Average Error: 9.4 → 0.1
Time: 3.0s
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 r822023 = x;
        double r822024 = y;
        double r822025 = r822023 / r822024;
        double r822026 = 2.0;
        double r822027 = z;
        double r822028 = r822027 * r822026;
        double r822029 = 1.0;
        double r822030 = t;
        double r822031 = r822029 - r822030;
        double r822032 = r822028 * r822031;
        double r822033 = r822026 + r822032;
        double r822034 = r822030 * r822027;
        double r822035 = r822033 / r822034;
        double r822036 = r822025 + r822035;
        return r822036;
}
↓
double f(double x, double y, double z, double t) {
        double r822037 = x;
        double r822038 = 1.0;
        double r822039 = y;
        double r822040 = r822038 / r822039;
        double r822041 = 2.0;
        double r822042 = t;
        double r822043 = z;
        double r822044 = r822042 * r822043;
        double r822045 = r822038 / r822044;
        double r822046 = r822038 / r822042;
        double r822047 = r822041 * r822046;
        double r822048 = r822047 - r822041;
        double r822049 = fma(r822041, r822045, r822048);
        double r822050 = fma(r822037, r822040, r822049);
        return r822050;
}

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

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

  5. Applied div-inv0.1

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

  6. Applied fma-def0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 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))))