Average Error: 9.4 → 0.2
Time: 4.5s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(x, \frac{1}{y}, \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)\right)\]
\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}, \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)\right)
double f(double x, double y, double z, double t) {
        double r878404 = x;
        double r878405 = y;
        double r878406 = r878404 / r878405;
        double r878407 = 2.0;
        double r878408 = z;
        double r878409 = r878408 * r878407;
        double r878410 = 1.0;
        double r878411 = t;
        double r878412 = r878410 - r878411;
        double r878413 = r878409 * r878412;
        double r878414 = r878407 + r878413;
        double r878415 = r878411 * r878408;
        double r878416 = r878414 / r878415;
        double r878417 = r878406 + r878416;
        return r878417;
}
↓
double f(double x, double y, double z, double t) {
        double r878418 = x;
        double r878419 = 1.0;
        double r878420 = y;
        double r878421 = r878419 / r878420;
        double r878422 = 2.0;
        double r878423 = t;
        double r878424 = z;
        double r878425 = r878423 * r878424;
        double r878426 = r878419 / r878425;
        double r878427 = r878419 / r878423;
        double r878428 = r878422 * r878427;
        double r878429 = r878428 - r878422;
        double r878430 = fma(r878422, r878426, r878429);
        double r878431 = fma(r878418, r878421, r878430);
        return r878431;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

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

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

  5. Applied div-inv0.2

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

  6. Applied fma-def0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020083 +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))))