Average Error: 28.4 → 0.3
Time: 8.8s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{-3 \cdot a}{-1}}{\frac{3 \cdot a}{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{-3 \cdot a}{-1}}{\frac{3 \cdot a}{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}
double f(double a, double b, double c) {
        double r212274 = b;
        double r212275 = -r212274;
        double r212276 = r212274 * r212274;
        double r212277 = 3.0;
        double r212278 = a;
        double r212279 = r212277 * r212278;
        double r212280 = c;
        double r212281 = r212279 * r212280;
        double r212282 = r212276 - r212281;
        double r212283 = sqrt(r212282);
        double r212284 = r212275 + r212283;
        double r212285 = r212284 / r212279;
        return r212285;
}
↓
double f(double a, double b, double c) {
        double r212286 = 3.0;
        double r212287 = a;
        double r212288 = r212286 * r212287;
        double r212289 = -r212288;
        double r212290 = -1.0;
        double r212291 = r212289 / r212290;
        double r212292 = c;
        double r212293 = b;
        double r212294 = -r212293;
        double r212295 = r212293 * r212293;
        double r212296 = r212288 * r212292;
        double r212297 = r212295 - r212296;
        double r212298 = sqrt(r212297);
        double r212299 = r212294 - r212298;
        double r212300 = r212292 / r212299;
        double r212301 = r212288 / r212300;
        double r212302 = r212291 / r212301;
        return r212302;
}

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 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-+28.5

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

  4. Simplified0.6

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

  6. Applied frac-2neg0.6

    \[\leadsto \frac{\color{blue}{\frac{-\left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right)}{-\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

  7. Simplified0.5

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

  9. Applied neg-mul-10.5

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

  10. Applied distribute-lft-neg-in0.5

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

  11. Applied times-frac0.3

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

  12. Applied associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020024 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :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)))