Average Error: 12.5 → 2.9
Time: 16.2s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.987697433686588590285611969201440389173 \cdot 10^{-172}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;x \le 1.318326588609397900170163534445278958933 \cdot 10^{-303}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{y - z} \cdot y}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.987697433686588590285611969201440389173 \cdot 10^{-172}:\\
\;\;\;\;x \cdot \frac{y - z}{y}\\

\mathbf{elif}\;x \le 1.318326588609397900170163534445278958933 \cdot 10^{-303}:\\
\;\;\;\;x - \frac{z \cdot x}{y}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{1}{y - z} \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r31486447 = x;
        double r31486448 = y;
        double r31486449 = z;
        double r31486450 = r31486448 - r31486449;
        double r31486451 = r31486447 * r31486450;
        double r31486452 = r31486451 / r31486448;
        return r31486452;
}
↓
double f(double x, double y, double z) {
        double r31486453 = x;
        double r31486454 = -7.987697433686589e-172;
        bool r31486455 = r31486453 <= r31486454;
        double r31486456 = y;
        double r31486457 = z;
        double r31486458 = r31486456 - r31486457;
        double r31486459 = r31486458 / r31486456;
        double r31486460 = r31486453 * r31486459;
        double r31486461 = 1.3183265886093979e-303;
        bool r31486462 = r31486453 <= r31486461;
        double r31486463 = r31486457 * r31486453;
        double r31486464 = r31486463 / r31486456;
        double r31486465 = r31486453 - r31486464;
        double r31486466 = 1.0;
        double r31486467 = r31486466 / r31486458;
        double r31486468 = r31486467 * r31486456;
        double r31486469 = r31486453 / r31486468;
        double r31486470 = r31486462 ? r31486465 : r31486469;
        double r31486471 = r31486455 ? r31486460 : r31486470;
        return r31486471;
}

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

Original12.5
Target2.9
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -7.987697433686589e-172

    1. Initial program 14.3

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

    3. Applied *-un-lft-identity14.3

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

    4. Applied times-frac1.8

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

    5. Simplified1.8

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

    if -7.987697433686589e-172 < x < 1.3183265886093979e-303

    1. Initial program 8.3

      \[\frac{x \cdot \left(y - z\right)}{y}\]

    2. Taylor expanded around 0 4.9

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

    if 1.3183265886093979e-303 < x

    1. Initial program 12.3

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

    3. Applied associate-/l*3.1

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}}\]
    4. Using strategy rm

    5. Applied div-inv3.3

      \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{1}{y - z}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.987697433686588590285611969201440389173 \cdot 10^{-172}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;x \le 1.318326588609397900170163534445278958933 \cdot 10^{-303}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{y - z} \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"

  :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))