Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2

Average Error: 9.3 → 0.1

Time: 23.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r474946 = x;
        double r474947 = y;
        double r474948 = r474946 / r474947;
        double r474949 = 2.0;
        double r474950 = z;
        double r474951 = r474950 * r474949;
        double r474952 = 1.0;
        double r474953 = t;
        double r474954 = r474952 - r474953;
        double r474955 = r474951 * r474954;
        double r474956 = r474949 + r474955;
        double r474957 = r474953 * r474950;
        double r474958 = r474956 / r474957;
        double r474959 = r474948 + r474958;
        return r474959;
}

↓

double f(double x, double y, double z, double t) {
        double r474960 = 2.0;
        double r474961 = 1.0;
        double r474962 = z;
        double r474963 = r474960 / r474962;
        double r474964 = fma(r474960, r474961, r474963);
        double r474965 = t;
        double r474966 = r474964 / r474965;
        double r474967 = x;
        double r474968 = y;
        double r474969 = r474967 / r474968;
        double r474970 = r474969 - r474960;
        double r474971 = r474966 + r474970;
        return r474971;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.3
Target	0.1
Herbie	0.1

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

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. Simplified 0.1

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

3. Final simplification 0.1

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

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))))