Average Error: 8.8 → 0.1
Time: 12.6s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\left(\frac{2}{t} + \frac{2}{t \cdot z}\right) + \left(\frac{x}{y} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\left(\frac{2}{t} + \frac{2}{t \cdot z}\right) + \left(\frac{x}{y} - 2\right)
double f(double x, double y, double z, double t) {
        double r1030921 = x;
        double r1030922 = y;
        double r1030923 = r1030921 / r1030922;
        double r1030924 = 2.0;
        double r1030925 = z;
        double r1030926 = r1030925 * r1030924;
        double r1030927 = 1.0;
        double r1030928 = t;
        double r1030929 = r1030927 - r1030928;
        double r1030930 = r1030926 * r1030929;
        double r1030931 = r1030924 + r1030930;
        double r1030932 = r1030928 * r1030925;
        double r1030933 = r1030931 / r1030932;
        double r1030934 = r1030923 + r1030933;
        return r1030934;
}
↓
double f(double x, double y, double z, double t) {
        double r1030935 = 2.0;
        double r1030936 = t;
        double r1030937 = r1030935 / r1030936;
        double r1030938 = z;
        double r1030939 = r1030936 * r1030938;
        double r1030940 = r1030935 / r1030939;
        double r1030941 = r1030937 + r1030940;
        double r1030942 = x;
        double r1030943 = y;
        double r1030944 = r1030942 / r1030943;
        double r1030945 = r1030944 - r1030935;
        double r1030946 = r1030941 + r1030945;
        return r1030946;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.8
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 8.8

    \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, 1, \frac{2}{z}\right)}{t} + \left(\frac{x}{y} - 2\right)}\]

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

    \[\leadsto \left(\frac{2}{t} + \frac{2}{t \cdot z}\right) + \left(\frac{x}{y} - 2\right)\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :herbie-target
  (- (/ (+ (/ 2 z) 2) t) (- 2 (/ x y)))

  (+ (/ x y) (/ (+ 2 (* (* z 2) (- 1 t))) (* t z))))