Average Error: 1.8 → 0.2
Time: 15.2s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 1.1771995155236972 \cdot 10^{+116}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \left(\frac{1}{y} \cdot z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

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 (fabs (- (/ (+ x 4) y) (* (/ x y) z))) < 1.1771995155236972e+116

    1. Initial program 2.9

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

    3. Applied div-inv2.9

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

    4. Applied associate-*l*0.3

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

    if 1.1771995155236972e+116 < (fabs (- (/ (+ x 4) y) (* (/ x y) z)))

    1. Initial program 0.1

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

    2. Taylor expanded around -inf 6.7

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

    3. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 1.1771995155236972 \cdot 10^{+116}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \left(\frac{1}{y} \cdot z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Runtime

Time bar (total: 15.2s)Debug logProfile

herbie shell --seed 2018230 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))