Average Error: 1.6 → 0.2
Time: 30.7s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - z \cdot \frac{x}{y} \le -6.450355433078854 \cdot 10^{-167}:\\ \;\;\;\;\left|(\left(\frac{x}{y}\right) \cdot \left(-z\right) + \left(\frac{4}{y} + \frac{x}{y}\right))_*\right|\\ \mathbf{if}\;\frac{4 + x}{y} - z \cdot \frac{x}{y} \le 6.628463670396576 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (+ x 4) y) (* (/ x y) z)) < -6.450355433078854e-167

    1. Initial program 0.4

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

    2. Taylor expanded around 0 3.7

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

    3. Applied simplify0.4

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

    if -6.450355433078854e-167 < (- (/ (+ x 4) y) (* (/ x y) z)) < 6.628463670396576e+46

    1. Initial program 4.5

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

    3. Applied *-un-lft-identity4.5

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

    4. Applied prod-diff4.5

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

    5. Applied simplify0.1

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

    6. Applied simplify0.1

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

    if 6.628463670396576e+46 < (- (/ (+ x 4) y) (* (/ x y) z))

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.9

      \[\leadsto \left|\color{blue}{\left(\sqrt[3]{\frac{x + 4}{y}} \cdot \sqrt[3]{\frac{x + 4}{y}}\right) \cdot \sqrt[3]{\frac{x + 4}{y}}} - \frac{x}{y} \cdot z\right|\]

    4. Applied prod-diff0.9

      \[\leadsto \left|\color{blue}{(\left(\sqrt[3]{\frac{x + 4}{y}} \cdot \sqrt[3]{\frac{x + 4}{y}}\right) \cdot \left(\sqrt[3]{\frac{x + 4}{y}}\right) + \left(-z \cdot \frac{x}{y}\right))_* + (\left(-z\right) \cdot \left(\frac{x}{y}\right) + \left(z \cdot \frac{x}{y}\right))_*}\right|\]

    5. Applied simplify0.1

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

    6. Applied simplify0.1

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

  4. Applied simplify0.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - z \cdot \frac{x}{y} \le -6.450355433078854 \cdot 10^{-167}:\\ \;\;\;\;\left|(\left(\frac{x}{y}\right) \cdot \left(-z\right) + \left(\frac{4}{y} + \frac{x}{y}\right))_*\right|\\ \mathbf{if}\;\frac{4 + x}{y} - z \cdot \frac{x}{y} \le 6.628463670396576 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}}\]

Runtime

Time bar (total: 30.7s)Debug logProfile

herbie shell --seed 2018201 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))