Average Error: 28.6 → 0.3
Time: 2.0m
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot \left(a \cdot c\right)\right)\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot \left(a \cdot c\right)\right)\right)}}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r10295208 = b;
        double r10295209 = -r10295208;
        double r10295210 = r10295208 * r10295208;
        double r10295211 = 3.0;
        double r10295212 = a;
        double r10295213 = r10295211 * r10295212;
        double r10295214 = c;
        double r10295215 = r10295213 * r10295214;
        double r10295216 = r10295210 - r10295215;
        double r10295217 = sqrt(r10295216);
        double r10295218 = r10295209 + r10295217;
        double r10295219 = r10295218 / r10295213;
        return r10295219;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r10295220 = c;
        double r10295221 = b;
        double r10295222 = -r10295221;
        double r10295223 = -3.0;
        double r10295224 = a;
        double r10295225 = r10295224 * r10295220;
        double r10295226 = r10295223 * r10295225;
        double r10295227 = fma(r10295221, r10295221, r10295226);
        double r10295228 = sqrt(r10295227);
        double r10295229 = r10295222 - r10295228;
        double r10295230 = r10295220 / r10295229;
        return r10295230;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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 times-frac0.6

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

  8. Simplified0.6

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

  10. Applied pow10.6

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

  11. Applied pow10.6

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

  12. Applied pow-prod-down0.6

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019120 +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)))