Average Error: 9.8 → 0.1
Time: 9.6s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\left(\left(\frac{2}{t} - 2\right) + \frac{\frac{2}{t}}{z}\right) + \frac{x}{y}\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\left(\left(\frac{2}{t} - 2\right) + \frac{\frac{2}{t}}{z}\right) + \frac{x}{y}
double f(double x, double y, double z, double t) {
        double r786472 = x;
        double r786473 = y;
        double r786474 = r786472 / r786473;
        double r786475 = 2.0;
        double r786476 = z;
        double r786477 = r786476 * r786475;
        double r786478 = 1.0;
        double r786479 = t;
        double r786480 = r786478 - r786479;
        double r786481 = r786477 * r786480;
        double r786482 = r786475 + r786481;
        double r786483 = r786479 * r786476;
        double r786484 = r786482 / r786483;
        double r786485 = r786474 + r786484;
        return r786485;
}
↓
double f(double x, double y, double z, double t) {
        double r786486 = 2.0;
        double r786487 = t;
        double r786488 = r786486 / r786487;
        double r786489 = r786488 - r786486;
        double r786490 = z;
        double r786491 = r786488 / r786490;
        double r786492 = r786489 + r786491;
        double r786493 = x;
        double r786494 = y;
        double r786495 = r786493 / r786494;
        double r786496 = r786492 + r786495;
        return r786496;
}

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

Original9.8
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.8

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

  2. Simplified9.7

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

  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. Using strategy rm

  5. Applied fma-udef0.1

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

  6. Simplified0.1

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

  7. Taylor expanded around 0 0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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))))