Average Error: 34.1 → 0.6
Time: 3.9s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\frac{z}{t} \cdot \frac{z}{t} + {\left(\sqrt{\left|\frac{x}{y}\right|}\right)}^{3} \cdot \sqrt{\left|\frac{x}{y}\right|}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\frac{z}{t} \cdot \frac{z}{t} + {\left(\sqrt{\left|\frac{x}{y}\right|}\right)}^{3} \cdot \sqrt{\left|\frac{x}{y}\right|}
double f(double x, double y, double z, double t) {
        double r655935 = x;
        double r655936 = r655935 * r655935;
        double r655937 = y;
        double r655938 = r655937 * r655937;
        double r655939 = r655936 / r655938;
        double r655940 = z;
        double r655941 = r655940 * r655940;
        double r655942 = t;
        double r655943 = r655942 * r655942;
        double r655944 = r655941 / r655943;
        double r655945 = r655939 + r655944;
        return r655945;
}
↓
double f(double x, double y, double z, double t) {
        double r655946 = z;
        double r655947 = t;
        double r655948 = r655946 / r655947;
        double r655949 = r655948 * r655948;
        double r655950 = x;
        double r655951 = y;
        double r655952 = r655950 / r655951;
        double r655953 = fabs(r655952);
        double r655954 = sqrt(r655953);
        double r655955 = 3.0;
        double r655956 = pow(r655954, r655955);
        double r655957 = r655956 * r655954;
        double r655958 = r655949 + r655957;
        return r655958;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.1
Target0.4
Herbie0.6
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 34.1

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

  2. Simplified19.4

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

  4. Applied add-sqr-sqrt19.4

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

  5. Simplified19.4

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

  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. Using strategy rm

  8. Applied fma-udef0.4

    \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t} + \left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right|}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.5

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

  11. Applied associate-*r*0.5

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

  12. Simplified0.6

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

  13. Final simplification0.6

    \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + {\left(\sqrt{\left|\frac{x}{y}\right|}\right)}^{3} \cdot \sqrt{\left|\frac{x}{y}\right|}\]

Reproduce

herbie shell --seed 2020027 +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))))