Average Error: 33.6 → 0.4
Time: 3.4s
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 r698813 = x;
        double r698814 = r698813 * r698813;
        double r698815 = y;
        double r698816 = r698815 * r698815;
        double r698817 = r698814 / r698816;
        double r698818 = z;
        double r698819 = r698818 * r698818;
        double r698820 = t;
        double r698821 = r698820 * r698820;
        double r698822 = r698819 / r698821;
        double r698823 = r698817 + r698822;
        return r698823;
}
↓
double f(double x, double y, double z, double t) {
        double r698824 = z;
        double r698825 = t;
        double r698826 = r698824 / r698825;
        double r698827 = x;
        double r698828 = y;
        double r698829 = r698827 / r698828;
        double r698830 = fabs(r698829);
        double r698831 = r698830 * r698830;
        double r698832 = fma(r698826, r698826, r698831);
        return r698832;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.6
Target0.4
Herbie0.4
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.6

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

  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-sqrt19.1

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

  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 2020025 +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))))