Average Error: 1.6 → 0.8
Time: 13.1s
Precision: 64
Internal Precision: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.982244575544609 \cdot 10^{-225}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 9.198166240003989 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{\left(\frac{x}{y} + \frac{-4}{y}\right) \cdot \left(\left(\frac{4}{y} + \frac{x}{y}\right) \cdot y - x \cdot z\right)}{x - 4}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -7.982244575544609e-225

    1. Initial program 1.5

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

    2. Taylor expanded around -inf 1.5

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

    3. Simplified1.5

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

    if -7.982244575544609e-225 < x < 9.198166240003989e-54

    1. Initial program 2.6

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

    2. Taylor expanded around -inf 2.6

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

    3. Simplified2.6

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

    5. Applied associate-*l/0.1

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

    6. Applied flip-+26.5

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

    7. Applied frac-sub26.6

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

    8. Simplified0.1

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

    9. Simplified0.2

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

    if 9.198166240003989e-54 < x

    1. Initial program 0.3

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

    2. Initial simplification0.3

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.982244575544609 \cdot 10^{-225}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 9.198166240003989 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{\left(\frac{x}{y} + \frac{-4}{y}\right) \cdot \left(\left(\frac{4}{y} + \frac{x}{y}\right) \cdot y - x \cdot z\right)}{x - 4}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Runtime

Time bar (total: 13.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.60.80.01.652.4%
herbie shell --seed 2018352 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))