Average Error: 33.5 → 0.5
Time: 4.9s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}} \cdot \sqrt{\left|\frac{x}{y}\right|}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}} \cdot \sqrt{\left|\frac{x}{y}\right|}\right)
double f(double x, double y, double z, double t) {
        double r556543 = x;
        double r556544 = r556543 * r556543;
        double r556545 = y;
        double r556546 = r556545 * r556545;
        double r556547 = r556544 / r556546;
        double r556548 = z;
        double r556549 = r556548 * r556548;
        double r556550 = t;
        double r556551 = r556550 * r556550;
        double r556552 = r556549 / r556551;
        double r556553 = r556547 + r556552;
        return r556553;
}
↓
double f(double x, double y, double z, double t) {
        double r556554 = z;
        double r556555 = t;
        double r556556 = r556554 / r556555;
        double r556557 = x;
        double r556558 = y;
        double r556559 = r556557 / r556558;
        double r556560 = fabs(r556559);
        double r556561 = 1.5;
        double r556562 = pow(r556560, r556561);
        double r556563 = sqrt(r556560);
        double r556564 = r556562 * r556563;
        double r556565 = fma(r556556, r556556, r556564);
        return r556565;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

Derivation

  1. Initial program 33.5

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

  2. Simplified18.7

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

  4. Applied add-sqr-sqrt18.7

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

  5. Simplified18.7

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

  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

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

  9. Applied associate-*r*0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020035 +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))))