Average Error: 1.9 → 0.1
Time: 2.7m
Precision: 64
Internal Precision: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.126552356689793 \cdot 10^{-49} \lor \neg \left(y \le 6.082729899443606 \cdot 10^{-54}\right):\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if y < -3.126552356689793e-49 or 6.082729899443606e-54 < y

    1. Initial program 2.7

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

    2. Initial simplification0.2

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

    if -3.126552356689793e-49 < y < 6.082729899443606e-54

    1. Initial program 0.1

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

    2. Initial simplification8.4

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

    4. Applied clear-num8.4

      \[\leadsto \left|\frac{4 + x}{y} - \color{blue}{\frac{1}{\frac{\frac{y}{z}}{x}}}\right|\]
    5. Using strategy rm

    6. Applied div-inv8.4

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

    7. Applied associate-/r*9.4

      \[\leadsto \left|\frac{4 + x}{y} - \color{blue}{\frac{\frac{1}{\frac{y}{z}}}{\frac{1}{x}}}\right|\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt33.7

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

    10. Applied *-un-lft-identity33.7

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

    11. Applied prod-diff33.7

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

    12. Simplified33.7

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

    13. Simplified0.1

      \[\leadsto \left|\frac{\left(4 + x\right) - z \cdot x}{y} + \color{blue}{0}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.126552356689793 \cdot 10^{-49} \lor \neg \left(y \le 6.082729899443606 \cdot 10^{-54}\right):\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes2.60.10.02.595.9%
herbie shell --seed 2018354 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))