Average Error: 12.0 → 2.6
Time: 10.2s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.727842335792455 \cdot 10^{-115}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;x \le 6.94029728708334437 \cdot 10^{-305}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{y - z}}{x}}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.727842335792455 \cdot 10^{-115}:\\
\;\;\;\;x \cdot \frac{y - z}{y}\\

\mathbf{elif}\;x \le 6.94029728708334437 \cdot 10^{-305}:\\
\;\;\;\;x - \frac{x \cdot z}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r908508 = x;
        double r908509 = y;
        double r908510 = z;
        double r908511 = r908509 - r908510;
        double r908512 = r908508 * r908511;
        double r908513 = r908512 / r908509;
        return r908513;
}
↓
double f(double x, double y, double z) {
        double r908514 = x;
        double r908515 = -4.727842335792455e-115;
        bool r908516 = r908514 <= r908515;
        double r908517 = y;
        double r908518 = z;
        double r908519 = r908517 - r908518;
        double r908520 = r908519 / r908517;
        double r908521 = r908514 * r908520;
        double r908522 = 6.940297287083344e-305;
        bool r908523 = r908514 <= r908522;
        double r908524 = r908514 * r908518;
        double r908525 = r908524 / r908517;
        double r908526 = r908514 - r908525;
        double r908527 = 1.0;
        double r908528 = r908517 / r908519;
        double r908529 = r908528 / r908514;
        double r908530 = r908527 / r908529;
        double r908531 = r908523 ? r908526 : r908530;
        double r908532 = r908516 ? r908521 : r908531;
        return r908532;
}

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

Original12.0
Target3.3
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 < -4.727842335792455e-115

    1. Initial program 14.6

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

    3. Applied *-un-lft-identity14.6

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

    4. Applied times-frac0.9

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

    5. Simplified0.9

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

    if -4.727842335792455e-115 < x < 6.940297287083344e-305

    1. Initial program 7.2

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

    3. Applied associate-/l*7.3

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

    4. Taylor expanded around 0 4.0

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

    if 6.940297287083344e-305 < x

    1. Initial program 12.1

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

    3. Applied associate-/l*3.0

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

    5. Applied div-inv3.2

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

    7. Applied clear-num3.2

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

    8. Simplified3.2

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.727842335792455 \cdot 10^{-115}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;x \le 6.94029728708334437 \cdot 10^{-305}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{y - z}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))