Average Error: 33.8 → 0.4
Time: 3.9s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right|\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right|\right)
double f(double x, double y, double z, double t) {
        double r640832 = x;
        double r640833 = r640832 * r640832;
        double r640834 = y;
        double r640835 = r640834 * r640834;
        double r640836 = r640833 / r640835;
        double r640837 = z;
        double r640838 = r640837 * r640837;
        double r640839 = t;
        double r640840 = r640839 * r640839;
        double r640841 = r640838 / r640840;
        double r640842 = r640836 + r640841;
        return r640842;
}
↓
double f(double x, double y, double z, double t) {
        double r640843 = z;
        double r640844 = t;
        double r640845 = r640843 / r640844;
        double r640846 = x;
        double r640847 = y;
        double r640848 = r640846 / r640847;
        double r640849 = fabs(r640848);
        double r640850 = r640849 * r640849;
        double r640851 = fma(r640845, r640845, r640850);
        return r640851;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.8
Target0.4
Herbie0.4
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.8

    \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

  2. Simplified19.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt19.0

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\sqrt{\frac{x \cdot x}{y \cdot y}} \cdot \sqrt{\frac{x \cdot x}{y \cdot y}}}\right)\]

  5. Simplified19.0

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

  6. Simplified0.4

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

  7. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right|\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :herbie-target
  (+ (pow (/ x y) 2) (pow (/ z t) 2))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))