Average Error: 52.7 → 0.5
Time: 20.8s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{4 \cdot \left(a \cdot c\right)}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{4 \cdot \left(a \cdot c\right)}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}
double f(double a, double b, double c) {
        double r32340 = b;
        double r32341 = -r32340;
        double r32342 = r32340 * r32340;
        double r32343 = 4.0;
        double r32344 = a;
        double r32345 = r32343 * r32344;
        double r32346 = c;
        double r32347 = r32345 * r32346;
        double r32348 = r32342 - r32347;
        double r32349 = sqrt(r32348);
        double r32350 = r32341 + r32349;
        double r32351 = 2.0;
        double r32352 = r32351 * r32344;
        double r32353 = r32350 / r32352;
        return r32353;
}
↓
double f(double a, double b, double c) {
        double r32354 = 4.0;
        double r32355 = a;
        double r32356 = c;
        double r32357 = r32355 * r32356;
        double r32358 = r32354 * r32357;
        double r32359 = 2.0;
        double r32360 = r32358 / r32359;
        double r32361 = 1.0;
        double r32362 = b;
        double r32363 = -r32362;
        double r32364 = r32362 * r32362;
        double r32365 = r32354 * r32355;
        double r32366 = r32365 * r32356;
        double r32367 = r32364 - r32366;
        double r32368 = sqrt(r32367);
        double r32369 = r32363 - r32368;
        double r32370 = r32361 / r32369;
        double r32371 = r32370 / r32355;
        double r32372 = r32360 * r32371;
        return r32372;
}

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 52.7

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

  3. Applied flip-+52.7

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

  4. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{0 + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  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 times-frac0.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))