Average Error: 28.5 → 0.6
Time: 16.1s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3 \cdot \left(a \cdot c\right)}{a \cdot \left(3 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{3 \cdot \left(a \cdot c\right)}{a \cdot \left(3 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}
double f(double a, double b, double c) {
        double r54373 = b;
        double r54374 = -r54373;
        double r54375 = r54373 * r54373;
        double r54376 = 3.0;
        double r54377 = a;
        double r54378 = r54376 * r54377;
        double r54379 = c;
        double r54380 = r54378 * r54379;
        double r54381 = r54375 - r54380;
        double r54382 = sqrt(r54381);
        double r54383 = r54374 + r54382;
        double r54384 = r54383 / r54378;
        return r54384;
}
↓
double f(double a, double b, double c) {
        double r54385 = 3.0;
        double r54386 = a;
        double r54387 = c;
        double r54388 = r54386 * r54387;
        double r54389 = r54385 * r54388;
        double r54390 = b;
        double r54391 = -r54390;
        double r54392 = r54390 * r54390;
        double r54393 = r54385 * r54386;
        double r54394 = r54393 * r54387;
        double r54395 = r54392 - r54394;
        double r54396 = sqrt(r54395);
        double r54397 = r54391 - r54396;
        double r54398 = r54385 * r54397;
        double r54399 = r54386 * r54398;
        double r54400 = r54389 / r54399;
        return r54400;
}

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 28.5

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

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

  6. Applied div-inv0.6

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

  7. Applied associate-/l*0.6

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

  8. Simplified0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019306 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))