Average Error: 28.8 → 0.3
Time: 18.0s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(-3 \cdot a\right) \cdot \frac{c}{\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} + b}}{3 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\left(-3 \cdot a\right) \cdot \frac{c}{\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} + b}}{3 \cdot a}
double f(double a, double b, double c) {
        double r103731 = b;
        double r103732 = -r103731;
        double r103733 = r103731 * r103731;
        double r103734 = 3.0;
        double r103735 = a;
        double r103736 = r103734 * r103735;
        double r103737 = c;
        double r103738 = r103736 * r103737;
        double r103739 = r103733 - r103738;
        double r103740 = sqrt(r103739);
        double r103741 = r103732 + r103740;
        double r103742 = r103741 / r103736;
        return r103742;
}
↓
double f(double a, double b, double c) {
        double r103743 = -3.0;
        double r103744 = a;
        double r103745 = r103743 * r103744;
        double r103746 = c;
        double r103747 = 3.0;
        double r103748 = r103747 * r103744;
        double r103749 = -r103748;
        double r103750 = b;
        double r103751 = r103750 * r103750;
        double r103752 = fma(r103746, r103749, r103751);
        double r103753 = sqrt(r103752);
        double r103754 = r103753 + r103750;
        double r103755 = r103746 / r103754;
        double r103756 = r103745 * r103755;
        double r103757 = r103756 / r103748;
        return r103757;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.8

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

  2. Simplified28.8

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

  4. Applied flip--28.8

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

  5. Simplified0.6

    \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-c \cdot a, 3, 0\right)}}{\sqrt{\mathsf{fma}\left(3 \cdot a, -c, b \cdot b\right)} + b}}{3 \cdot a}\]

  6. Simplified0.6

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-c \cdot a, 3, 0\right)}{\color{blue}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}}}{3 \cdot a}\]

  7. Taylor expanded around 0 0.6

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

  8. Simplified0.5

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

  10. Applied *-un-lft-identity0.5

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

  11. Applied times-frac0.3

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

  12. Simplified0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b c)
  :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) (* (* 3.0 a) c)))) (* 3.0 a)))