Average Error: 33.6 → 0.4
Time: 20.1s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)
double f(double x, double y, double z, double t) {
        double r509731 = x;
        double r509732 = r509731 * r509731;
        double r509733 = y;
        double r509734 = r509733 * r509733;
        double r509735 = r509732 / r509734;
        double r509736 = z;
        double r509737 = r509736 * r509736;
        double r509738 = t;
        double r509739 = r509738 * r509738;
        double r509740 = r509737 / r509739;
        double r509741 = r509735 + r509740;
        return r509741;
}
↓
double f(double x, double y, double z, double t) {
        double r509742 = x;
        double r509743 = y;
        double r509744 = r509742 / r509743;
        double r509745 = z;
        double r509746 = t;
        double r509747 = r509745 / r509746;
        double r509748 = r509746 / r509745;
        double r509749 = r509747 / r509748;
        double r509750 = fma(r509744, r509744, r509749);
        return r509750;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

Derivation

  1. Initial program 33.6

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

  2. Simplified18.7

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

  4. Applied associate-/l*12.9

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

  5. Simplified3.9

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

  7. Applied associate-/r*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 +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))))