Average Error: 0.1 → 0.1
Time: 16.0s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, x, -z \cdot \left(y \cdot 4\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(x, x, -z \cdot \left(y \cdot 4\right)\right)
double f(double x, double y, double z) {
        double r94493 = x;
        double r94494 = r94493 * r94493;
        double r94495 = y;
        double r94496 = 4.0;
        double r94497 = r94495 * r94496;
        double r94498 = z;
        double r94499 = r94497 * r94498;
        double r94500 = r94494 - r94499;
        return r94500;
}
↓
double f(double x, double y, double z) {
        double r94501 = x;
        double r94502 = z;
        double r94503 = y;
        double r94504 = 4.0;
        double r94505 = r94503 * r94504;
        double r94506 = r94502 * r94505;
        double r94507 = -r94506;
        double r94508 = fma(r94501, r94501, r94507);
        return r94508;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[x \cdot x - \left(y \cdot 4\right) \cdot z\]
  2. Using strategy rm

  3. Applied fma-neg0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot z\right)}\]

  4. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{-z \cdot \left(y \cdot 4\right)}\right)\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, x, -z \cdot \left(y \cdot 4\right)\right)\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  (- (* x x) (* (* y 4.0) z)))