Average Error: 2.0 → 2.1
Time: 14.1s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.84981784046249300200636436486704057411 \cdot 10^{-181} \lor \neg \left(z \le 1.036800313662513774601491787735915867558 \cdot 10^{-190}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right) + t\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.84981784046249300200636436486704057411 \cdot 10^{-181} \lor \neg \left(z \le 1.036800313662513774601491787735915867558 \cdot 10^{-190}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r536976 = x;
        double r536977 = y;
        double r536978 = r536976 / r536977;
        double r536979 = z;
        double r536980 = t;
        double r536981 = r536979 - r536980;
        double r536982 = r536978 * r536981;
        double r536983 = r536982 + r536980;
        return r536983;
}
↓
double f(double x, double y, double z, double t) {
        double r536984 = z;
        double r536985 = -1.849817840462493e-181;
        bool r536986 = r536984 <= r536985;
        double r536987 = 1.0368003136625138e-190;
        bool r536988 = r536984 <= r536987;
        double r536989 = !r536988;
        bool r536990 = r536986 || r536989;
        double r536991 = x;
        double r536992 = y;
        double r536993 = r536991 / r536992;
        double r536994 = t;
        double r536995 = r536984 - r536994;
        double r536996 = fma(r536993, r536995, r536994);
        double r536997 = r536991 * r536984;
        double r536998 = r536997 / r536992;
        double r536999 = r536994 * r536991;
        double r537000 = r536999 / r536992;
        double r537001 = r536998 - r537000;
        double r537002 = r537001 + r536994;
        double r537003 = r536990 ? r536996 : r537002;
        return r537003;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.0
Target2.1
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.849817840462493e-181 or 1.0368003136625138e-190 < z

    1. Initial program 1.6

      \[\frac{x}{y} \cdot \left(z - t\right) + t\]

    2. Simplified1.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)}\]

    if -1.849817840462493e-181 < z < 1.0368003136625138e-190

    1. Initial program 3.5

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

    3. Applied div-inv3.5

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

    4. Applied associate-*l*4.6

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

    5. Simplified4.5

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

    7. Applied add-cube-cbrt4.9

      \[\leadsto x \cdot \frac{z - t}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} + t\]

    8. Applied *-un-lft-identity4.9

      \[\leadsto x \cdot \frac{\color{blue}{1 \cdot \left(z - t\right)}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} + t\]

    9. Applied times-frac4.9

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z - t}{\sqrt[3]{y}}\right)} + t\]

    10. Applied associate-*r*2.9

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{z - t}{\sqrt[3]{y}}} + t\]

    11. Simplified2.9

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{z - t}{\sqrt[3]{y}} + t\]

    12. Taylor expanded around 0 3.9

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.84981784046249300200636436486704057411 \cdot 10^{-181} \lor \neg \left(z \le 1.036800313662513774601491787735915867558 \cdot 10^{-190}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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