Average Error: 1.5 → 0.1
Time: 14.1s
Precision: 64
Internal Precision: 320
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.1305845923678062 \cdot 10^{+54}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \le 3.624279614891611 \cdot 10^{-46}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Error

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Bits error versus y

Bits error versus z

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Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -2.1305845923678062e+54

    1. Initial program 0.1

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

    3. Applied associate-*l/10.7

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

    4. Applied sub-div10.7

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

    5. Taylor expanded around -inf 10.7

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

    6. Simplified0.1

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

    if -2.1305845923678062e+54 < x < 3.624279614891611e-46

    1. Initial program 2.4

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

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

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

    if 3.624279614891611e-46 < x

    1. Initial program 0.2

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.1305845923678062 \cdot 10^{+54}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \le 3.624279614891611 \cdot 10^{-46}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Runtime

Time bar (total: 14.1s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))