Average Error: 9.1 → 0.1
Time: 2.6s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, \frac{2}{t}\right) - 2\right)\]
\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{\frac{1}{t}}{z}, \frac{2}{t}\right) - 2\right)
double f(double x, double y, double z, double t) {
        double r982304 = x;
        double r982305 = y;
        double r982306 = r982304 / r982305;
        double r982307 = 2.0;
        double r982308 = z;
        double r982309 = r982308 * r982307;
        double r982310 = 1.0;
        double r982311 = t;
        double r982312 = r982310 - r982311;
        double r982313 = r982309 * r982312;
        double r982314 = r982307 + r982313;
        double r982315 = r982311 * r982308;
        double r982316 = r982314 / r982315;
        double r982317 = r982306 + r982316;
        return r982317;
}
↓
double f(double x, double y, double z, double t) {
        double r982318 = x;
        double r982319 = y;
        double r982320 = r982318 / r982319;
        double r982321 = 2.0;
        double r982322 = 1.0;
        double r982323 = t;
        double r982324 = r982322 / r982323;
        double r982325 = z;
        double r982326 = r982324 / r982325;
        double r982327 = r982321 / r982323;
        double r982328 = fma(r982321, r982326, r982327);
        double r982329 = r982328 - r982321;
        double r982330 = r982320 + r982329;
        return r982330;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

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

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

  5. Applied associate-/r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 +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))))