Average Error: 10.1 → 0.1
Time: 17.5s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\left(\frac{\frac{1}{t}}{z} - 1\right) + \frac{1}{t}, 2, \frac{x}{y}\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\left(\frac{\frac{1}{t}}{z} - 1\right) + \frac{1}{t}, 2, \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r566831 = x;
        double r566832 = y;
        double r566833 = r566831 / r566832;
        double r566834 = 2.0;
        double r566835 = z;
        double r566836 = r566835 * r566834;
        double r566837 = 1.0;
        double r566838 = t;
        double r566839 = r566837 - r566838;
        double r566840 = r566836 * r566839;
        double r566841 = r566834 + r566840;
        double r566842 = r566838 * r566835;
        double r566843 = r566841 / r566842;
        double r566844 = r566833 + r566843;
        return r566844;
}
↓
double f(double x, double y, double z, double t) {
        double r566845 = 1.0;
        double r566846 = t;
        double r566847 = r566845 / r566846;
        double r566848 = z;
        double r566849 = r566847 / r566848;
        double r566850 = r566849 - r566845;
        double r566851 = 1.0;
        double r566852 = r566851 / r566846;
        double r566853 = r566850 + r566852;
        double r566854 = 2.0;
        double r566855 = x;
        double r566856 = y;
        double r566857 = r566855 / r566856;
        double r566858 = fma(r566853, r566854, r566857);
        return r566858;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original10.1
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 10.1

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

  2. Simplified0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"

  :herbie-target
  (- (/ (+ (/ 2.0 z) 2.0) t) (- 2.0 (/ x y)))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))