Average Error: 1.6 → 1.6
Time: 16.1s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\mathsf{fma}\left(z, -\frac{x}{y}, \frac{4}{y} + \frac{x}{y}\right) + \left(0 \cdot \frac{x}{y}\right) \cdot z\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\mathsf{fma}\left(z, -\frac{x}{y}, \frac{4}{y} + \frac{x}{y}\right) + \left(0 \cdot \frac{x}{y}\right) \cdot z\right|
double f(double x, double y, double z) {
        double r33994 = x;
        double r33995 = 4.0;
        double r33996 = r33994 + r33995;
        double r33997 = y;
        double r33998 = r33996 / r33997;
        double r33999 = r33994 / r33997;
        double r34000 = z;
        double r34001 = r33999 * r34000;
        double r34002 = r33998 - r34001;
        double r34003 = fabs(r34002);
        return r34003;
}
↓
double f(double x, double y, double z) {
        double r34004 = z;
        double r34005 = x;
        double r34006 = y;
        double r34007 = r34005 / r34006;
        double r34008 = -r34007;
        double r34009 = 4.0;
        double r34010 = r34009 / r34006;
        double r34011 = r34010 + r34007;
        double r34012 = fma(r34004, r34008, r34011);
        double r34013 = 0.0;
        double r34014 = r34013 * r34007;
        double r34015 = r34014 * r34004;
        double r34016 = r34012 + r34015;
        double r34017 = fabs(r34016);
        return r34017;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.6

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
  2. Using strategy rm

  3. Applied add-cube-cbrt2.6

    \[\leadsto \left|\color{blue}{\left(\sqrt[3]{\frac{x + 4}{y}} \cdot \sqrt[3]{\frac{x + 4}{y}}\right) \cdot \sqrt[3]{\frac{x + 4}{y}}} - \frac{x}{y} \cdot z\right|\]

  4. Applied prod-diff2.6

    \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{x + 4}{y}} \cdot \sqrt[3]{\frac{x + 4}{y}}, \sqrt[3]{\frac{x + 4}{y}}, -z \cdot \frac{x}{y}\right) + \mathsf{fma}\left(-z, \frac{x}{y}, z \cdot \frac{x}{y}\right)}\right|\]

  5. Simplified1.6

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

  6. Simplified1.6

    \[\leadsto \left|\mathsf{fma}\left(z, -\frac{x}{y}, \frac{x + 4}{y}\right) + \color{blue}{\left(0 \cdot \frac{x}{y}\right) \cdot z}\right|\]

  7. Taylor expanded around 0 1.6

    \[\leadsto \left|\mathsf{fma}\left(z, -\frac{x}{y}, \color{blue}{4 \cdot \frac{1}{y} + \frac{x}{y}}\right) + \left(0 \cdot \frac{x}{y}\right) \cdot z\right|\]

  8. Simplified1.6

    \[\leadsto \left|\mathsf{fma}\left(z, -\frac{x}{y}, \color{blue}{\frac{4}{y} + \frac{x}{y}}\right) + \left(0 \cdot \frac{x}{y}\right) \cdot z\right|\]

  9. Final simplification1.6

    \[\leadsto \left|\mathsf{fma}\left(z, -\frac{x}{y}, \frac{4}{y} + \frac{x}{y}\right) + \left(0 \cdot \frac{x}{y}\right) \cdot z\right|\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))