Average Error: 0.1 → 0.1
Time: 2.6s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\left(\left(\sqrt[3]{3} \cdot {\left(\sqrt{\sqrt[3]{3}}\right)}^{3}\right) \cdot z\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot z\right) + x \cdot y\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\left(\left(\sqrt[3]{3} \cdot {\left(\sqrt{\sqrt[3]{3}}\right)}^{3}\right) \cdot z\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot z\right) + x \cdot y
double f(double x, double y, double z) {
        double r476181 = x;
        double r476182 = y;
        double r476183 = r476181 * r476182;
        double r476184 = z;
        double r476185 = r476184 * r476184;
        double r476186 = r476183 + r476185;
        double r476187 = r476186 + r476185;
        double r476188 = r476187 + r476185;
        return r476188;
}
↓
double f(double x, double y, double z) {
        double r476189 = 3.0;
        double r476190 = cbrt(r476189);
        double r476191 = sqrt(r476190);
        double r476192 = pow(r476191, r476189);
        double r476193 = r476190 * r476192;
        double r476194 = z;
        double r476195 = r476193 * r476194;
        double r476196 = r476191 * r476194;
        double r476197 = r476195 * r476196;
        double r476198 = x;
        double r476199 = y;
        double r476200 = r476198 * r476199;
        double r476201 = r476197 + r476200;
        return r476201;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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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. Simplified0.1

    \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right) + x \cdot y}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.1

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

  5. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \left(z \cdot z\right)\right)} + x \cdot y\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.2

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

  8. Applied unswap-sqr0.2

    \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \color{blue}{\left(\left(\sqrt{\sqrt[3]{3}} \cdot z\right) \cdot \left(\sqrt{\sqrt[3]{3}} \cdot z\right)\right)} + x \cdot y\]
  9. Using strategy rm

  10. Applied associate-*r*0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 
(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)))