Average Error: 28.7 → 0.4
Time: 7.1s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{-1 \cdot \left(a \cdot b\right) + a \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{-1 \cdot \left(a \cdot b\right) + a \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}
double f(double a, double b, double c) {
        double r37675 = b;
        double r37676 = -r37675;
        double r37677 = r37675 * r37675;
        double r37678 = 4.0;
        double r37679 = a;
        double r37680 = r37678 * r37679;
        double r37681 = c;
        double r37682 = r37680 * r37681;
        double r37683 = r37677 - r37682;
        double r37684 = sqrt(r37683);
        double r37685 = r37676 + r37684;
        double r37686 = 2.0;
        double r37687 = r37686 * r37679;
        double r37688 = r37685 / r37687;
        return r37688;
}
↓
double f(double a, double b, double c) {
        double r37689 = 1.0;
        double r37690 = 2.0;
        double r37691 = r37689 / r37690;
        double r37692 = 4.0;
        double r37693 = a;
        double r37694 = c;
        double r37695 = r37693 * r37694;
        double r37696 = r37692 * r37695;
        double r37697 = -1.0;
        double r37698 = b;
        double r37699 = r37693 * r37698;
        double r37700 = r37697 * r37699;
        double r37701 = r37698 * r37698;
        double r37702 = r37692 * r37693;
        double r37703 = r37702 * r37694;
        double r37704 = r37701 - r37703;
        double r37705 = sqrt(r37704);
        double r37706 = -r37705;
        double r37707 = r37693 * r37706;
        double r37708 = r37700 + r37707;
        double r37709 = r37696 / r37708;
        double r37710 = r37691 * r37709;
        return r37710;
}

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 28.7

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+28.7

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

  7. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

  8. Applied times-frac0.5

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  9. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}}\]

  10. Simplified0.5

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]

  11. Simplified0.5

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
  12. Using strategy rm

  13. Applied sub-neg0.5

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{a \cdot \color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right)}}\]

  14. Applied distribute-lft-in0.4

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{a \cdot \left(-b\right) + a \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

  15. Simplified0.4

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{-1 \cdot \left(a \cdot b\right)} + a \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

  16. Final simplification0.4

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{-1 \cdot \left(a \cdot b\right) + a \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))