Average Error: 1.7 → 0.3
Time: 4.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.09954904008270574 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 4360323.74267882574:\\ \;\;\;\;\left|\frac{x + 4}{y} - 1 \cdot \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.09954904008270574 \cdot 10^{-121}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 4360323.74267882574:\\
\;\;\;\;\left|\frac{x + 4}{y} - 1 \cdot \frac{x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r35157 = x;
        double r35158 = 4.0;
        double r35159 = r35157 + r35158;
        double r35160 = y;
        double r35161 = r35159 / r35160;
        double r35162 = r35157 / r35160;
        double r35163 = z;
        double r35164 = r35162 * r35163;
        double r35165 = r35161 - r35164;
        double r35166 = fabs(r35165);
        return r35166;
}
↓
double f(double x, double y, double z) {
        double r35167 = x;
        double r35168 = -1.0995490400827057e-121;
        bool r35169 = r35167 <= r35168;
        double r35170 = 4.0;
        double r35171 = r35167 + r35170;
        double r35172 = y;
        double r35173 = r35171 / r35172;
        double r35174 = z;
        double r35175 = r35174 / r35172;
        double r35176 = r35167 * r35175;
        double r35177 = r35173 - r35176;
        double r35178 = fabs(r35177);
        double r35179 = 4360323.742678826;
        bool r35180 = r35167 <= r35179;
        double r35181 = 1.0;
        double r35182 = r35167 * r35174;
        double r35183 = r35182 / r35172;
        double r35184 = r35181 * r35183;
        double r35185 = r35173 - r35184;
        double r35186 = fabs(r35185);
        double r35187 = r35167 / r35172;
        double r35188 = r35187 * r35174;
        double r35189 = r35173 - r35188;
        double r35190 = fabs(r35189);
        double r35191 = r35180 ? r35186 : r35190;
        double r35192 = r35169 ? r35178 : r35191;
        return r35192;
}

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 < -1.0995490400827057e-121

    1. Initial program 0.9

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

    3. Applied div-inv0.9

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

    4. Applied associate-*l*1.0

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

    5. Simplified0.9

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

    if -1.0995490400827057e-121 < x < 4360323.742678826

    1. Initial program 2.8

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

    3. Applied add-cube-cbrt3.0

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

    4. Applied *-un-lft-identity3.0

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

    5. Applied times-frac3.0

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

    6. Applied associate-*l*0.5

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

    8. Applied *-un-lft-identity0.5

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

    9. Applied associate-*l*0.5

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

    10. Simplified0.1

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

    if 4360323.742678826 < x

    1. Initial program 0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.09954904008270574 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 4360323.74267882574:\\ \;\;\;\;\left|\frac{x + 4}{y} - 1 \cdot \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))