Average Error: 9.6 → 0.1
Time: 5.2s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r981396 = x;
        double r981397 = y;
        double r981398 = r981396 / r981397;
        double r981399 = 2.0;
        double r981400 = z;
        double r981401 = r981400 * r981399;
        double r981402 = 1.0;
        double r981403 = t;
        double r981404 = r981402 - r981403;
        double r981405 = r981401 * r981404;
        double r981406 = r981399 + r981405;
        double r981407 = r981403 * r981400;
        double r981408 = r981406 / r981407;
        double r981409 = r981398 + r981408;
        return r981409;
}
↓
double f(double x, double y, double z, double t) {
        double r981410 = 1.0;
        double r981411 = z;
        double r981412 = r981410 / r981411;
        double r981413 = 1.0;
        double r981414 = r981412 + r981413;
        double r981415 = t;
        double r981416 = r981414 - r981415;
        double r981417 = 2.0;
        double r981418 = r981417 / r981415;
        double r981419 = x;
        double r981420 = y;
        double r981421 = r981419 / r981420;
        double r981422 = fma(r981416, r981418, r981421);
        return r981422;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.6

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

  2. Simplified9.5

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

  3. Taylor expanded around 0 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020057 +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))))