Average Error: 1.6 → 0.1
Time: 10.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 1.2452244086873697 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + \frac{4}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 1.2452244086873697 \cdot 10^{-13}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + \frac{4}{y}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r25934 = x;
        double r25935 = 4.0;
        double r25936 = r25934 + r25935;
        double r25937 = y;
        double r25938 = r25936 / r25937;
        double r25939 = r25934 / r25937;
        double r25940 = z;
        double r25941 = r25939 * r25940;
        double r25942 = r25938 - r25941;
        double r25943 = fabs(r25942);
        return r25943;
}
↓
double f(double x, double y, double z) {
        double r25944 = x;
        double r25945 = -1.384698452851939e-17;
        bool r25946 = r25944 <= r25945;
        double r25947 = 4.0;
        double r25948 = r25944 + r25947;
        double r25949 = y;
        double r25950 = r25948 / r25949;
        double r25951 = z;
        double r25952 = r25951 / r25949;
        double r25953 = r25944 * r25952;
        double r25954 = r25950 - r25953;
        double r25955 = fabs(r25954);
        double r25956 = 1.2452244086873697e-13;
        bool r25957 = r25944 <= r25956;
        double r25958 = r25944 * r25951;
        double r25959 = r25948 - r25958;
        double r25960 = r25959 / r25949;
        double r25961 = fabs(r25960);
        double r25962 = r25944 / r25949;
        double r25963 = 1.0;
        double r25964 = r25963 - r25951;
        double r25965 = r25962 * r25964;
        double r25966 = r25947 / r25949;
        double r25967 = r25965 + r25966;
        double r25968 = fabs(r25967);
        double r25969 = r25957 ? r25961 : r25968;
        double r25970 = r25946 ? r25955 : r25969;
        return r25970;
}

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 < -1.384698452851939e-17

    1. Initial program 0.1

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

    3. Applied div-inv0.2

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

    4. Applied associate-*l*0.2

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

    5. Simplified0.2

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

    if -1.384698452851939e-17 < x < 1.2452244086873697e-13

    1. Initial program 2.7

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

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

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

    if 1.2452244086873697e-13 < x

    1. Initial program 0.1

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

    2. Taylor expanded around 0 7.7

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 1.2452244086873697 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + \frac{4}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))