Average Error: 2.7 → 0.2
Time: 2.9s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[x + \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{\frac{e^{z}}{y}} \cdot \sqrt[3]{\frac{e^{z}}{y}}\right) \cdot \sqrt[3]{\frac{e^{z}}{y}}, 1.12837916709551256, -x\right)}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
x + \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{\frac{e^{z}}{y}} \cdot \sqrt[3]{\frac{e^{z}}{y}}\right) \cdot \sqrt[3]{\frac{e^{z}}{y}}, 1.12837916709551256, -x\right)}
double f(double x, double y, double z) {
        double r381878 = x;
        double r381879 = y;
        double r381880 = 1.1283791670955126;
        double r381881 = z;
        double r381882 = exp(r381881);
        double r381883 = r381880 * r381882;
        double r381884 = r381878 * r381879;
        double r381885 = r381883 - r381884;
        double r381886 = r381879 / r381885;
        double r381887 = r381878 + r381886;
        return r381887;
}
↓
double f(double x, double y, double z) {
        double r381888 = x;
        double r381889 = 1.0;
        double r381890 = z;
        double r381891 = exp(r381890);
        double r381892 = y;
        double r381893 = r381891 / r381892;
        double r381894 = cbrt(r381893);
        double r381895 = r381894 * r381894;
        double r381896 = r381895 * r381894;
        double r381897 = 1.1283791670955126;
        double r381898 = -r381888;
        double r381899 = fma(r381896, r381897, r381898);
        double r381900 = r381889 / r381899;
        double r381901 = r381888 + r381900;
        return r381901;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original2.7
Target0.0
Herbie0.2
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Initial program 2.7

    \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
  2. Using strategy rm

  3. Applied clear-num2.7

    \[\leadsto x + \color{blue}{\frac{1}{\frac{1.12837916709551256 \cdot e^{z} - x \cdot y}{y}}}\]

  4. Simplified0.0

    \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{z}}{y}, 1.12837916709551256, -x\right)}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.2

    \[\leadsto x + \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\frac{e^{z}}{y}} \cdot \sqrt[3]{\frac{e^{z}}{y}}\right) \cdot \sqrt[3]{\frac{e^{z}}{y}}}, 1.12837916709551256, -x\right)}\]

  7. Final simplification0.2

    \[\leadsto x + \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{\frac{e^{z}}{y}} \cdot \sqrt[3]{\frac{e^{z}}{y}}\right) \cdot \sqrt[3]{\frac{e^{z}}{y}}, 1.12837916709551256, -x\right)}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))