Average Error: 9.7 → 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 r936796 = x;
        double r936797 = y;
        double r936798 = r936796 / r936797;
        double r936799 = 2.0;
        double r936800 = z;
        double r936801 = r936800 * r936799;
        double r936802 = 1.0;
        double r936803 = t;
        double r936804 = r936802 - r936803;
        double r936805 = r936801 * r936804;
        double r936806 = r936799 + r936805;
        double r936807 = r936803 * r936800;
        double r936808 = r936806 / r936807;
        double r936809 = r936798 + r936808;
        return r936809;
}
↓
double f(double x, double y, double z, double t) {
        double r936810 = x;
        double r936811 = y;
        double r936812 = r936810 / r936811;
        double r936813 = 2.0;
        double r936814 = 1.0;
        double r936815 = t;
        double r936816 = z;
        double r936817 = r936815 * r936816;
        double r936818 = r936814 / r936817;
        double r936819 = r936814 / r936815;
        double r936820 = r936813 * r936819;
        double r936821 = r936820 - r936813;
        double r936822 = fma(r936813, r936818, r936821);
        double r936823 = r936812 + r936822;
        return r936823;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.7
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.7

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

  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 2019353 +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))))