Average Error: 1.7 → 0.1
Time: 22.2s
Precision: 64
Internal Precision: 320
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le -1130508328.6653645:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le 47917721.40686438:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(4 + x\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (+ x 4) y) (* (/ x y) z)) < -1130508328.6653645

    1. Initial program 0.1

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

    2. Taylor expanded around 0 5.1

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

    3. Simplified0.1

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

    if -1130508328.6653645 < (- (/ (+ x 4) y) (* (/ x y) z)) < 47917721.40686438

    1. Initial program 3.9

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

    2. Initial simplification0.1

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

    if 47917721.40686438 < (- (/ (+ 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 div-inv0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le -1130508328.6653645:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le 47917721.40686438:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(4 + x\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Runtime

Time bar (total: 22.2s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))