Average Error: 2.2 → 1.6
Time: 11.1s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.3965576020640518 \cdot 10^{99}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}} + t\\ \mathbf{elif}\;y \le 4.2531001424521584 \cdot 10^{-33}:\\ \;\;\;\;\left(t + \frac{x \cdot z}{y}\right) - \frac{t \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.3965576020640518 \cdot 10^{99}:\\
\;\;\;\;\frac{x}{\frac{y}{z - t}} + t\\

\mathbf{elif}\;y \le 4.2531001424521584 \cdot 10^{-33}:\\
\;\;\;\;\left(t + \frac{x \cdot z}{y}\right) - \frac{t \cdot x}{y}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r542397 = x;
        double r542398 = y;
        double r542399 = r542397 / r542398;
        double r542400 = z;
        double r542401 = t;
        double r542402 = r542400 - r542401;
        double r542403 = r542399 * r542402;
        double r542404 = r542403 + r542401;
        return r542404;
}
↓
double f(double x, double y, double z, double t) {
        double r542405 = y;
        double r542406 = -2.396557602064052e+99;
        bool r542407 = r542405 <= r542406;
        double r542408 = x;
        double r542409 = z;
        double r542410 = t;
        double r542411 = r542409 - r542410;
        double r542412 = r542405 / r542411;
        double r542413 = r542408 / r542412;
        double r542414 = r542413 + r542410;
        double r542415 = 4.2531001424521584e-33;
        bool r542416 = r542405 <= r542415;
        double r542417 = r542408 * r542409;
        double r542418 = r542417 / r542405;
        double r542419 = r542410 + r542418;
        double r542420 = r542410 * r542408;
        double r542421 = r542420 / r542405;
        double r542422 = r542419 - r542421;
        double r542423 = r542408 / r542405;
        double r542424 = fma(r542423, r542411, r542410);
        double r542425 = r542416 ? r542422 : r542424;
        double r542426 = r542407 ? r542414 : r542425;
        return r542426;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.2
Target2.5
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.396557602064052e+99

    1. Initial program 1.4

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

    2. Simplified1.4

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

    4. Applied fma-udef1.4

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

    5. Simplified1.2

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

    if -2.396557602064052e+99 < y < 4.2531001424521584e-33

    1. Initial program 3.5

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

    2. Simplified3.5

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

    4. Applied fma-udef3.5

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

    5. Simplified11.8

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

    6. Taylor expanded around 0 2.1

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

    if 4.2531001424521584e-33 < y

    1. Initial program 1.1

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

    2. Simplified1.1

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.3965576020640518 \cdot 10^{99}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}} + t\\ \mathbf{elif}\;y \le 4.2531001424521584 \cdot 10^{-33}:\\ \;\;\;\;\left(t + \frac{x \cdot z}{y}\right) - \frac{t \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))