Average Error: 10.5 → 0.2
Time: 11.5s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.042091540010478640221144318013957849902 \cdot 10^{-28}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y}{z} \cdot x\\ \mathbf{elif}\;z \le 3.580267717967572005052386658208804661473 \cdot 10^{-87}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y}{z} \cdot x\\ \end{array}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.042091540010478640221144318013957849902 \cdot 10^{-28}:\\
\;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y}{z} \cdot x\\

\mathbf{elif}\;z \le 3.580267717967572005052386658208804661473 \cdot 10^{-87}:\\
\;\;\;\;\left(y + \frac{x}{z}\right) - \frac{1}{\frac{z}{y \cdot x}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r38990365 = x;
        double r38990366 = y;
        double r38990367 = z;
        double r38990368 = r38990367 - r38990365;
        double r38990369 = r38990366 * r38990368;
        double r38990370 = r38990365 + r38990369;
        double r38990371 = r38990370 / r38990367;
        return r38990371;
}
↓
double f(double x, double y, double z) {
        double r38990372 = z;
        double r38990373 = -6.042091540010479e-28;
        bool r38990374 = r38990372 <= r38990373;
        double r38990375 = y;
        double r38990376 = x;
        double r38990377 = r38990376 / r38990372;
        double r38990378 = r38990375 + r38990377;
        double r38990379 = r38990375 / r38990372;
        double r38990380 = r38990379 * r38990376;
        double r38990381 = r38990378 - r38990380;
        double r38990382 = 3.580267717967572e-87;
        bool r38990383 = r38990372 <= r38990382;
        double r38990384 = 1.0;
        double r38990385 = r38990375 * r38990376;
        double r38990386 = r38990372 / r38990385;
        double r38990387 = r38990384 / r38990386;
        double r38990388 = r38990378 - r38990387;
        double r38990389 = r38990383 ? r38990388 : r38990381;
        double r38990390 = r38990374 ? r38990381 : r38990389;
        return r38990390;
}

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

Original10.5
Target0.0
Herbie0.2
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.042091540010479e-28 or 3.580267717967572e-87 < z

    1. Initial program 15.0

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

    2. Taylor expanded around 0 4.9

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

    4. Applied *-un-lft-identity4.9

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

    5. Applied times-frac0.3

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

    6. Simplified0.3

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

    if -6.042091540010479e-28 < z < 3.580267717967572e-87

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    4. Applied clear-num0.1

      \[\leadsto \left(y + \frac{x}{z}\right) - \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.042091540010478640221144318013957849902 \cdot 10^{-28}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y}{z} \cdot x\\ \mathbf{elif}\;z \le 3.580267717967572005052386658208804661473 \cdot 10^{-87}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y}{z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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