Average Error: 9.5 → 0.1
Time: 3.8s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, \frac{2}{t}\right) - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, \frac{2}{t}\right) - 2\right)
double f(double x, double y, double z, double t) {
        double r759243 = x;
        double r759244 = y;
        double r759245 = r759243 / r759244;
        double r759246 = 2.0;
        double r759247 = z;
        double r759248 = r759247 * r759246;
        double r759249 = 1.0;
        double r759250 = t;
        double r759251 = r759249 - r759250;
        double r759252 = r759248 * r759251;
        double r759253 = r759246 + r759252;
        double r759254 = r759250 * r759247;
        double r759255 = r759253 / r759254;
        double r759256 = r759245 + r759255;
        return r759256;
}
↓
double f(double x, double y, double z, double t) {
        double r759257 = x;
        double r759258 = y;
        double r759259 = r759257 / r759258;
        double r759260 = 2.0;
        double r759261 = 1.0;
        double r759262 = t;
        double r759263 = r759261 / r759262;
        double r759264 = z;
        double r759265 = r759263 / r759264;
        double r759266 = r759260 / r759262;
        double r759267 = fma(r759260, r759265, r759266);
        double r759268 = r759267 - r759260;
        double r759269 = r759259 + r759268;
        return r759269;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.5
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.5

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

  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

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

  5. Applied associate-/r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020039 +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))))