Average Error: 12.2 → 2.6
Time: 2.6s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 5.8561005025320592 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;x \le 7.80513442966517965 \cdot 10^{-43}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le 5.8561005025320592 \cdot 10^{-177}:\\
\;\;\;\;\frac{x}{\frac{y}{y - z}}\\

\mathbf{elif}\;x \le 7.80513442966517965 \cdot 10^{-43}:\\
\;\;\;\;x - \frac{x \cdot z}{y}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{y - z}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r883430 = x;
        double r883431 = y;
        double r883432 = z;
        double r883433 = r883431 - r883432;
        double r883434 = r883430 * r883433;
        double r883435 = r883434 / r883431;
        return r883435;
}
↓
double f(double x, double y, double z) {
        double r883436 = x;
        double r883437 = 5.856100502532059e-177;
        bool r883438 = r883436 <= r883437;
        double r883439 = y;
        double r883440 = z;
        double r883441 = r883439 - r883440;
        double r883442 = r883439 / r883441;
        double r883443 = r883436 / r883442;
        double r883444 = 7.80513442966518e-43;
        bool r883445 = r883436 <= r883444;
        double r883446 = r883436 * r883440;
        double r883447 = r883446 / r883439;
        double r883448 = r883436 - r883447;
        double r883449 = r883441 / r883439;
        double r883450 = r883436 * r883449;
        double r883451 = r883445 ? r883448 : r883450;
        double r883452 = r883438 ? r883443 : r883451;
        return r883452;
}

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

Target

Original12.2
Target3.0
Herbie2.6
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.69397660138285259 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < 5.856100502532059e-177

    1. Initial program 11.4

      \[\frac{x \cdot \left(y - z\right)}{y}\]
    2. Using strategy rm

    3. Applied associate-/l*3.8

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}}\]

    if 5.856100502532059e-177 < x < 7.80513442966518e-43

    1. Initial program 1.9

      \[\frac{x \cdot \left(y - z\right)}{y}\]
    2. Using strategy rm

    3. Applied associate-/l*4.9

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}}\]

    4. Taylor expanded around 0 1.0

      \[\leadsto \color{blue}{x - \frac{x \cdot z}{y}}\]

    if 7.80513442966518e-43 < x

    1. Initial program 19.0

      \[\frac{x \cdot \left(y - z\right)}{y}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity19.0

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

    4. Applied times-frac0.4

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y - z}{y}}\]

    5. Simplified0.4

      \[\leadsto \color{blue}{x} \cdot \frac{y - z}{y}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.8561005025320592 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;x \le 7.80513442966517965 \cdot 10^{-43}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))