Average Error: 1.7 → 0.3
Time: 3.8s
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 r31551 = x;
        double r31552 = 4.0;
        double r31553 = r31551 + r31552;
        double r31554 = y;
        double r31555 = r31553 / r31554;
        double r31556 = r31551 / r31554;
        double r31557 = z;
        double r31558 = r31556 * r31557;
        double r31559 = r31555 - r31558;
        double r31560 = fabs(r31559);
        return r31560;
}
↓
double f(double x, double y, double z) {
        double r31561 = x;
        double r31562 = -1.0995490400827057e-121;
        bool r31563 = r31561 <= r31562;
        double r31564 = 4.0;
        double r31565 = r31561 + r31564;
        double r31566 = y;
        double r31567 = r31565 / r31566;
        double r31568 = z;
        double r31569 = r31568 / r31566;
        double r31570 = r31561 * r31569;
        double r31571 = r31567 - r31570;
        double r31572 = fabs(r31571);
        double r31573 = 4360323.742678826;
        bool r31574 = r31561 <= r31573;
        double r31575 = 1.0;
        double r31576 = r31561 * r31568;
        double r31577 = r31576 / r31566;
        double r31578 = r31575 * r31577;
        double r31579 = r31567 - r31578;
        double r31580 = fabs(r31579);
        double r31581 = r31561 / r31566;
        double r31582 = r31581 * r31568;
        double r31583 = r31567 - r31582;
        double r31584 = fabs(r31583);
        double r31585 = r31574 ? r31580 : r31584;
        double r31586 = r31563 ? r31572 : r31585;
        return r31586;
}

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