Average Error: 12.4 → 2.9
Time: 17.8s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.699929332195363087572042602326430898574 \cdot 10^{-162} \lor \neg \left(y \le 2.724407101896751811416130978286132894297 \cdot 10^{-107}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\left(y - z\right) \cdot x}}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.699929332195363087572042602326430898574 \cdot 10^{-162} \lor \neg \left(y \le 2.724407101896751811416130978286132894297 \cdot 10^{-107}\right):\\
\;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r594389 = x;
        double r594390 = y;
        double r594391 = z;
        double r594392 = r594390 - r594391;
        double r594393 = r594389 * r594392;
        double r594394 = r594393 / r594390;
        return r594394;
}
↓
double f(double x, double y, double z) {
        double r594395 = y;
        double r594396 = -7.699929332195363e-162;
        bool r594397 = r594395 <= r594396;
        double r594398 = 2.7244071018967518e-107;
        bool r594399 = r594395 <= r594398;
        double r594400 = !r594399;
        bool r594401 = r594397 || r594400;
        double r594402 = x;
        double r594403 = 1.0;
        double r594404 = z;
        double r594405 = r594404 / r594395;
        double r594406 = r594403 - r594405;
        double r594407 = r594402 * r594406;
        double r594408 = r594395 - r594404;
        double r594409 = r594408 * r594402;
        double r594410 = r594395 / r594409;
        double r594411 = r594403 / r594410;
        double r594412 = r594401 ? r594407 : r594411;
        return r594412;
}

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

Original12.4
Target3.2
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 2 regimes

  2. if y < -7.699929332195363e-162 or 2.7244071018967518e-107 < y

    1. Initial program 13.2

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

    3. Applied associate-/l*0.9

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

    5. Applied div-inv1.2

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

    6. Simplified1.1

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

    if -7.699929332195363e-162 < y < 2.7244071018967518e-107

    1. Initial program 9.5

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

    3. Applied associate-/l*11.1

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

    5. Applied clear-num11.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{y - z}}{x}}}\]
    6. Using strategy rm

    7. Applied div-inv11.2

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

    8. Applied associate-/l*9.6

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

    9. Simplified9.5

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.699929332195363087572042602326430898574 \cdot 10^{-162} \lor \neg \left(y \le 2.724407101896751811416130978286132894297 \cdot 10^{-107}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\left(y - z\right) \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))