Average Error: 11.9 → 2.4
Time: 3.5s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{t - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.30440867097708453 \cdot 10^{-106} \lor \neg \left(z \le 4.5683127974833349 \cdot 10^{-53}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{t - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.30440867097708453 \cdot 10^{-106} \lor \neg \left(z \le 4.5683127974833349 \cdot 10^{-53}\right):\\
\;\;\;\;x \cdot \frac{y - z}{t - z}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r548490 = x;
        double r548491 = y;
        double r548492 = z;
        double r548493 = r548491 - r548492;
        double r548494 = r548490 * r548493;
        double r548495 = t;
        double r548496 = r548495 - r548492;
        double r548497 = r548494 / r548496;
        return r548497;
}
↓
double f(double x, double y, double z, double t) {
        double r548498 = z;
        double r548499 = -4.3044086709770845e-106;
        bool r548500 = r548498 <= r548499;
        double r548501 = 4.568312797483335e-53;
        bool r548502 = r548498 <= r548501;
        double r548503 = !r548502;
        bool r548504 = r548500 || r548503;
        double r548505 = x;
        double r548506 = y;
        double r548507 = r548506 - r548498;
        double r548508 = t;
        double r548509 = r548508 - r548498;
        double r548510 = r548507 / r548509;
        double r548511 = r548505 * r548510;
        double r548512 = r548505 / r548509;
        double r548513 = r548512 * r548507;
        double r548514 = r548504 ? r548511 : r548513;
        return r548514;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.9
Target2.2
Herbie2.4
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.3044086709770845e-106 or 4.568312797483335e-53 < z

    1. Initial program 15.3

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

    3. Applied *-un-lft-identity15.3

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

    4. Applied times-frac0.5

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

    5. Simplified0.5

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

    if -4.3044086709770845e-106 < z < 4.568312797483335e-53

    1. Initial program 5.6

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

    3. Applied associate-/l*5.3

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

    5. Applied associate-/r/6.0

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.30440867097708453 \cdot 10^{-106} \lor \neg \left(z \le 4.5683127974833349 \cdot 10^{-53}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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