Average Error: 1.7 → 0.3
Time: 15.7s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.300080753561144 \cdot 10^{+100} \lor \neg \left(x \le 197728541030068.12\right):\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array}\]

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 2 regimes

  2. if x < -4.300080753561144e+100 or 197728541030068.12 < x

    1. Initial program 0.1

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

    2. Taylor expanded around 0 9.8

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

    3. Simplified0.1

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

    if -4.300080753561144e+100 < x < 197728541030068.12

    1. Initial program 2.4

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

    3. Applied associate-*l/0.4

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

    4. Applied sub-div0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.300080753561144 \cdot 10^{+100} \lor \neg \left(x \le 197728541030068.12\right):\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array}\]

Runtime

Time bar (total: 15.7s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))