\right), c, {b}^{2}\right)} + b}{c} \cdot \frac{a \cdot 3}{3 \cdot \left(-a\right)}}
double f(double a, double b, double c) {
        double r98859 = b;
        double r98860 = -r98859;
        double r98861 = r98859 * r98859;
        double r98862 = 3.0;
        double r98863 = a;
        double r98864 = r98862 * r98863;
        double r98865 = c;
        double r98866 = r98864 * r98865;
        double r98867 = r98861 - r98866;
        double r98868 = sqrt(r98867);
        double r98869 = r98860 + r98868;
        double r98870 = r98869 / r98864;
        return r98870;
}
↓
double f(double a, double b, double c) {
        double r98871 = 1.0;
        double r98872 = 3.0;
        double r98873 = a;
        double r98874 = -r98873;
        double r98875 = r98872 * r98874;
        double r98876 = c;
        double r98877 = b;
        double r98878 = 2.0;
        double r98879 = pow(r98877, r98878);
        double r98880 = fma(r98875, r98876, r98879);
        double r98881 = sqrt(r98880);
        double r98882 = r98881 + r98877;
        double r98883 = r98882 / r98876;
        double r98884 = r98873 * r98872;
        double r98885 = r98884 / r98875;
        double r98886 = r98883 * r98885;
        double r98887 = r98871 / r98886;
        return r98887;
}

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 *-un-lft-identity0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Applied times-frac0.5

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

  11. Applied associate-/l*0.6

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

  12. Simplified0.3

    \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{a \cdot 3}{\left(-3\right) \cdot a} \cdot \frac{b + \sqrt{\mathsf{fma}\left(\left(-3\right) \cdot a, c, {b}^{2}\right)}}{c}}}\]

  13. Final simplification0.3

    \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(3 \cdot \left(-a\right), c, {b}^{2}\right)} + b}{c} \cdot \frac{a \cdot 3}{3 \cdot \left(-a\right)}}\]

Reproduce

herbie shell --seed 2019179 +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)))