Average Error: 1.6 → 0.4
Time: 21.0s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.685923375054369 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.468024746810054 \cdot 10^{-140}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Your Program's Arguments

Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -6.685923375054369e-63

    1. Initial program 0.5

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

    2. Initial simplification0.4

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

    4. Applied add-sqr-sqrt25.2

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

    5. Applied div-inv25.3

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

    6. Applied prod-diff25.3

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

    7. Simplified25.1

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

    8. Simplified0.4

      \[\leadsto \left|\left(\frac{x + 4}{y} - \frac{z}{y} \cdot x\right) + \color{blue}{0}\right|\]

    if -6.685923375054369e-63 < x < 8.468024746810054e-140

    1. Initial program 2.6

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

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

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

    if 8.468024746810054e-140 < x

    1. Initial program 1.1

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

    2. Initial simplification0.9

      \[\leadsto \left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.685923375054369 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.468024746810054 \cdot 10^{-140}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Runtime

Time bar (total: 21.0s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))