Average Error: 44.2 → 0.4
Time: 17.2s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\left(a \cdot 4\right) \cdot c}{2 \cdot \left(-\left(b \cdot a + a \cdot \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\right)\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\left(a \cdot 4\right) \cdot c}{2 \cdot \left(-\left(b \cdot a + a \cdot \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\right)\right)}
double f(double a, double b, double c) {
        double r54436 = b;
        double r54437 = -r54436;
        double r54438 = r54436 * r54436;
        double r54439 = 4.0;
        double r54440 = a;
        double r54441 = r54439 * r54440;
        double r54442 = c;
        double r54443 = r54441 * r54442;
        double r54444 = r54438 - r54443;
        double r54445 = sqrt(r54444);
        double r54446 = r54437 + r54445;
        double r54447 = 2.0;
        double r54448 = r54447 * r54440;
        double r54449 = r54446 / r54448;
        return r54449;
}
↓
double f(double a, double b, double c) {
        double r54450 = a;
        double r54451 = 4.0;
        double r54452 = r54450 * r54451;
        double r54453 = c;
        double r54454 = r54452 * r54453;
        double r54455 = 2.0;
        double r54456 = b;
        double r54457 = r54456 * r54450;
        double r54458 = r54456 * r54456;
        double r54459 = r54451 * r54453;
        double r54460 = r54459 * r54450;
        double r54461 = r54458 - r54460;
        double r54462 = sqrt(r54461);
        double r54463 = r54450 * r54462;
        double r54464 = r54457 + r54463;
        double r54465 = -r54464;
        double r54466 = r54455 * r54465;
        double r54467 = r54454 / r54466;
        return r54467;
}

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 44.2

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

  3. Applied flip-+44.2

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

  4. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{0 + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  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. Simplified0.4

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

  10. Applied sub-neg0.4

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

  11. Applied distribute-rgt-in0.4

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

  12. Simplified0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))