Average Error: 9.4 → 0.1
Time: 7.9s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(x, \frac{1}{y}, \left(\frac{2}{t} - 2\right) + \frac{2}{t \cdot z}\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(x, \frac{1}{y}, \left(\frac{2}{t} - 2\right) + \frac{2}{t \cdot z}\right)
double f(double x, double y, double z, double t) {
        double r875207 = x;
        double r875208 = y;
        double r875209 = r875207 / r875208;
        double r875210 = 2.0;
        double r875211 = z;
        double r875212 = r875211 * r875210;
        double r875213 = 1.0;
        double r875214 = t;
        double r875215 = r875213 - r875214;
        double r875216 = r875212 * r875215;
        double r875217 = r875210 + r875216;
        double r875218 = r875214 * r875211;
        double r875219 = r875217 / r875218;
        double r875220 = r875209 + r875219;
        return r875220;
}
↓
double f(double x, double y, double z, double t) {
        double r875221 = x;
        double r875222 = 1.0;
        double r875223 = y;
        double r875224 = r875222 / r875223;
        double r875225 = 2.0;
        double r875226 = t;
        double r875227 = r875225 / r875226;
        double r875228 = r875227 - r875225;
        double r875229 = z;
        double r875230 = r875226 * r875229;
        double r875231 = r875225 / r875230;
        double r875232 = r875228 + r875231;
        double r875233 = fma(r875221, r875224, r875232);
        return r875233;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.4
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.4

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

  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

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

  5. Applied div-inv0.1

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

  6. Applied fma-def0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +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))))