Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Average Error: 2.8 → 1.1

Time: 25.5s

Precision: 64

Math

\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x + \left(\frac{-1}{x} - \frac{\frac{\frac{e^{z} \cdot 1.1283791670955126}{y}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\mathsf{fma}\left(1.1283791670955126, -e^{z}, x \cdot y\right)} + x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r17401852 = x;
        double r17401853 = y;
        double r17401854 = 1.1283791670955126;
        double r17401855 = z;
        double r17401856 = exp(r17401855);
        double r17401857 = r17401854 * r17401856;
        double r17401858 = r17401852 * r17401853;
        double r17401859 = r17401857 - r17401858;
        double r17401860 = r17401853 / r17401859;
        double r17401861 = r17401852 + r17401860;
        return r17401861;
}

↓

double f(double x, double y, double z) {
        double r17401862 = z;
        double r17401863 = exp(r17401862);
        double r17401864 = 0.0;
        bool r17401865 = r17401863 <= r17401864;
        double r17401866 = x;
        double r17401867 = -1.0;
        double r17401868 = r17401867 / r17401866;
        double r17401869 = 1.1283791670955126;
        double r17401870 = r17401863 * r17401869;
        double r17401871 = y;
        double r17401872 = r17401870 / r17401871;
        double r17401873 = r17401872 / r17401866;
        double r17401874 = r17401873 / r17401866;
        double r17401875 = r17401868 - r17401874;
        double r17401876 = r17401866 + r17401875;
        double r17401877 = -r17401871;
        double r17401878 = -r17401863;
        double r17401879 = r17401866 * r17401871;
        double r17401880 = fma(r17401869, r17401878, r17401879);
        double r17401881 = r17401877 / r17401880;
        double r17401882 = r17401881 + r17401866;
        double r17401883 = r17401865 ? r17401876 : r17401882;
        return r17401883;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.8
Target	0.0
Herbie	1.1

\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x}\]

Derivation

1. Split input into 2 regimes

2. if (exp z) < 0.0

   1. Initial program 7.5

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]

   2. Taylor expanded around inf 18.8

      \[\leadsto x + \color{blue}{\left(-\left(\frac{1}{x} + 1.1283791670955126 \cdot \frac{e^{z}}{{x}^{2} \cdot y}\right)\right)}\]

   3. Simplified 0.0

      \[\leadsto x + \color{blue}{\left(\frac{-1}{x} - \frac{\frac{\frac{e^{z} \cdot 1.1283791670955126}{y}}{x}}{x}\right)}\]

   if 0.0 < (exp z)

   1. Initial program 1.4

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]
   2. Using strategy rm

   3. Applied frac-2neg 1.4

      \[\leadsto x + \color{blue}{\frac{-y}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}}\]

   4. Simplified 1.4

      \[\leadsto x + \frac{-y}{\color{blue}{\mathsf{fma}\left(1.1283791670955126, -e^{z}, x \cdot y\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x + \left(\frac{-1}{x} - \frac{\frac{\frac{e^{z} \cdot 1.1283791670955126}{y}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\mathsf{fma}\left(1.1283791670955126, -e^{z}, x \cdot y\right)} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"

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

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