Average Error: 1.6 → 0.7
Time: 20.4s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.64776357827141 \cdot 10^{+153}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;y \le 1.678180763428303 \cdot 10^{-58}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Derivation

  1. Split input into 3 regimes

  2. if y < -8.64776357827141e+153

    1. Initial program 4.2

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

    2. Initial simplification0.1

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

    if -8.64776357827141e+153 < y < 1.678180763428303e-58

    1. Initial program 0.2

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied associate-*l/1.1

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

    4. Applied sub-div1.1

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

    if 1.678180763428303e-58 < y

    1. Initial program 2.3

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied div-inv2.3

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

    4. Applied prod-diff2.3

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

    5. Simplified0.4

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

    6. Simplified0.4

      \[\leadsto \left|\left(\frac{4 + x}{y} - \frac{z}{y} \cdot x\right) + \color{blue}{0}\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.64776357827141 \cdot 10^{+153}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;y \le 1.678180763428303 \cdot 10^{-58}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Runtime

Time bar (total: 20.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes2.50.70.02.573%
herbie shell --seed 2018295 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))