Average Error: 1.6 → 1.5
Time: 33.8s
Precision: 64
Ground Truth: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -9.37854704553904 \cdot 10^{+142}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{z}{y} \cdot x\right|\\ \mathbf{if}\;x \le 6.635871491323079 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(\frac{4}{1} - \left(z \cdot x - \frac{x}{1}\right)\right)}{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 3 regimes.

  2. if x < -9.37854704553904e+142

    1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

    2. Applied taylor 14.5

      \[\leadsto \left|\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}\right|\]

    3. Taylor expanded around 0 14.5

      \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}}\right|\]

    4. Applied simplify 0.1

      \[\leadsto \color{blue}{\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{z}{y} \cdot x\right|}\]

    if -9.37854704553904e+142 < x < 6.635871491323079e-24

    1. Initial program 2.3

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

    2. Applied taylor 0.7

      \[\leadsto \left|\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}\right|\]

    3. Taylor expanded around 0 0.7

      \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}}\right|\]

    4. Applied simplify 4.6

      \[\leadsto \color{blue}{\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{z}{y} \cdot x\right|}\]
    5. Using strategy rm

    6. Applied associate-*l/ 0.7

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \color{blue}{\frac{z \cdot x}{y}}\right|\]

    7. Applied flip-+ 24.3

      \[\leadsto \left|\color{blue}{\frac{{\left(\frac{x}{y}\right)}^2 - {\left(\frac{4}{y}\right)}^2}{\frac{x}{y} - \frac{4}{y}}} - \frac{z \cdot x}{y}\right|\]

    8. Applied frac-sub 24.6

      \[\leadsto \left|\color{blue}{\frac{\left({\left(\frac{x}{y}\right)}^2 - {\left(\frac{4}{y}\right)}^2\right) \cdot y - \left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(z \cdot x\right)}{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot y}}\right|\]

    9. Applied simplify 2.1

      \[\leadsto \left|\frac{\color{blue}{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(\frac{4}{1} - \left(z \cdot x - \frac{x}{1}\right)\right)}}{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot y}\right|\]

    if 6.635871491323079e-24 < x

    1. Initial program 0.2

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Total time: 33.8s Debug log

Please include this information when filing a bug report:

herbie --seed '#(2720959114 4054591762 382387250 3127969484 1647930059 3330391360)'
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))