Average Error: 1.6 → 0.1
Time: 50.8s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{z}{\frac{y}{x}} \le -8.329407476394978 \cdot 10^{-64}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{if}\;\frac{x + 4}{y} - \frac{z}{\frac{y}{x}} \le 4.7820266681580235 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - z \cdot \frac{x}{y}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 3 regimes

  2. if (+ (- (/ (+ 4 x) y) (/ z (/ y x))) 0) < -8.329407476394978e-64

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt1.0

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

    4. Applied prod-diff1.0

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

    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

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

    if -8.329407476394978e-64 < (+ (- (/ (+ 4 x) y) (/ z (/ y x))) 0) < 4.7820266681580235e+26

    1. Initial program 4.2

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

    3. Applied *-un-lft-identity4.2

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

    4. Applied prod-diff4.2

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

    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 4.7820266681580235e+26 < (+ (- (/ (+ 4 x) y) (/ z (/ y x))) 0)

    1. Initial program 0.1

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

    3. Applied clear-num0.2

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{y}{x + 4}}} - \frac{x}{y} \cdot z\right|\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{z}{\frac{y}{x}} \le -8.329407476394978 \cdot 10^{-64}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{if}\;\frac{x + 4}{y} - \frac{z}{\frac{y}{x}} \le 4.7820266681580235 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - z \cdot \frac{x}{y}\right|\\ \end{array}}\]

Runtime

Time bar (total: 50.8s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))