Average Error: 1.5 → 1.2
Time: 31.1s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 2.4249468621417827 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \left(\sqrt[3]{\frac{1}{y}} \cdot x\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{\frac{y}{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}}{z}}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le 2.4249468621417827 \cdot 10^{-83}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \left(\sqrt[3]{\frac{1}{y}} \cdot x\right)\right)\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{\frac{y}{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}}{z}}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r1452220 = x;
        double r1452221 = 4.0;
        double r1452222 = r1452220 + r1452221;
        double r1452223 = y;
        double r1452224 = r1452222 / r1452223;
        double r1452225 = r1452220 / r1452223;
        double r1452226 = z;
        double r1452227 = r1452225 * r1452226;
        double r1452228 = r1452224 - r1452227;
        double r1452229 = fabs(r1452228);
        return r1452229;
}
↓
double f(double x, double y, double z) {
        double r1452230 = y;
        double r1452231 = 2.4249468621417827e-83;
        bool r1452232 = r1452230 <= r1452231;
        double r1452233 = x;
        double r1452234 = 4.0;
        double r1452235 = r1452233 + r1452234;
        double r1452236 = r1452235 / r1452230;
        double r1452237 = 1.0;
        double r1452238 = cbrt(r1452230);
        double r1452239 = r1452238 * r1452238;
        double r1452240 = r1452237 / r1452239;
        double r1452241 = z;
        double r1452242 = r1452237 / r1452230;
        double r1452243 = cbrt(r1452242);
        double r1452244 = r1452243 * r1452233;
        double r1452245 = r1452241 * r1452244;
        double r1452246 = r1452240 * r1452245;
        double r1452247 = r1452236 - r1452246;
        double r1452248 = fabs(r1452247);
        double r1452249 = -1.0;
        double r1452250 = cbrt(r1452249);
        double r1452251 = r1452250 * r1452250;
        double r1452252 = r1452230 / r1452251;
        double r1452253 = r1452252 / r1452241;
        double r1452254 = r1452233 / r1452253;
        double r1452255 = r1452236 - r1452254;
        double r1452256 = fabs(r1452255);
        double r1452257 = r1452232 ? r1452248 : r1452256;
        return r1452257;
}

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 2 regimes

  2. if y < 2.4249468621417827e-83

    1. Initial program 1.3

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

    3. Applied add-cube-cbrt1.5

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

    4. Applied *-un-lft-identity1.5

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

    5. Applied times-frac1.6

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

    6. Applied associate-*l*1.8

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

    7. Taylor expanded around inf 48.9

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

    8. Simplified1.7

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

    if 2.4249468621417827e-83 < y

    1. Initial program 1.9

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

    3. Applied add-cube-cbrt2.2

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

    4. Applied *-un-lft-identity2.2

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

    5. Applied times-frac2.2

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

    6. Applied associate-*l*2.7

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

    7. Taylor expanded around inf 5.4

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

    8. Simplified2.7

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

    10. Applied add-cube-cbrt2.7

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

    11. Applied associate-*r*2.7

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

    12. Taylor expanded around -inf 4.3

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot \left({\left(\sqrt[3]{-1}\right)}^{2} \cdot z\right)}{y}}\right|\]

    13. Simplified0.4

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x}{\frac{\frac{y}{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}}{z}}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 2.4249468621417827 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \left(\sqrt[3]{\frac{1}{y}} \cdot x\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{\frac{y}{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}}{z}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))