Average Error: 1.6 → 0.2
Time: 13.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.4332739855837574 \cdot 10^{73}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;y \le 1.4318648069754203 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.4332739855837574 \cdot 10^{73}:\\
\;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\

\mathbf{elif}\;y \le 1.4318648069754203 \cdot 10^{-23}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r24219 = x;
        double r24220 = 4.0;
        double r24221 = r24219 + r24220;
        double r24222 = y;
        double r24223 = r24221 / r24222;
        double r24224 = r24219 / r24222;
        double r24225 = z;
        double r24226 = r24224 * r24225;
        double r24227 = r24223 - r24226;
        double r24228 = fabs(r24227);
        return r24228;
}
↓
double f(double x, double y, double z) {
        double r24229 = y;
        double r24230 = -5.4332739855837574e+73;
        bool r24231 = r24229 <= r24230;
        double r24232 = x;
        double r24233 = r24232 / r24229;
        double r24234 = 4.0;
        double r24235 = r24234 / r24229;
        double r24236 = r24233 + r24235;
        double r24237 = z;
        double r24238 = r24229 / r24237;
        double r24239 = r24232 / r24238;
        double r24240 = r24236 - r24239;
        double r24241 = fabs(r24240);
        double r24242 = 1.4318648069754203e-23;
        bool r24243 = r24229 <= r24242;
        double r24244 = r24232 + r24234;
        double r24245 = r24244 / r24229;
        double r24246 = r24232 * r24237;
        double r24247 = r24246 / r24229;
        double r24248 = r24245 - r24247;
        double r24249 = fabs(r24248);
        double r24250 = r24237 / r24229;
        double r24251 = r24232 * r24250;
        double r24252 = r24236 - r24251;
        double r24253 = fabs(r24252);
        double r24254 = r24243 ? r24249 : r24253;
        double r24255 = r24231 ? r24241 : r24254;
        return r24255;
}

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 y < -5.4332739855837574e+73

    1. Initial program 3.7

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

    2. Taylor expanded around 0 3.7

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

    3. Simplified3.7

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

    5. Applied pow13.7

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

    6. Applied pow13.7

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

    7. Applied pow-prod-down3.7

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

    8. Simplified0.1

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

    if -5.4332739855837574e+73 < y < 1.4318648069754203e-23

    1. Initial program 0.1

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

    3. Applied associate-*l/0.2

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

    if 1.4318648069754203e-23 < y

    1. Initial program 2.1

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

    2. Taylor expanded around 0 2.1

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

    3. Simplified2.1

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

    5. Applied div-inv2.1

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

    6. Applied associate-*l*0.2

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

    7. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.4332739855837574 \cdot 10^{73}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;y \le 1.4318648069754203 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))