Average Error: 44.0 → 0.2
Time: 1.8m
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{(b \cdot b + \left(-3 \cdot \left(a \cdot c\right)\right))_*}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{\left(-b\right) - \sqrt{(b \cdot b + \left(-3 \cdot \left(a \cdot c\right)\right))_*}}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r21689901 = b;
        double r21689902 = -r21689901;
        double r21689903 = r21689901 * r21689901;
        double r21689904 = 3.0;
        double r21689905 = a;
        double r21689906 = r21689904 * r21689905;
        double r21689907 = c;
        double r21689908 = r21689906 * r21689907;
        double r21689909 = r21689903 - r21689908;
        double r21689910 = sqrt(r21689909);
        double r21689911 = r21689902 + r21689910;
        double r21689912 = r21689911 / r21689906;
        return r21689912;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r21689913 = c;
        double r21689914 = b;
        double r21689915 = -r21689914;
        double r21689916 = -3.0;
        double r21689917 = a;
        double r21689918 = r21689917 * r21689913;
        double r21689919 = r21689916 * r21689918;
        double r21689920 = fma(r21689914, r21689914, r21689919);
        double r21689921 = sqrt(r21689920);
        double r21689922 = r21689915 - r21689921;
        double r21689923 = r21689913 / r21689922;
        return r21689923;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 44.0

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+44.0

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

  4. Applied associate-/l/44.0

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

  5. Simplified0.5

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

  7. Applied times-frac0.5

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

  8. Simplified0.5

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

  10. Applied pow10.5

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

  11. Applied pow10.5

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

  12. Applied pow-prod-down0.5

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))