Average Error: 33.0 → 0.4
Time: 10.9s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left|\frac{z}{t}\right| \cdot \left|\frac{z}{t}\right|\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left|\frac{z}{t}\right| \cdot \left|\frac{z}{t}\right|\right)
double f(double x, double y, double z, double t) {
        double r504744 = x;
        double r504745 = r504744 * r504744;
        double r504746 = y;
        double r504747 = r504746 * r504746;
        double r504748 = r504745 / r504747;
        double r504749 = z;
        double r504750 = r504749 * r504749;
        double r504751 = t;
        double r504752 = r504751 * r504751;
        double r504753 = r504750 / r504752;
        double r504754 = r504748 + r504753;
        return r504754;
}
↓
double f(double x, double y, double z, double t) {
        double r504755 = x;
        double r504756 = y;
        double r504757 = r504755 / r504756;
        double r504758 = z;
        double r504759 = t;
        double r504760 = r504758 / r504759;
        double r504761 = fabs(r504760);
        double r504762 = r504761 * r504761;
        double r504763 = fma(r504757, r504757, r504762);
        return r504763;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.0
Target0.4
Herbie0.4
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.0

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

  2. Simplified18.9

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

  4. Applied add-sqr-sqrt18.9

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

  5. Simplified18.9

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 +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))))