Average Error: 9.4 → 0.1
Time: 4.5s
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 r851473 = x;
        double r851474 = y;
        double r851475 = r851473 / r851474;
        double r851476 = 2.0;
        double r851477 = z;
        double r851478 = r851477 * r851476;
        double r851479 = 1.0;
        double r851480 = t;
        double r851481 = r851479 - r851480;
        double r851482 = r851478 * r851481;
        double r851483 = r851476 + r851482;
        double r851484 = r851480 * r851477;
        double r851485 = r851483 / r851484;
        double r851486 = r851475 + r851485;
        return r851486;
}
↓
double f(double x, double y, double z, double t) {
        double r851487 = x;
        double r851488 = y;
        double r851489 = r851487 / r851488;
        double r851490 = 2.0;
        double r851491 = 1.0;
        double r851492 = t;
        double r851493 = z;
        double r851494 = r851492 * r851493;
        double r851495 = r851491 / r851494;
        double r851496 = r851490 / r851492;
        double r851497 = fma(r851490, r851495, r851496);
        double r851498 = r851497 - r851490;
        double r851499 = r851489 + r851498;
        return r851499;
}

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))))