Cubic critical, wide range

Average Error: 52.4 → 51.6

Time: 21.8s

Precision: 64

\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r2226966 = b;
        double r2226967 = -r2226966;
        double r2226968 = r2226966 * r2226966;
        double r2226969 = 3.0;
        double r2226970 = a;
        double r2226971 = r2226969 * r2226970;
        double r2226972 = c;
        double r2226973 = r2226971 * r2226972;
        double r2226974 = r2226968 - r2226973;
        double r2226975 = sqrt(r2226974);
        double r2226976 = r2226967 + r2226975;
        double r2226977 = r2226976 / r2226971;
        return r2226977;
}

↓

double f(double a, double b, double c) {
        double r2226978 = -3.0;
        double r2226979 = a;
        double r2226980 = c;
        double r2226981 = r2226979 * r2226980;
        double r2226982 = b;
        double r2226983 = r2226982 * r2226982;
        double r2226984 = fma(r2226978, r2226981, r2226983);
        double r2226985 = sqrt(r2226984);
        double r2226986 = sqrt(r2226985);
        double r2226987 = -r2226982;
        double r2226988 = fma(r2226986, r2226986, r2226987);
        double r2226989 = 0.3333333333333333;
        double r2226990 = cbrt(r2226989);
        double r2226991 = r2226990 * r2226990;
        double r2226992 = sqrt(r2226979);
        double r2226993 = r2226991 / r2226992;
        double r2226994 = sqrt(r2226993);
        double r2226995 = r2226989 / r2226979;
        double r2226996 = sqrt(r2226995);
        double r2226997 = r2226990 / r2226992;
        double r2226998 = sqrt(r2226997);
        double r2226999 = r2226996 * r2226998;
        double r2227000 = r2226994 * r2226999;
        double r2227001 = r2226988 * r2227000;
        return r2227001;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

2. Simplified 52.4

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

4. Applied add-sqr-sqrt 52.4

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

5. Applied sqrt-prod 52.2

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

6. Applied fma-neg 51.6

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

8. Applied div-inv 51.6

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

9. Simplified 51.6

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

11. Applied add-sqr-sqrt 51.6

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

13. Applied add-sqr-sqrt 51.6

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

14. Applied add-cube-cbrt 51.6

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

15. Applied times-frac 51.6

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

16. Applied sqrt-prod 51.6

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

17. Applied associate-*l* 51.6

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

18. Final simplification 51.6

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

Reproduce

herbie shell --seed 2019156 +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 a) c)))) (* 3 a)))