Average Error: 2.8 → 0.1
Time: 8.6s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[x + \frac{1}{1.12837916709551256 \cdot \frac{\sqrt{e^{z}}}{\frac{y}{\sqrt{e^{z}}}} - x}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
x + \frac{1}{1.12837916709551256 \cdot \frac{\sqrt{e^{z}}}{\frac{y}{\sqrt{e^{z}}}} - x}
double f(double x, double y, double z) {
        double r455165 = x;
        double r455166 = y;
        double r455167 = 1.1283791670955126;
        double r455168 = z;
        double r455169 = exp(r455168);
        double r455170 = r455167 * r455169;
        double r455171 = r455165 * r455166;
        double r455172 = r455170 - r455171;
        double r455173 = r455166 / r455172;
        double r455174 = r455165 + r455173;
        return r455174;
}
↓
double f(double x, double y, double z) {
        double r455175 = x;
        double r455176 = 1.0;
        double r455177 = 1.1283791670955126;
        double r455178 = z;
        double r455179 = exp(r455178);
        double r455180 = sqrt(r455179);
        double r455181 = y;
        double r455182 = r455181 / r455180;
        double r455183 = r455180 / r455182;
        double r455184 = r455177 * r455183;
        double r455185 = r455184 - r455175;
        double r455186 = r455176 / r455185;
        double r455187 = r455175 + r455186;
        return r455187;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original2.8
Target0.0
Herbie0.1
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Initial program 2.8

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

  3. Applied clear-num2.8

    \[\leadsto x + \color{blue}{\frac{1}{\frac{1.12837916709551256 \cdot e^{z} - x \cdot y}{y}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity2.8

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

  6. Applied *-un-lft-identity2.8

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

  7. Applied times-frac2.8

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

  8. Simplified2.8

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

  9. Simplified0.1

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

  11. Applied add-sqr-sqrt0.1

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

  12. Applied associate-/l*0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 
(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)))))