Average Error: 1.8 → 1.8
Time: 8.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|
double f(double x, double y, double z) {
        double r26768 = x;
        double r26769 = 4.0;
        double r26770 = r26768 + r26769;
        double r26771 = y;
        double r26772 = r26770 / r26771;
        double r26773 = r26768 / r26771;
        double r26774 = z;
        double r26775 = r26773 * r26774;
        double r26776 = r26772 - r26775;
        double r26777 = fabs(r26776);
        return r26777;
}
↓
double f(double x, double y, double z) {
        double r26778 = x;
        double r26779 = y;
        double r26780 = r26778 / r26779;
        double r26781 = 1.0;
        double r26782 = z;
        double r26783 = r26781 - r26782;
        double r26784 = 4.0;
        double r26785 = r26784 / r26779;
        double r26786 = fma(r26780, r26783, r26785);
        double r26787 = fabs(r26786);
        return r26787;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.8

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

  2. Taylor expanded around 0 3.2

    \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right|\]

  3. Simplified1.8

    \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)}\right|\]

  4. Final simplification1.8

    \[\leadsto \left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))