Average Error: 9.4 → 0.1
Time: 22.1s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\left(\frac{2}{t \cdot z} + \frac{2}{t}\right) + \left(\frac{x}{y} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\left(\frac{2}{t \cdot z} + \frac{2}{t}\right) + \left(\frac{x}{y} - 2\right)
double f(double x, double y, double z, double t) {
        double r98603 = x;
        double r98604 = y;
        double r98605 = r98603 / r98604;
        double r98606 = 2.0;
        double r98607 = z;
        double r98608 = r98607 * r98606;
        double r98609 = 1.0;
        double r98610 = t;
        double r98611 = r98609 - r98610;
        double r98612 = r98608 * r98611;
        double r98613 = r98606 + r98612;
        double r98614 = r98610 * r98607;
        double r98615 = r98613 / r98614;
        double r98616 = r98605 + r98615;
        return r98616;
}
↓
double f(double x, double y, double z, double t) {
        double r98617 = 2.0;
        double r98618 = t;
        double r98619 = z;
        double r98620 = r98618 * r98619;
        double r98621 = r98617 / r98620;
        double r98622 = r98617 / r98618;
        double r98623 = r98621 + r98622;
        double r98624 = x;
        double r98625 = y;
        double r98626 = r98624 / r98625;
        double r98627 = r98626 - r98617;
        double r98628 = r98623 + r98627;
        return r98628;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019310 +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))))