Average Error: 9.1 → 0.2
Time: 15.3s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(x, \frac{1}{y}, \frac{2}{t \cdot z} + \left(\frac{2}{t} - 2\right)\right)\]
\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}, \frac{2}{t \cdot z} + \left(\frac{2}{t} - 2\right)\right)
double f(double x, double y, double z, double t) {
        double r555449 = x;
        double r555450 = y;
        double r555451 = r555449 / r555450;
        double r555452 = 2.0;
        double r555453 = z;
        double r555454 = r555453 * r555452;
        double r555455 = 1.0;
        double r555456 = t;
        double r555457 = r555455 - r555456;
        double r555458 = r555454 * r555457;
        double r555459 = r555452 + r555458;
        double r555460 = r555456 * r555453;
        double r555461 = r555459 / r555460;
        double r555462 = r555451 + r555461;
        return r555462;
}
↓
double f(double x, double y, double z, double t) {
        double r555463 = x;
        double r555464 = 1.0;
        double r555465 = y;
        double r555466 = r555464 / r555465;
        double r555467 = 2.0;
        double r555468 = t;
        double r555469 = z;
        double r555470 = r555468 * r555469;
        double r555471 = r555467 / r555470;
        double r555472 = r555467 / r555468;
        double r555473 = r555472 - r555467;
        double r555474 = r555471 + r555473;
        double r555475 = fma(r555463, r555466, r555474);
        return r555475;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

Derivation

  1. Initial program 9.1

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

  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

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

  5. Applied div-inv0.2

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

  6. Applied fma-def0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019208 +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))))