Average Error: 2.8 → 2.5
Time: 11.2s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(1.128379167095512558560699289955664426088 \cdot z + \left(0.5641895835477562792803496449778322130442 \cdot {z}^{2} + 1.128379167095512558560699289955664426088\right)\right) - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.0:\\
\;\;\;\;x - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\left(1.128379167095512558560699289955664426088 \cdot z + \left(0.5641895835477562792803496449778322130442 \cdot {z}^{2} + 1.128379167095512558560699289955664426088\right)\right) - x \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r394766 = x;
        double r394767 = y;
        double r394768 = 1.1283791670955126;
        double r394769 = z;
        double r394770 = exp(r394769);
        double r394771 = r394768 * r394770;
        double r394772 = r394766 * r394767;
        double r394773 = r394771 - r394772;
        double r394774 = r394767 / r394773;
        double r394775 = r394766 + r394774;
        return r394775;
}
↓
double f(double x, double y, double z) {
        double r394776 = z;
        double r394777 = exp(r394776);
        double r394778 = 0.0;
        bool r394779 = r394777 <= r394778;
        double r394780 = x;
        double r394781 = 1.0;
        double r394782 = r394781 / r394780;
        double r394783 = r394780 - r394782;
        double r394784 = y;
        double r394785 = 1.1283791670955126;
        double r394786 = r394785 * r394776;
        double r394787 = 0.5641895835477563;
        double r394788 = 2.0;
        double r394789 = pow(r394776, r394788);
        double r394790 = r394787 * r394789;
        double r394791 = r394790 + r394785;
        double r394792 = r394786 + r394791;
        double r394793 = r394780 * r394784;
        double r394794 = r394792 - r394793;
        double r394795 = r394784 / r394794;
        double r394796 = r394780 + r394795;
        double r394797 = r394779 ? r394783 : r394796;
        return r394797;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.8
Target0.0
Herbie2.5
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.0

    1. Initial program 7.0

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x - \frac{1}{x}}\]

    if 0.0 < (exp z)

    1. Initial program 1.3

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]

    2. Taylor expanded around 0 3.3

      \[\leadsto x + \frac{y}{\color{blue}{\left(1.128379167095512558560699289955664426088 \cdot z + \left(0.5641895835477562792803496449778322130442 \cdot {z}^{2} + 1.128379167095512558560699289955664426088\right)\right)} - x \cdot y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\left(1.128379167095512558560699289955664426088 \cdot z + \left(0.5641895835477562792803496449778322130442 \cdot {z}^{2} + 1.128379167095512558560699289955664426088\right)\right) - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.12837916709551256 y) (exp z)) x)))

  (+ x (/ y (- (* 1.12837916709551256 (exp z)) (* x y)))))