Average Error: 9.3 → 0.1
Time: 26.1s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{\mathsf{fma}\left(2, 1, \frac{2}{z}\right)}{t} + \left(\frac{x}{y} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{\mathsf{fma}\left(2, 1, \frac{2}{z}\right)}{t} + \left(\frac{x}{y} - 2\right)
double f(double x, double y, double z, double t) {
        double r469741 = x;
        double r469742 = y;
        double r469743 = r469741 / r469742;
        double r469744 = 2.0;
        double r469745 = z;
        double r469746 = r469745 * r469744;
        double r469747 = 1.0;
        double r469748 = t;
        double r469749 = r469747 - r469748;
        double r469750 = r469746 * r469749;
        double r469751 = r469744 + r469750;
        double r469752 = r469748 * r469745;
        double r469753 = r469751 / r469752;
        double r469754 = r469743 + r469753;
        return r469754;
}
↓
double f(double x, double y, double z, double t) {
        double r469755 = 2.0;
        double r469756 = 1.0;
        double r469757 = z;
        double r469758 = r469755 / r469757;
        double r469759 = fma(r469755, r469756, r469758);
        double r469760 = t;
        double r469761 = r469759 / r469760;
        double r469762 = x;
        double r469763 = y;
        double r469764 = r469762 / r469763;
        double r469765 = r469764 - r469755;
        double r469766 = r469761 + r469765;
        return r469766;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.3
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.3

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 +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))))