Average Error: 1.5 → 0.4
Time: 12.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 6.465235870316558323202068264318582009221 \cdot 10^{-120}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -16246640829.1726360321044921875:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 6.465235870316558323202068264318582009221 \cdot 10^{-120}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r1675362 = x;
        double r1675363 = 4.0;
        double r1675364 = r1675362 + r1675363;
        double r1675365 = y;
        double r1675366 = r1675364 / r1675365;
        double r1675367 = r1675362 / r1675365;
        double r1675368 = z;
        double r1675369 = r1675367 * r1675368;
        double r1675370 = r1675366 - r1675369;
        double r1675371 = fabs(r1675370);
        return r1675371;
}
↓
double f(double x, double y, double z) {
        double r1675372 = x;
        double r1675373 = -16246640829.172636;
        bool r1675374 = r1675372 <= r1675373;
        double r1675375 = 4.0;
        double r1675376 = y;
        double r1675377 = r1675375 / r1675376;
        double r1675378 = r1675372 / r1675376;
        double r1675379 = r1675377 + r1675378;
        double r1675380 = z;
        double r1675381 = r1675378 * r1675380;
        double r1675382 = r1675379 - r1675381;
        double r1675383 = fabs(r1675382);
        double r1675384 = 6.465235870316558e-120;
        bool r1675385 = r1675372 <= r1675384;
        double r1675386 = r1675375 + r1675372;
        double r1675387 = r1675372 * r1675380;
        double r1675388 = r1675386 - r1675387;
        double r1675389 = r1675388 / r1675376;
        double r1675390 = fabs(r1675389);
        double r1675391 = r1675376 / r1675380;
        double r1675392 = r1675372 / r1675391;
        double r1675393 = r1675379 - r1675392;
        double r1675394 = fabs(r1675393);
        double r1675395 = r1675385 ? r1675390 : r1675394;
        double r1675396 = r1675374 ? r1675383 : r1675395;
        return r1675396;
}

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 < -16246640829.172636

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if -16246640829.172636 < x < 6.465235870316558e-120

    1. Initial program 2.6

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 6.465235870316558e-120 < x

    1. Initial program 0.7

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.7

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

    5. Applied add-cube-cbrt1.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{y} \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right|\]

    6. Applied associate-*r*1.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \color{blue}{\left(\frac{x}{y} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}}\right|\]
    7. Using strategy rm

    8. Applied pow11.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \left(\frac{x}{y} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{z}\right)}^{1}}\right|\]

    9. Applied pow11.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \left(\frac{x}{y} \cdot \left(\sqrt[3]{z} \cdot \color{blue}{{\left(\sqrt[3]{z}\right)}^{1}}\right)\right) \cdot {\left(\sqrt[3]{z}\right)}^{1}\right|\]

    10. Applied pow11.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \left(\frac{x}{y} \cdot \left(\color{blue}{{\left(\sqrt[3]{z}\right)}^{1}} \cdot {\left(\sqrt[3]{z}\right)}^{1}\right)\right) \cdot {\left(\sqrt[3]{z}\right)}^{1}\right|\]

    11. Applied pow-prod-down1.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \left(\frac{x}{y} \cdot \color{blue}{{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{z}\right)}^{1}\right|\]

    12. Applied pow11.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} \cdot {\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}^{1}\right) \cdot {\left(\sqrt[3]{z}\right)}^{1}\right|\]

    13. Applied pow-prod-down1.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \color{blue}{{\left(\frac{x}{y} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)}^{1}} \cdot {\left(\sqrt[3]{z}\right)}^{1}\right|\]

    14. Applied pow-prod-down1.0

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \color{blue}{{\left(\left(\frac{x}{y} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}\right)}^{1}}\right|\]

    15. Simplified1.1

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - {\color{blue}{\left(\frac{x}{\frac{y}{z}}\right)}}^{1}\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 6.465235870316558323202068264318582009221 \cdot 10^{-120}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))