Average Error: 52.5 → 6.2
Time: 18.1s
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-\frac{c}{b}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-\frac{c}{b}
double f(double a, double b, double c) {
        double r490463 = b;
        double r490464 = -r490463;
        double r490465 = r490463 * r490463;
        double r490466 = 4.0;
        double r490467 = a;
        double r490468 = r490466 * r490467;
        double r490469 = c;
        double r490470 = r490468 * r490469;
        double r490471 = r490465 - r490470;
        double r490472 = sqrt(r490471);
        double r490473 = r490464 + r490472;
        double r490474 = 2.0;
        double r490475 = r490474 * r490467;
        double r490476 = r490473 / r490475;
        return r490476;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r490477 = c;
        double r490478 = b;
        double r490479 = r490477 / r490478;
        double r490480 = -r490479;
        return r490480;
}

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 52.5

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

  2. Simplified52.5

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

  3. Taylor expanded around inf 6.2

    \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

  4. Simplified6.2

    \[\leadsto \color{blue}{-\frac{c}{b}}\]

  5. Final simplification6.2

    \[\leadsto -\frac{c}{b}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))