Average Error: 1.6 → 0.3
Time: 20.5s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(4 + x\right) - z \cdot x \le -4.3979325261402567 \cdot 10^{+272}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;\left(4 + x\right) - z \cdot x \le 3.0192466468152004 \cdot 10^{+96}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

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Derivation

  1. Split input into 3 regimes

  2. if (- (+ x 4) (* x z)) < -4.3979325261402567e+272

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt1.1

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

    4. Applied associate-*l*1.1

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

    5. Taylor expanded around 0 50.6

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

    6. Simplified0.2

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

    if -4.3979325261402567e+272 < (- (+ x 4) (* x z)) < 3.0192466468152004e+96

    1. Initial program 1.8

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 3.0192466468152004e+96 < (- (+ x 4) (* x z))

    1. Initial program 1.1

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

    2. Initial simplification1.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 + x\right) - z \cdot x \le -4.3979325261402567 \cdot 10^{+272}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;\left(4 + x\right) - z \cdot x \le 3.0192466468152004 \cdot 10^{+96}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Runtime

Time bar (total: 20.5s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))