Average Error: 3.0 → 0.1
Time: 10.6s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\frac{1}{\mathsf{fma}\left(\frac{1.128379167095512558560699289955664426088}{y}, e^{z}, -x\right)} + x\]
x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}
↓
\frac{1}{\mathsf{fma}\left(\frac{1.128379167095512558560699289955664426088}{y}, e^{z}, -x\right)} + x
double f(double x, double y, double z) {
        double r350683 = x;
        double r350684 = y;
        double r350685 = 1.1283791670955126;
        double r350686 = z;
        double r350687 = exp(r350686);
        double r350688 = r350685 * r350687;
        double r350689 = r350683 * r350684;
        double r350690 = r350688 - r350689;
        double r350691 = r350684 / r350690;
        double r350692 = r350683 + r350691;
        return r350692;
}
↓
double f(double x, double y, double z) {
        double r350693 = 1.0;
        double r350694 = 1.1283791670955126;
        double r350695 = y;
        double r350696 = r350694 / r350695;
        double r350697 = z;
        double r350698 = exp(r350697);
        double r350699 = x;
        double r350700 = -r350699;
        double r350701 = fma(r350696, r350698, r350700);
        double r350702 = r350693 / r350701;
        double r350703 = r350702 + r350699;
        return r350703;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.0
Target0.1
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. Using strategy rm

  5. Applied add-cube-cbrt3.0

    \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}{y}}\]

  6. Applied associate-/l*3.0

    \[\leadsto x + \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}{y}}{\sqrt[3]{1}}}}\]

  7. Simplified0.1

    \[\leadsto x + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\mathsf{fma}\left(\frac{1.128379167095512558560699289955664426088}{y}, e^{z}, -x\right)}}\]

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 +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 (- (* (/ 1.12837916709551256 y) (exp z)) x)))

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