Average Error: 12.1 → 2.3
Time: 11.2s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{t - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.155924047883603900871162775757218637321 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{elif}\;z \le 8.757644994032034378013442251697187719343 \cdot 10^{-217}:\\ \;\;\;\;\left(\left(y - z\right) \cdot x\right) \cdot \frac{1}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{t - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.155924047883603900871162775757218637321 \cdot 10^{-263}:\\
\;\;\;\;x \cdot \frac{y - z}{t - z}\\

\mathbf{elif}\;z \le 8.757644994032034378013442251697187719343 \cdot 10^{-217}:\\
\;\;\;\;\left(\left(y - z\right) \cdot x\right) \cdot \frac{1}{t - z}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r591599 = x;
        double r591600 = y;
        double r591601 = z;
        double r591602 = r591600 - r591601;
        double r591603 = r591599 * r591602;
        double r591604 = t;
        double r591605 = r591604 - r591601;
        double r591606 = r591603 / r591605;
        return r591606;
}
↓
double f(double x, double y, double z, double t) {
        double r591607 = z;
        double r591608 = -2.155924047883604e-263;
        bool r591609 = r591607 <= r591608;
        double r591610 = x;
        double r591611 = y;
        double r591612 = r591611 - r591607;
        double r591613 = t;
        double r591614 = r591613 - r591607;
        double r591615 = r591612 / r591614;
        double r591616 = r591610 * r591615;
        double r591617 = 8.757644994032034e-217;
        bool r591618 = r591607 <= r591617;
        double r591619 = r591612 * r591610;
        double r591620 = 1.0;
        double r591621 = r591620 / r591614;
        double r591622 = r591619 * r591621;
        double r591623 = r591614 / r591612;
        double r591624 = r591610 / r591623;
        double r591625 = r591618 ? r591622 : r591624;
        double r591626 = r591609 ? r591616 : r591625;
        return r591626;
}

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

Original12.1
Target2.2
Herbie2.3
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.155924047883604e-263

    1. Initial program 12.6

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

    3. Applied *-un-lft-identity12.6

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

    4. Applied times-frac1.9

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

    5. Simplified1.9

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

    if -2.155924047883604e-263 < z < 8.757644994032034e-217

    1. Initial program 7.4

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

    3. Applied *-un-lft-identity7.4

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

    4. Applied times-frac6.6

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

    5. Simplified6.6

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

    7. Applied div-inv6.7

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

    8. Applied associate-*r*7.4

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

    9. Simplified7.4

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

    if 8.757644994032034e-217 < z

    1. Initial program 12.6

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

    3. Applied associate-/l*1.6

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.155924047883603900871162775757218637321 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{elif}\;z \le 8.757644994032034378013442251697187719343 \cdot 10^{-217}:\\ \;\;\;\;\left(\left(y - z\right) \cdot x\right) \cdot \frac{1}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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)))