Average Error: 32.3 → 0.6
Time: 21.1s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)} \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)} \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}}\right)
double f(double x, double y, double z, double t) {
        double r33391006 = x;
        double r33391007 = r33391006 * r33391006;
        double r33391008 = y;
        double r33391009 = r33391008 * r33391008;
        double r33391010 = r33391007 / r33391009;
        double r33391011 = z;
        double r33391012 = r33391011 * r33391011;
        double r33391013 = t;
        double r33391014 = r33391013 * r33391013;
        double r33391015 = r33391012 / r33391014;
        double r33391016 = r33391010 + r33391015;
        return r33391016;
}
↓
double f(double x, double y, double z, double t) {
        double r33391017 = z;
        double r33391018 = t;
        double r33391019 = r33391017 / r33391018;
        double r33391020 = x;
        double r33391021 = y;
        double r33391022 = r33391020 / r33391021;
        double r33391023 = r33391022 * r33391022;
        double r33391024 = fma(r33391019, r33391019, r33391023);
        double r33391025 = sqrt(r33391024);
        double r33391026 = sqrt(r33391025);
        double r33391027 = r33391026 * r33391026;
        double r33391028 = r33391025 * r33391027;
        return r33391028;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original32.3
Target0.4
Herbie0.6
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 32.3

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

  2. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied sqrt-prod0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"

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

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