Average Error: 33.8 → 0.6
Time: 35.8s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\sqrt{\sqrt{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right)}} \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right)}} \cdot \sqrt{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right)}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\sqrt{\sqrt{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right)}} \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right)}} \cdot \sqrt{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right)}\right)
double f(double x, double y, double z, double t) {
        double r28994019 = x;
        double r28994020 = r28994019 * r28994019;
        double r28994021 = y;
        double r28994022 = r28994021 * r28994021;
        double r28994023 = r28994020 / r28994022;
        double r28994024 = z;
        double r28994025 = r28994024 * r28994024;
        double r28994026 = t;
        double r28994027 = r28994026 * r28994026;
        double r28994028 = r28994025 / r28994027;
        double r28994029 = r28994023 + r28994028;
        return r28994029;
}
↓
double f(double x, double y, double z, double t) {
        double r28994030 = x;
        double r28994031 = y;
        double r28994032 = r28994030 / r28994031;
        double r28994033 = z;
        double r28994034 = t;
        double r28994035 = r28994033 / r28994034;
        double r28994036 = r28994035 * r28994035;
        double r28994037 = fma(r28994032, r28994032, r28994036);
        double r28994038 = sqrt(r28994037);
        double r28994039 = sqrt(r28994038);
        double r28994040 = r28994039 * r28994038;
        double r28994041 = r28994039 * r28994040;
        return r28994041;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.8
Target0.4
Herbie0.6
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.8

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

  2. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied sqrt-prod0.6

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

  8. Applied associate-*r*0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"

  :herbie-target
  (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))