Average Error: 52.5 → 50.7
Time: 42.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{(\left(\sqrt{\sqrt{{\left((-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}} \cdot \left(\left(\sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right)\right)}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14234923 = b;
        double r14234924 = -r14234923;
        double r14234925 = r14234923 * r14234923;
        double r14234926 = 3.0;
        double r14234927 = a;
        double r14234928 = r14234926 * r14234927;
        double r14234929 = c;
        double r14234930 = r14234928 * r14234929;
        double r14234931 = r14234925 - r14234930;
        double r14234932 = sqrt(r14234931);
        double r14234933 = r14234924 + r14234932;
        double r14234934 = r14234933 / r14234928;
        return r14234934;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14234935 = -3.0;
        double r14234936 = a;
        double r14234937 = c;
        double r14234938 = r14234936 * r14234937;
        double r14234939 = b;
        double r14234940 = r14234939 * r14234939;
        double r14234941 = fma(r14234935, r14234938, r14234940);
        double r14234942 = 0.3333333333333333;
        double r14234943 = pow(r14234941, r14234942);
        double r14234944 = sqrt(r14234941);
        double r14234945 = cbrt(r14234944);
        double r14234946 = r14234945 * r14234945;
        double r14234947 = r14234946 * r14234946;
        double r14234948 = r14234943 * r14234947;
        double r14234949 = sqrt(r14234948);
        double r14234950 = sqrt(r14234949);
        double r14234951 = sqrt(r14234944);
        double r14234952 = -r14234939;
        double r14234953 = fma(r14234950, r14234951, r14234952);
        double r14234954 = 3.0;
        double r14234955 = r14234954 * r14234936;
        double r14234956 = r14234953 / r14234955;
        return r14234956;
}


\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{(\left(\sqrt{\sqrt{{\left((-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}} \cdot \left(\left(\sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right)\right)}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 52.5

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

  2. Simplified52.5

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

  4. Applied add-sqr-sqrt52.5

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

  5. Applied sqrt-prod52.3

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

  6. Applied fma-neg51.7

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

  8. Applied add-cube-cbrt51.9

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

  10. Applied pow1/350.7

    \[\leadsto \frac{(\left(\sqrt{\sqrt{\left(\sqrt[3]{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}\right) \cdot \color{blue}{{\left((-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt50.7

    \[\leadsto \frac{(\left(\sqrt{\sqrt{\left(\sqrt[3]{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{\color{blue}{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}\right) \cdot {\left((-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]

  13. Applied cbrt-prod50.7

    \[\leadsto \frac{(\left(\sqrt{\sqrt{\left(\sqrt[3]{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \color{blue}{\left(\sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)}\right) \cdot {\left((-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]

  14. Applied add-sqr-sqrt50.7

    \[\leadsto \frac{(\left(\sqrt{\sqrt{\left(\sqrt[3]{\color{blue}{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}} \cdot \left(\sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)\right) \cdot {\left((-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]

  15. Applied cbrt-prod50.7

    \[\leadsto \frac{(\left(\sqrt{\sqrt{\left(\color{blue}{\left(\sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)} \cdot \left(\sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)\right) \cdot {\left((-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]

  16. Applied swap-sqr50.7

    \[\leadsto \frac{(\left(\sqrt{\sqrt{\color{blue}{\left(\left(\sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)\right)} \cdot {\left((-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]

  17. Final simplification50.7

    \[\leadsto \frac{(\left(\sqrt{\sqrt{{\left((-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}} \cdot \left(\left(\sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right)\right)}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a b c d)
  :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)))