Average Error: 28.4 → 0.5
Time: 18.9s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\left(4 \cdot a\right) \cdot c}{-\left(\left(a \cdot 2\right) \cdot \sqrt{\sqrt[3]{{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}^{3}}} + \left(a \cdot 2\right) \cdot b\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\left(4 \cdot a\right) \cdot c}{-\left(\left(a \cdot 2\right) \cdot \sqrt{\sqrt[3]{{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}^{3}}} + \left(a \cdot 2\right) \cdot b\right)}
double f(double a, double b, double c) {
        double r41563 = b;
        double r41564 = -r41563;
        double r41565 = r41563 * r41563;
        double r41566 = 4.0;
        double r41567 = a;
        double r41568 = r41566 * r41567;
        double r41569 = c;
        double r41570 = r41568 * r41569;
        double r41571 = r41565 - r41570;
        double r41572 = sqrt(r41571);
        double r41573 = r41564 + r41572;
        double r41574 = 2.0;
        double r41575 = r41574 * r41567;
        double r41576 = r41573 / r41575;
        return r41576;
}
↓
double f(double a, double b, double c) {
        double r41577 = 4.0;
        double r41578 = a;
        double r41579 = r41577 * r41578;
        double r41580 = c;
        double r41581 = r41579 * r41580;
        double r41582 = 2.0;
        double r41583 = r41578 * r41582;
        double r41584 = b;
        double r41585 = 2.0;
        double r41586 = pow(r41584, r41585);
        double r41587 = r41578 * r41580;
        double r41588 = r41577 * r41587;
        double r41589 = r41586 - r41588;
        double r41590 = 3.0;
        double r41591 = pow(r41589, r41590);
        double r41592 = cbrt(r41591);
        double r41593 = sqrt(r41592);
        double r41594 = r41583 * r41593;
        double r41595 = r41583 * r41584;
        double r41596 = r41594 + r41595;
        double r41597 = -r41596;
        double r41598 = r41581 / r41597;
        return r41598;
}

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

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

  3. Applied flip-+28.4

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

  4. Simplified0.5

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

  6. Applied div-inv0.5

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.5

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

  10. Applied sub-neg0.5

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

  11. Applied distribute-lft-in0.4

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

  13. Applied add-cbrt-cube0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019208 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))