Average Error: 9.7 → 0.1
Time: 3.3s
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 r788946 = x;
        double r788947 = y;
        double r788948 = r788946 / r788947;
        double r788949 = 2.0;
        double r788950 = z;
        double r788951 = r788950 * r788949;
        double r788952 = 1.0;
        double r788953 = t;
        double r788954 = r788952 - r788953;
        double r788955 = r788951 * r788954;
        double r788956 = r788949 + r788955;
        double r788957 = r788953 * r788950;
        double r788958 = r788956 / r788957;
        double r788959 = r788948 + r788958;
        return r788959;
}
↓
double f(double x, double y, double z, double t) {
        double r788960 = x;
        double r788961 = y;
        double r788962 = r788960 / r788961;
        double r788963 = 2.0;
        double r788964 = 1.0;
        double r788965 = t;
        double r788966 = z;
        double r788967 = r788965 * r788966;
        double r788968 = r788964 / r788967;
        double r788969 = r788964 / r788965;
        double r788970 = r788963 * r788969;
        double r788971 = r788970 - r788963;
        double r788972 = fma(r788963, r788968, r788971);
        double r788973 = r788962 + r788972;
        return r788973;
}

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))))