Average Error: 10.1 → 0.1
Time: 21.8s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \left(\frac{\frac{2}{t}}{z} + \left(\frac{2}{t} - 2\right)\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \left(\frac{\frac{2}{t}}{z} + \left(\frac{2}{t} - 2\right)\right)
double f(double x, double y, double z, double t) {
        double r53067550 = x;
        double r53067551 = y;
        double r53067552 = r53067550 / r53067551;
        double r53067553 = 2.0;
        double r53067554 = z;
        double r53067555 = r53067554 * r53067553;
        double r53067556 = 1.0;
        double r53067557 = t;
        double r53067558 = r53067556 - r53067557;
        double r53067559 = r53067555 * r53067558;
        double r53067560 = r53067553 + r53067559;
        double r53067561 = r53067557 * r53067554;
        double r53067562 = r53067560 / r53067561;
        double r53067563 = r53067552 + r53067562;
        return r53067563;
}
↓
double f(double x, double y, double z, double t) {
        double r53067564 = x;
        double r53067565 = y;
        double r53067566 = r53067564 / r53067565;
        double r53067567 = 2.0;
        double r53067568 = t;
        double r53067569 = r53067567 / r53067568;
        double r53067570 = z;
        double r53067571 = r53067569 / r53067570;
        double r53067572 = r53067569 - r53067567;
        double r53067573 = r53067571 + r53067572;
        double r53067574 = r53067566 + r53067573;
        return r53067574;
}

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

Original10.1
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 10.1

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

  2. Taylor expanded around 0 0.1

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

  3. Simplified0.1

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

  4. Taylor expanded around 0 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"

  :herbie-target
  (- (/ (+ (/ 2.0 z) 2.0) t) (- 2.0 (/ x y)))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))