Average Error: 33.6 → 0.4
Time: 17.1s
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 r447177 = x;
        double r447178 = r447177 * r447177;
        double r447179 = y;
        double r447180 = r447179 * r447179;
        double r447181 = r447178 / r447180;
        double r447182 = z;
        double r447183 = r447182 * r447182;
        double r447184 = t;
        double r447185 = r447184 * r447184;
        double r447186 = r447183 / r447185;
        double r447187 = r447181 + r447186;
        return r447187;
}
↓
double f(double x, double y, double z, double t) {
        double r447188 = x;
        double r447189 = y;
        double r447190 = r447188 / r447189;
        double r447191 = z;
        double r447192 = t;
        double r447193 = r447191 / r447192;
        double r447194 = r447192 / r447191;
        double r447195 = r447193 / r447194;
        double r447196 = fma(r447190, r447190, r447195);
        return r447196;
}

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))))