Average Error: 43.6 → 0.2
Time: 21.6s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2} \cdot \frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]
\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 c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}
double f(double a, double b, double c) {
        double r55485 = b;
        double r55486 = -r55485;
        double r55487 = r55485 * r55485;
        double r55488 = 4.0;
        double r55489 = a;
        double r55490 = r55488 * r55489;
        double r55491 = c;
        double r55492 = r55490 * r55491;
        double r55493 = r55487 - r55492;
        double r55494 = sqrt(r55493);
        double r55495 = r55486 + r55494;
        double r55496 = 2.0;
        double r55497 = r55496 * r55489;
        double r55498 = r55495 / r55497;
        return r55498;
}
↓
double f(double a, double b, double c) {
        double r55499 = 1.0;
        double r55500 = 2.0;
        double r55501 = r55499 / r55500;
        double r55502 = 4.0;
        double r55503 = c;
        double r55504 = r55502 * r55503;
        double r55505 = b;
        double r55506 = -r55505;
        double r55507 = r55505 * r55505;
        double r55508 = a;
        double r55509 = r55502 * r55508;
        double r55510 = r55509 * r55503;
        double r55511 = r55507 - r55510;
        double r55512 = sqrt(r55511);
        double r55513 = r55506 - r55512;
        double r55514 = r55504 / r55513;
        double r55515 = r55501 * r55514;
        return r55515;
}

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 43.6

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

  3. Applied flip-+43.6

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

  4. Simplified0.4

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

  6. Applied *-un-lft-identity0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied times-frac0.4

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

  9. Applied times-frac0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  13. Applied associate-/r*0.2

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

  14. Taylor expanded around 0 0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +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)))