Average Error: 1.5 → 0.2
Time: 6.8m
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.292148201334136 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \le 4.010878860327887 \cdot 10^{-41}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if x < -6.292148201334136e+31 or 4.010878860327887e-41 < x

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity0.2

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

    4. Applied add-cube-cbrt0.9

      \[\leadsto \left|\frac{\color{blue}{\left(\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}\right) \cdot \sqrt[3]{x + 4}}}{1 \cdot y} - \frac{x}{y} \cdot z\right|\]

    5. Applied times-frac1.0

      \[\leadsto \left|\color{blue}{\frac{\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}}{1} \cdot \frac{\sqrt[3]{x + 4}}{y}} - \frac{x}{y} \cdot z\right|\]

    6. Applied prod-diff0.9

      \[\leadsto \left|\color{blue}{(\left(\frac{\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}}{1}\right) \cdot \left(\frac{\sqrt[3]{x + 4}}{y}\right) + \left(-z \cdot \frac{x}{y}\right))_* + (\left(-z\right) \cdot \left(\frac{x}{y}\right) + \left(z \cdot \frac{x}{y}\right))_*}\right|\]

    7. Simplified0.2

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

    8. Simplified0.2

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

    if -6.292148201334136e+31 < x < 4.010878860327887e-41

    1. Initial program 2.4

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

    2. Taylor expanded around inf 0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.292148201334136 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \le 4.010878860327887 \cdot 10^{-41}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))