Average Error: 28.6 → 0.3
Time: 2.5m
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{\left(-b\right) - \sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r13207804 = b;
        double r13207805 = -r13207804;
        double r13207806 = r13207804 * r13207804;
        double r13207807 = 3.0;
        double r13207808 = a;
        double r13207809 = r13207807 * r13207808;
        double r13207810 = c;
        double r13207811 = r13207809 * r13207810;
        double r13207812 = r13207806 - r13207811;
        double r13207813 = sqrt(r13207812);
        double r13207814 = r13207805 + r13207813;
        double r13207815 = r13207814 / r13207809;
        return r13207815;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r13207816 = c;
        double r13207817 = b;
        double r13207818 = -r13207817;
        double r13207819 = r13207817 * r13207817;
        double r13207820 = r13207819 * r13207819;
        double r13207821 = 3.0;
        double r13207822 = a;
        double r13207823 = r13207821 * r13207822;
        double r13207824 = r13207823 * r13207816;
        double r13207825 = r13207824 * r13207824;
        double r13207826 = r13207820 - r13207825;
        double r13207827 = r13207819 + r13207824;
        double r13207828 = r13207826 / r13207827;
        double r13207829 = sqrt(r13207828);
        double r13207830 = r13207818 - r13207829;
        double r13207831 = r13207816 / r13207830;
        return r13207831;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.6

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

  3. Applied flip-+28.6

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

  4. Applied associate-/l/28.6

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

  5. Simplified0.6

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

  7. Applied associate-/r*0.5

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

  8. Taylor expanded around inf 0.3

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

  10. Applied flip--0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))