Average Error: 12.4 → 2.9
Time: 13.0s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.699929332195363087572042602326430898574 \cdot 10^{-162} \lor \neg \left(y \le 2.724407101896751811416130978286132894297 \cdot 10^{-107}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(y - z\right)}}\\ \end{array}\]
\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}{x \cdot \left(y - z\right)}}\\

\end{array}
double f(double x, double y, double z) {
        double r500920 = x;
        double r500921 = y;
        double r500922 = z;
        double r500923 = r500921 - r500922;
        double r500924 = r500920 * r500923;
        double r500925 = r500924 / r500921;
        return r500925;
}
↓
double f(double x, double y, double z) {
        double r500926 = y;
        double r500927 = -7.699929332195363e-162;
        bool r500928 = r500926 <= r500927;
        double r500929 = 2.7244071018967518e-107;
        bool r500930 = r500926 <= r500929;
        double r500931 = !r500930;
        bool r500932 = r500928 || r500931;
        double r500933 = x;
        double r500934 = 1.0;
        double r500935 = z;
        double r500936 = r500935 / r500926;
        double r500937 = r500934 - r500936;
        double r500938 = r500933 * r500937;
        double r500939 = r500926 - r500935;
        double r500940 = r500933 * r500939;
        double r500941 = r500926 / r500940;
        double r500942 = r500934 / r500941;
        double r500943 = r500932 ? r500938 : r500942;
        return r500943;
}

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 *-un-lft-identity11.1

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

    8. Applied *-un-lft-identity11.1

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

    9. Applied *-un-lft-identity11.1

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

    10. Applied times-frac11.1

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

    11. Applied times-frac11.1

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

    12. Simplified11.1

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

    13. Simplified9.5

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

  4. Final simplification2.9

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

Reproduce

herbie shell --seed 2019325 
(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))