Average Error: 12.0 → 2.4
Time: 3.4s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{t - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.5411015698060821 \cdot 10^{-184} \lor \neg \left(z \le 2.77988840679091214 \cdot 10^{-239}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t - z} \cdot \left(x \cdot \left(y - z\right)\right)\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{t - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.5411015698060821 \cdot 10^{-184} \lor \neg \left(z \le 2.77988840679091214 \cdot 10^{-239}\right):\\
\;\;\;\;x \cdot \frac{y - z}{t - z}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r709654 = x;
        double r709655 = y;
        double r709656 = z;
        double r709657 = r709655 - r709656;
        double r709658 = r709654 * r709657;
        double r709659 = t;
        double r709660 = r709659 - r709656;
        double r709661 = r709658 / r709660;
        return r709661;
}
↓
double f(double x, double y, double z, double t) {
        double r709662 = z;
        double r709663 = -7.541101569806082e-184;
        bool r709664 = r709662 <= r709663;
        double r709665 = 2.779888406790912e-239;
        bool r709666 = r709662 <= r709665;
        double r709667 = !r709666;
        bool r709668 = r709664 || r709667;
        double r709669 = x;
        double r709670 = y;
        double r709671 = r709670 - r709662;
        double r709672 = t;
        double r709673 = r709672 - r709662;
        double r709674 = r709671 / r709673;
        double r709675 = r709669 * r709674;
        double r709676 = 1.0;
        double r709677 = r709676 / r709673;
        double r709678 = r709669 * r709671;
        double r709679 = r709677 * r709678;
        double r709680 = r709668 ? r709675 : r709679;
        return r709680;
}

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

Original12.0
Target2.4
Herbie2.4
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.541101569806082e-184 or 2.779888406790912e-239 < z

    1. Initial program 12.8

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

    3. Applied *-un-lft-identity12.8

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

    4. Applied times-frac1.6

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

    5. Simplified1.6

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

    if -7.541101569806082e-184 < z < 2.779888406790912e-239

    1. Initial program 7.2

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

    3. Applied associate-/l*7.0

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

    5. Applied div-inv7.1

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

    6. Applied *-un-lft-identity7.1

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

    7. Applied times-frac7.3

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

    8. Simplified7.3

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.5411015698060821 \cdot 10^{-184} \lor \neg \left(z \le 2.77988840679091214 \cdot 10^{-239}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t - z} \cdot \left(x \cdot \left(y - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))