Average Error: 3.0 → 0.1
Time: 10.3s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[x + \frac{1}{\mathsf{fma}\left(\frac{e^{z}}{y}, 1.128379167095512558560699289955664426088, -x\right)}\]
x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}
↓
x + \frac{1}{\mathsf{fma}\left(\frac{e^{z}}{y}, 1.128379167095512558560699289955664426088, -x\right)}
double f(double x, double y, double z) {
        double r341917 = x;
        double r341918 = y;
        double r341919 = 1.1283791670955126;
        double r341920 = z;
        double r341921 = exp(r341920);
        double r341922 = r341919 * r341921;
        double r341923 = r341917 * r341918;
        double r341924 = r341922 - r341923;
        double r341925 = r341918 / r341924;
        double r341926 = r341917 + r341925;
        return r341926;
}
↓
double f(double x, double y, double z) {
        double r341927 = x;
        double r341928 = 1.0;
        double r341929 = z;
        double r341930 = exp(r341929);
        double r341931 = y;
        double r341932 = r341930 / r341931;
        double r341933 = 1.1283791670955126;
        double r341934 = -r341927;
        double r341935 = fma(r341932, r341933, r341934);
        double r341936 = r341928 / r341935;
        double r341937 = r341927 + r341936;
        return r341937;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.0
Target0.0
Herbie0.1
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Initial program 3.0

    \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
  2. Using strategy rm

  3. Applied clear-num3.0

    \[\leadsto x + \color{blue}{\frac{1}{\frac{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}{y}}}\]

  4. Simplified3.0

    \[\leadsto x + \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(e^{z}, 1.128379167095512558560699289955664426088, -y \cdot x\right)}{y}}}\]

  5. Taylor expanded around inf 0.1

    \[\leadsto x + \frac{1}{\color{blue}{1.128379167095512558560699289955664426088 \cdot \frac{e^{z}}{y} - x}}\]

  6. Simplified0.1

    \[\leadsto x + \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{z}}{y}, 1.128379167095512558560699289955664426088, -x\right)}}\]

  7. Final simplification0.1

    \[\leadsto x + \frac{1}{\mathsf{fma}\left(\frac{e^{z}}{y}, 1.128379167095512558560699289955664426088, -x\right)}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

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