Average Error: 1.6 → 0.4
Time: 8.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.685625371565923148712864066413058588036 \cdot 10^{53}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 9.897501881007222830200809530769256909371 \cdot 10^{-104}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.685625371565923148712864066413058588036 \cdot 10^{53}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 9.897501881007222830200809530769256909371 \cdot 10^{-104}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r39311 = x;
        double r39312 = 4.0;
        double r39313 = r39311 + r39312;
        double r39314 = y;
        double r39315 = r39313 / r39314;
        double r39316 = r39311 / r39314;
        double r39317 = z;
        double r39318 = r39316 * r39317;
        double r39319 = r39315 - r39318;
        double r39320 = fabs(r39319);
        return r39320;
}
↓
double f(double x, double y, double z) {
        double r39321 = x;
        double r39322 = -1.6856253715659231e+53;
        bool r39323 = r39321 <= r39322;
        double r39324 = 4.0;
        double r39325 = y;
        double r39326 = r39324 / r39325;
        double r39327 = r39321 / r39325;
        double r39328 = r39326 + r39327;
        double r39329 = z;
        double r39330 = r39329 / r39325;
        double r39331 = r39321 * r39330;
        double r39332 = r39328 - r39331;
        double r39333 = fabs(r39332);
        double r39334 = 9.897501881007223e-104;
        bool r39335 = r39321 <= r39334;
        double r39336 = r39321 * r39329;
        double r39337 = r39336 / r39325;
        double r39338 = r39328 - r39337;
        double r39339 = fabs(r39338);
        double r39340 = r39321 + r39324;
        double r39341 = r39340 / r39325;
        double r39342 = r39325 / r39329;
        double r39343 = r39321 / r39342;
        double r39344 = r39341 - r39343;
        double r39345 = fabs(r39344);
        double r39346 = r39335 ? r39339 : r39345;
        double r39347 = r39323 ? r39333 : r39346;
        return r39347;
}

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 < -1.6856253715659231e+53

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied div-inv0.1

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

    6. Applied associate-*l*0.1

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

    7. Simplified0.1

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

    if -1.6856253715659231e+53 < x < 9.897501881007223e-104

    1. Initial program 2.6

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

    2. Taylor expanded around 0 2.6

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

    3. Simplified2.6

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

    5. Applied associate-*l/0.3

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

    if 9.897501881007223e-104 < x

    1. Initial program 0.5

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

    3. Applied pow10.5

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

    4. Applied pow10.5

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

    5. Applied pow-prod-down0.5

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

    6. Simplified0.9

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.685625371565923148712864066413058588036 \cdot 10^{53}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 9.897501881007222830200809530769256909371 \cdot 10^{-104}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))