Average Error: 28.4 → 0.3
Time: 1.8m
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{(b \cdot b + \left(-3 \cdot \left(a \cdot c\right)\right))_*}}\]
\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 r15870404 = b;
        double r15870405 = -r15870404;
        double r15870406 = r15870404 * r15870404;
        double r15870407 = 3.0;
        double r15870408 = a;
        double r15870409 = r15870407 * r15870408;
        double r15870410 = c;
        double r15870411 = r15870409 * r15870410;
        double r15870412 = r15870406 - r15870411;
        double r15870413 = sqrt(r15870412);
        double r15870414 = r15870405 + r15870413;
        double r15870415 = r15870414 / r15870409;
        return r15870415;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15870416 = c;
        double r15870417 = b;
        double r15870418 = -r15870417;
        double r15870419 = -3.0;
        double r15870420 = a;
        double r15870421 = r15870420 * r15870416;
        double r15870422 = r15870419 * r15870421;
        double r15870423 = fma(r15870417, r15870417, r15870422);
        double r15870424 = sqrt(r15870423);
        double r15870425 = r15870418 - r15870424;
        double r15870426 = r15870416 / r15870425;
        return r15870426;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 28.4

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

  3. Applied flip-+28.4

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

  4. Applied associate-/l/28.4

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

  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

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

  10. Applied pow10.6

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

  11. Applied pow10.6

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

  12. Applied pow-prod-down0.6

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019104 +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)))