Average Error: 9.7 → 0.1
Time: 5.5s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r728817 = x;
        double r728818 = y;
        double r728819 = r728817 / r728818;
        double r728820 = 2.0;
        double r728821 = z;
        double r728822 = r728821 * r728820;
        double r728823 = 1.0;
        double r728824 = t;
        double r728825 = r728823 - r728824;
        double r728826 = r728822 * r728825;
        double r728827 = r728820 + r728826;
        double r728828 = r728824 * r728821;
        double r728829 = r728827 / r728828;
        double r728830 = r728819 + r728829;
        return r728830;
}
↓
double f(double x, double y, double z, double t) {
        double r728831 = 1.0;
        double r728832 = z;
        double r728833 = r728831 / r728832;
        double r728834 = 1.0;
        double r728835 = r728833 + r728834;
        double r728836 = t;
        double r728837 = r728835 - r728836;
        double r728838 = 2.0;
        double r728839 = r728838 / r728836;
        double r728840 = x;
        double r728841 = y;
        double r728842 = r728840 / r728841;
        double r728843 = fma(r728837, r728839, r728842);
        return r728843;
}

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. Simplified9.6

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

  3. Taylor expanded around 0 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020002 +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))))