Average Error: 44.0 → 0.4
Time: 8.6s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4}}}{2} \cdot \frac{a \cdot c}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4}}}{2} \cdot \frac{a \cdot c}{a}
double f(double a, double b, double c) {
        double r44707 = b;
        double r44708 = -r44707;
        double r44709 = r44707 * r44707;
        double r44710 = 4.0;
        double r44711 = a;
        double r44712 = r44710 * r44711;
        double r44713 = c;
        double r44714 = r44712 * r44713;
        double r44715 = r44709 - r44714;
        double r44716 = sqrt(r44715);
        double r44717 = r44708 + r44716;
        double r44718 = 2.0;
        double r44719 = r44718 * r44711;
        double r44720 = r44717 / r44719;
        return r44720;
}
↓
double f(double a, double b, double c) {
        double r44721 = 1.0;
        double r44722 = b;
        double r44723 = -r44722;
        double r44724 = r44722 * r44722;
        double r44725 = 4.0;
        double r44726 = a;
        double r44727 = r44725 * r44726;
        double r44728 = c;
        double r44729 = r44727 * r44728;
        double r44730 = r44724 - r44729;
        double r44731 = 3.0;
        double r44732 = pow(r44730, r44731);
        double r44733 = cbrt(r44732);
        double r44734 = sqrt(r44733);
        double r44735 = r44723 - r44734;
        double r44736 = r44735 / r44725;
        double r44737 = r44721 / r44736;
        double r44738 = 2.0;
        double r44739 = r44737 / r44738;
        double r44740 = r44726 * r44728;
        double r44741 = r44740 / r44726;
        double r44742 = r44739 * r44741;
        return r44742;
}

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 44.0

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

  3. Applied flip-+44.0

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

  4. Simplified0.4

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

  6. Applied add-cbrt-cube0.4

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}}}}{2 \cdot a}\]

  7. Simplified0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{\color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}}{2 \cdot a}\]
  8. Using strategy rm

  9. Applied clear-num0.5

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{0 + 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

  10. Simplified0.5

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4}}{a \cdot c}}}}{2 \cdot a}\]
  11. Using strategy rm

  12. Applied div-inv0.6

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4} \cdot \frac{1}{a \cdot c}}}}{2 \cdot a}\]

  13. Applied add-cube-cbrt0.6

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4} \cdot \frac{1}{a \cdot c}}}{2 \cdot a}\]

  14. Applied times-frac0.6

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{a \cdot c}}}}{2 \cdot a}\]

  15. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4}}}{2} \cdot \frac{\frac{\sqrt[3]{1}}{\frac{1}{a \cdot c}}}{a}}\]

  16. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4}}}{2}} \cdot \frac{\frac{\sqrt[3]{1}}{\frac{1}{a \cdot c}}}{a}\]

  17. Simplified0.4

    \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4}}}{2} \cdot \color{blue}{\frac{a \cdot c}{a}}\]

  18. Final simplification0.4

    \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}{4}}}{2} \cdot \frac{a \cdot c}{a}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))