Average Error: 33.6 → 0.5
Time: 4.0s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \left({\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}}\right)\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \left({\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}}\right)\right)
double f(double x, double y, double z, double t) {
        double r742283 = x;
        double r742284 = r742283 * r742283;
        double r742285 = y;
        double r742286 = r742285 * r742285;
        double r742287 = r742284 / r742286;
        double r742288 = z;
        double r742289 = r742288 * r742288;
        double r742290 = t;
        double r742291 = r742290 * r742290;
        double r742292 = r742289 / r742291;
        double r742293 = r742287 + r742292;
        return r742293;
}
↓
double f(double x, double y, double z, double t) {
        double r742294 = z;
        double r742295 = t;
        double r742296 = r742294 / r742295;
        double r742297 = x;
        double r742298 = y;
        double r742299 = r742297 / r742298;
        double r742300 = fabs(r742299);
        double r742301 = sqrt(r742300);
        double r742302 = 0.75;
        double r742303 = pow(r742300, r742302);
        double r742304 = r742303 * r742303;
        double r742305 = r742301 * r742304;
        double r742306 = fma(r742296, r742296, r742305);
        return r742306;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.6
Target0.4
Herbie0.5
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

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. Simplified19.1

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

  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 add-sqr-sqrt0.5

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

  9. Applied associate-*l*0.5

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

  10. Simplified0.5

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \color{blue}{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}}\right)\]
  11. Using strategy rm

  12. Applied sqr-pow0.5

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \color{blue}{\left({\left(\left|\frac{x}{y}\right|\right)}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}\right)}\right)\]

  13. Simplified0.5

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \left(\color{blue}{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}\right)\right)\]

  14. Simplified0.5

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \left({\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}} \cdot \color{blue}{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}}}\right)\right)\]

  15. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \left({\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{4}}\right)\right)\]

Reproduce

herbie shell --seed 2020057 +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))))