Average Error: 1.5 → 0.1
Time: 29.0s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - \frac{z \cdot x}{y} \le -1.5947771882232943 \cdot 10^{+306}:\\ \;\;\;\;\left|(\left(\sqrt[3]{\frac{4 + x}{y}} \cdot \sqrt[3]{\frac{4 + x}{y}}\right) \cdot \left(\sqrt[3]{\frac{4 + x}{y}}\right) + \left(\left(-z\right) \cdot \frac{x}{y}\right))_*\right|\\ \mathbf{elif}\;\frac{4 + x}{y} - \frac{z \cdot x}{y} \le 7.020074180516919 \cdot 10^{+223}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

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 z) y)) < -1.5947771882232943e+306

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.3

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

    4. Applied fma-neg0.3

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

    if -1.5947771882232943e+306 < (- (/ (+ x 4) y) (/ (* x z) y)) < 7.020074180516919e+223

    1. Initial program 1.7

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

    3. Applied associate-*l/0.1

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

    if 7.020074180516919e+223 < (- (/ (+ x 4) y) (/ (* x z) y))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt13.8

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

    4. Applied prod-diff13.8

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

    5. Simplified0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - \frac{z \cdot x}{y} \le -1.5947771882232943 \cdot 10^{+306}:\\ \;\;\;\;\left|(\left(\sqrt[3]{\frac{4 + x}{y}} \cdot \sqrt[3]{\frac{4 + x}{y}}\right) \cdot \left(\sqrt[3]{\frac{4 + x}{y}}\right) + \left(\left(-z\right) \cdot \frac{x}{y}\right))_*\right|\\ \mathbf{elif}\;\frac{4 + x}{y} - \frac{z \cdot x}{y} \le 7.020074180516919 \cdot 10^{+223}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Runtime

Time bar (total: 29.0s)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))