Average Error: 33.6 → 1.2
Time: 10.1s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\left({\left(\sqrt{\sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}}\right)}^{3} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\left({\left(\sqrt{\sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}}\right)}^{3} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}
double f(double x, double y, double z, double t) {
        double r3552 = x;
        double r3553 = r3552 * r3552;
        double r3554 = y;
        double r3555 = r3554 * r3554;
        double r3556 = r3553 / r3555;
        double r3557 = z;
        double r3558 = r3557 * r3557;
        double r3559 = t;
        double r3560 = r3559 * r3559;
        double r3561 = r3558 / r3560;
        double r3562 = r3556 + r3561;
        return r3562;
}
↓
double f(double x, double y, double z, double t) {
        double r3563 = z;
        double r3564 = t;
        double r3565 = r3563 / r3564;
        double r3566 = x;
        double r3567 = y;
        double r3568 = r3566 / r3567;
        double r3569 = r3568 * r3568;
        double r3570 = fma(r3565, r3565, r3569);
        double r3571 = cbrt(r3570);
        double r3572 = sqrt(r3571);
        double r3573 = 3.0;
        double r3574 = pow(r3572, r3573);
        double r3575 = r3574 * r3572;
        double r3576 = r3575 * r3571;
        return r3576;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.6
Target0.4
Herbie1.2
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

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 times-frac0.4

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

  6. Applied add-cube-cbrt1.2

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

  8. Applied add-sqr-sqrt1.2

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

  9. Applied associate-*r*1.2

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

  10. Simplified1.2

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

  11. Final simplification1.2

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

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))))