Average Error: 9.4 → 0.1
Time: 3.6s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{1}{t \cdot z}, \frac{2}{t}\right) - 2\right)\]
\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{1}{t \cdot z}, \frac{2}{t}\right) - 2\right)
double f(double x, double y, double z, double t) {
        double r794408 = x;
        double r794409 = y;
        double r794410 = r794408 / r794409;
        double r794411 = 2.0;
        double r794412 = z;
        double r794413 = r794412 * r794411;
        double r794414 = 1.0;
        double r794415 = t;
        double r794416 = r794414 - r794415;
        double r794417 = r794413 * r794416;
        double r794418 = r794411 + r794417;
        double r794419 = r794415 * r794412;
        double r794420 = r794418 / r794419;
        double r794421 = r794410 + r794420;
        return r794421;
}
↓
double f(double x, double y, double z, double t) {
        double r794422 = x;
        double r794423 = y;
        double r794424 = r794422 / r794423;
        double r794425 = 2.0;
        double r794426 = 1.0;
        double r794427 = t;
        double r794428 = z;
        double r794429 = r794427 * r794428;
        double r794430 = r794426 / r794429;
        double r794431 = r794425 / r794427;
        double r794432 = fma(r794425, r794430, r794431);
        double r794433 = r794432 - r794425;
        double r794434 = r794424 + r794433;
        return r794434;
}

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

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 +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))))