Average Error: 1.7 → 1.8
Time: 3.4s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 3.99075484149353127 \cdot 10^{-258} \lor \neg \left(t \le 3.67768671539096219 \cdot 10^{-96}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{z - t}{\sqrt[3]{y}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} + t\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;t \le 3.99075484149353127 \cdot 10^{-258} \lor \neg \left(t \le 3.67768671539096219 \cdot 10^{-96}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \frac{z - t}{\sqrt[3]{y}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} + t\\

\end{array}
double f(double x, double y, double z, double t) {
        double r485665 = x;
        double r485666 = y;
        double r485667 = r485665 / r485666;
        double r485668 = z;
        double r485669 = t;
        double r485670 = r485668 - r485669;
        double r485671 = r485667 * r485670;
        double r485672 = r485671 + r485669;
        return r485672;
}
↓
double f(double x, double y, double z, double t) {
        double r485673 = t;
        double r485674 = 3.9907548414935313e-258;
        bool r485675 = r485673 <= r485674;
        double r485676 = 3.677686715390962e-96;
        bool r485677 = r485673 <= r485676;
        double r485678 = !r485677;
        bool r485679 = r485675 || r485678;
        double r485680 = x;
        double r485681 = y;
        double r485682 = r485680 / r485681;
        double r485683 = z;
        double r485684 = r485683 - r485673;
        double r485685 = fma(r485682, r485684, r485673);
        double r485686 = cbrt(r485681);
        double r485687 = r485684 / r485686;
        double r485688 = r485680 * r485687;
        double r485689 = r485686 * r485686;
        double r485690 = r485688 / r485689;
        double r485691 = r485690 + r485673;
        double r485692 = r485679 ? r485685 : r485691;
        return r485692;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original1.7
Target1.8
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < 3.9907548414935313e-258 or 3.677686715390962e-96 < t

    1. Initial program 1.4

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

    2. Simplified1.4

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

    if 3.9907548414935313e-258 < t < 3.677686715390962e-96

    1. Initial program 3.6

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

    3. Applied div-inv3.6

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

    4. Applied associate-*l*4.3

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

    5. Simplified4.2

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

    7. Applied add-cube-cbrt4.8

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

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

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

    9. Applied times-frac4.8

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

    10. Applied associate-*r*3.2

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

    11. Simplified3.2

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{z - t}{\sqrt[3]{y}} + t\]
    12. Using strategy rm

    13. Applied associate-*l/3.9

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.99075484149353127 \cdot 10^{-258} \lor \neg \left(t \le 3.67768671539096219 \cdot 10^{-96}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{z - t}{\sqrt[3]{y}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))