Average Error: 9.1 → 0.1
Time: 3.2s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)
double f(double x, double y, double z, double t) {
        double r980795 = x;
        double r980796 = y;
        double r980797 = r980795 / r980796;
        double r980798 = 2.0;
        double r980799 = z;
        double r980800 = r980799 * r980798;
        double r980801 = 1.0;
        double r980802 = t;
        double r980803 = r980801 - r980802;
        double r980804 = r980800 * r980803;
        double r980805 = r980798 + r980804;
        double r980806 = r980802 * r980799;
        double r980807 = r980805 / r980806;
        double r980808 = r980797 + r980807;
        return r980808;
}
↓
double f(double x, double y, double z, double t) {
        double r980809 = x;
        double r980810 = y;
        double r980811 = r980809 / r980810;
        double r980812 = 2.0;
        double r980813 = 1.0;
        double r980814 = t;
        double r980815 = z;
        double r980816 = r980814 * r980815;
        double r980817 = r980813 / r980816;
        double r980818 = r980813 / r980814;
        double r980819 = r980812 * r980818;
        double r980820 = r980819 - r980812;
        double r980821 = fma(r980812, r980817, r980820);
        double r980822 = r980811 + r980821;
        return r980822;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.1
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.1

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

  2. Taylor expanded around 0 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019362 +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))))