Average Error: 52.4 → 0.4
Time: 17.5s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(\frac{a \cdot c}{a} \cdot \frac{4}{-\left(\sqrt{\mathsf{fma}\left(4, -a \cdot c, {b}^{2}\right)} + b\right)}\right) \cdot \frac{1}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(\frac{a \cdot c}{a} \cdot \frac{4}{-\left(\sqrt{\mathsf{fma}\left(4, -a \cdot c, {b}^{2}\right)} + b\right)}\right) \cdot \frac{1}{2}
double f(double a, double b, double c) {
        double r30962 = b;
        double r30963 = -r30962;
        double r30964 = r30962 * r30962;
        double r30965 = 4.0;
        double r30966 = a;
        double r30967 = r30965 * r30966;
        double r30968 = c;
        double r30969 = r30967 * r30968;
        double r30970 = r30964 - r30969;
        double r30971 = sqrt(r30970);
        double r30972 = r30963 + r30971;
        double r30973 = 2.0;
        double r30974 = r30973 * r30966;
        double r30975 = r30972 / r30974;
        return r30975;
}
↓
double f(double a, double b, double c) {
        double r30976 = a;
        double r30977 = c;
        double r30978 = r30976 * r30977;
        double r30979 = r30978 / r30976;
        double r30980 = 4.0;
        double r30981 = -r30978;
        double r30982 = b;
        double r30983 = 2.0;
        double r30984 = pow(r30982, r30983);
        double r30985 = fma(r30980, r30981, r30984);
        double r30986 = sqrt(r30985);
        double r30987 = r30986 + r30982;
        double r30988 = -r30987;
        double r30989 = r30980 / r30988;
        double r30990 = r30979 * r30989;
        double r30991 = 1.0;
        double r30992 = 2.0;
        double r30993 = r30991 / r30992;
        double r30994 = r30990 * r30993;
        return r30994;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.4

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

  3. Applied flip-+52.4

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

  4. Simplified0.4

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

  5. Simplified0.4

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

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

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

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

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

  9. Applied times-frac0.4

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

  10. Applied times-frac0.4

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

  11. Simplified0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))