Average Error: 52.1 → 0.4
Time: 11.9s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(a \cdot 4\right) \cdot \frac{\frac{c}{2}}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(a \cdot 4\right) \cdot \frac{\frac{c}{2}}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}
double f(double a, double b, double c) {
        double r38575 = b;
        double r38576 = -r38575;
        double r38577 = r38575 * r38575;
        double r38578 = 4.0;
        double r38579 = a;
        double r38580 = r38578 * r38579;
        double r38581 = c;
        double r38582 = r38580 * r38581;
        double r38583 = r38577 - r38582;
        double r38584 = sqrt(r38583);
        double r38585 = r38576 + r38584;
        double r38586 = 2.0;
        double r38587 = r38586 * r38579;
        double r38588 = r38585 / r38587;
        return r38588;
}
↓
double f(double a, double b, double c) {
        double r38589 = a;
        double r38590 = 4.0;
        double r38591 = r38589 * r38590;
        double r38592 = c;
        double r38593 = 2.0;
        double r38594 = r38592 / r38593;
        double r38595 = b;
        double r38596 = -r38595;
        double r38597 = r38595 * r38595;
        double r38598 = r38590 * r38589;
        double r38599 = r38598 * r38592;
        double r38600 = r38597 - r38599;
        double r38601 = sqrt(r38600);
        double r38602 = r38596 - r38601;
        double r38603 = r38589 * r38602;
        double r38604 = r38594 / r38603;
        double r38605 = r38591 * r38604;
        return r38605;
}

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 52.1

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

  3. Applied flip-+52.1

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

  4. Simplified0.4

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

  6. Applied div-inv0.5

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

  7. Applied times-frac0.5

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

  8. Simplified0.5

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

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

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

  11. Applied times-frac0.5

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

  12. Applied associate-*l*0.5

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))