Average Error: 28.7 → 0.5
Time: 17.3s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c \cdot \left(4 \cdot a\right)}{\left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right) \cdot 2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{c \cdot \left(4 \cdot a\right)}{\left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right) \cdot 2}
double f(double a, double b, double c) {
        double r43525 = b;
        double r43526 = -r43525;
        double r43527 = r43525 * r43525;
        double r43528 = 4.0;
        double r43529 = a;
        double r43530 = r43528 * r43529;
        double r43531 = c;
        double r43532 = r43530 * r43531;
        double r43533 = r43527 - r43532;
        double r43534 = sqrt(r43533);
        double r43535 = r43526 + r43534;
        double r43536 = 2.0;
        double r43537 = r43536 * r43529;
        double r43538 = r43535 / r43537;
        return r43538;
}
↓
double f(double a, double b, double c) {
        double r43539 = c;
        double r43540 = 4.0;
        double r43541 = a;
        double r43542 = r43540 * r43541;
        double r43543 = r43539 * r43542;
        double r43544 = b;
        double r43545 = -r43544;
        double r43546 = r43544 * r43544;
        double r43547 = r43542 * r43539;
        double r43548 = r43546 - r43547;
        double r43549 = sqrt(r43548);
        double r43550 = r43545 - r43549;
        double r43551 = r43541 * r43550;
        double r43552 = 2.0;
        double r43553 = r43551 * r43552;
        double r43554 = r43543 / r43553;
        return r43554;
}

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 28.7

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

  3. Applied flip-+28.7

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

  4. Simplified0.5

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

  6. Applied div-inv0.5

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))