Average Error: 0.1 → 0.1
Time: 2.5s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\left(3 \cdot \left|z\right|\right) \cdot \sqrt{{z}^{2}} + x \cdot y\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\left(3 \cdot \left|z\right|\right) \cdot \sqrt{{z}^{2}} + x \cdot y
double f(double x, double y, double z) {
        double r520856 = x;
        double r520857 = y;
        double r520858 = r520856 * r520857;
        double r520859 = z;
        double r520860 = r520859 * r520859;
        double r520861 = r520858 + r520860;
        double r520862 = r520861 + r520860;
        double r520863 = r520862 + r520860;
        return r520863;
}
↓
double f(double x, double y, double z) {
        double r520864 = 3.0;
        double r520865 = z;
        double r520866 = fabs(r520865);
        double r520867 = r520864 * r520866;
        double r520868 = 2.0;
        double r520869 = pow(r520865, r520868);
        double r520870 = sqrt(r520869);
        double r520871 = r520867 * r520870;
        double r520872 = x;
        double r520873 = y;
        double r520874 = r520872 * r520873;
        double r520875 = r520871 + r520874;
        return r520875;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

Enter valid numbers for all inputs

Target

Original0.1
Target0.1
Herbie0.1
\[\left(3 \cdot z\right) \cdot z + y \cdot x\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{3 \cdot {z}^{2} + x \cdot y}\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, {z}^{2}, x \cdot y\right)}\]
  4. Using strategy rm

  5. Applied fma-udef0.1

    \[\leadsto \color{blue}{3 \cdot {z}^{2} + x \cdot y}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

    \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{{z}^{2}} \cdot \sqrt{{z}^{2}}\right)} + x \cdot y\]

  8. Applied associate-*r*0.1

    \[\leadsto \color{blue}{\left(3 \cdot \sqrt{{z}^{2}}\right) \cdot \sqrt{{z}^{2}}} + x \cdot y\]

  9. Simplified0.1

    \[\leadsto \color{blue}{\left(3 \cdot \left|z\right|\right)} \cdot \sqrt{{z}^{2}} + x \cdot y\]

  10. Final simplification0.1

    \[\leadsto \left(3 \cdot \left|z\right|\right) \cdot \sqrt{{z}^{2}} + x \cdot y\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))