Average Error: 0.1 → 0.2
Time: 21.3s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(y + x\right) - \left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y} \cdot {y}^{\frac{1}{3}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right) + z\right)\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(y + x\right) - \left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y} \cdot {y}^{\frac{1}{3}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right) + z\right)\right)
double f(double x, double y, double z) {
        double r20957878 = x;
        double r20957879 = y;
        double r20957880 = 0.5;
        double r20957881 = r20957879 + r20957880;
        double r20957882 = log(r20957879);
        double r20957883 = r20957881 * r20957882;
        double r20957884 = r20957878 - r20957883;
        double r20957885 = r20957884 + r20957879;
        double r20957886 = z;
        double r20957887 = r20957885 - r20957886;
        return r20957887;
}
↓
double f(double x, double y, double z) {
        double r20957888 = y;
        double r20957889 = x;
        double r20957890 = r20957888 + r20957889;
        double r20957891 = 0.5;
        double r20957892 = r20957888 + r20957891;
        double r20957893 = cbrt(r20957888);
        double r20957894 = 0.3333333333333333;
        double r20957895 = pow(r20957888, r20957894);
        double r20957896 = r20957893 * r20957895;
        double r20957897 = log(r20957896);
        double r20957898 = r20957892 * r20957897;
        double r20957899 = log(r20957893);
        double r20957900 = r20957899 * r20957892;
        double r20957901 = z;
        double r20957902 = r20957900 + r20957901;
        double r20957903 = r20957898 + r20957902;
        double r20957904 = r20957890 - r20957903;
        return r20957904;
}

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
Herbie0.2
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(\left(x + y\right) - \left(y \cdot \log y + 0.5 \cdot \log y\right)\right)} - z\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\left(\left(x + y\right) - \left(y + 0.5\right) \cdot \log y\right)} - z\]
  4. Using strategy rm

  5. Applied associate--l-0.1

    \[\leadsto \color{blue}{\left(x + y\right) - \left(\left(y + 0.5\right) \cdot \log y + z\right)}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.1

    \[\leadsto \left(x + y\right) - \left(\left(y + 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z\right)\]

  8. Applied log-prod0.2

    \[\leadsto \left(x + y\right) - \left(\left(y + 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z\right)\]

  9. Applied distribute-lft-in0.2

    \[\leadsto \left(x + y\right) - \left(\color{blue}{\left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)} + z\right)\]

  10. Applied associate-+l+0.2

    \[\leadsto \left(x + y\right) - \color{blue}{\left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right) + z\right)\right)}\]
  11. Using strategy rm

  12. Applied pow1/30.2

    \[\leadsto \left(x + y\right) - \left(\left(y + 0.5\right) \cdot \log \left(\color{blue}{{y}^{\frac{1}{3}}} \cdot \sqrt[3]{y}\right) + \left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right) + z\right)\right)\]

  13. Final simplification0.2

    \[\leadsto \left(y + x\right) - \left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y} \cdot {y}^{\frac{1}{3}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right) + z\right)\right)\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))