Average Error: 44.1 → 0.5
Time: 21.2s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{3 \cdot a}\]
\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 c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{3 \cdot a}
double f(double a, double b, double c) {
        double r72306 = b;
        double r72307 = -r72306;
        double r72308 = r72306 * r72306;
        double r72309 = 3.0;
        double r72310 = a;
        double r72311 = r72309 * r72310;
        double r72312 = c;
        double r72313 = r72311 * r72312;
        double r72314 = r72308 - r72313;
        double r72315 = sqrt(r72314);
        double r72316 = r72307 + r72315;
        double r72317 = r72316 / r72311;
        return r72317;
}
↓
double f(double a, double b, double c) {
        double r72318 = 3.0;
        double r72319 = a;
        double r72320 = r72318 * r72319;
        double r72321 = c;
        double r72322 = r72320 * r72321;
        double r72323 = b;
        double r72324 = -r72323;
        double r72325 = r72323 * r72323;
        double r72326 = r72319 * r72321;
        double r72327 = r72318 * r72326;
        double r72328 = r72325 - r72327;
        double r72329 = sqrt(r72328);
        double r72330 = r72324 - r72329;
        double r72331 = r72322 / r72330;
        double r72332 = 1.0;
        double r72333 = r72332 / r72320;
        double r72334 = r72331 * r72333;
        return r72334;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 44.1

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

  3. Applied flip-+44.1

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

  4. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{0 + \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
  5. Using strategy rm

  6. Applied pow10.4

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot \color{blue}{{c}^{1}}}}}{3 \cdot a}\]

  7. Applied pow10.4

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot \color{blue}{{a}^{1}}\right) \cdot {c}^{1}}}}{3 \cdot a}\]

  8. Applied pow10.4

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(\color{blue}{{3}^{1}} \cdot {a}^{1}\right) \cdot {c}^{1}}}}{3 \cdot a}\]

  9. Applied pow-prod-down0.4

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{{\left(3 \cdot a\right)}^{1}} \cdot {c}^{1}}}}{3 \cdot a}\]

  10. Applied pow-prod-down0.4

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{{\left(\left(3 \cdot a\right) \cdot c\right)}^{1}}}}}{3 \cdot a}\]

  11. Simplified0.4

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{1}}}}{3 \cdot a}\]
  12. Using strategy rm

  13. Applied div-inv0.5

    \[\leadsto \color{blue}{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - {\left(3 \cdot \left(a \cdot c\right)\right)}^{1}}} \cdot \frac{1}{3 \cdot a}}\]

  14. Final simplification0.5

    \[\leadsto \frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{3 \cdot a}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))