Average Error: 2.2 → 2.0
Time: 3.5s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.173002117854331202174503352915459420841 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{y}, x, t\right)\\ \mathbf{elif}\;x \le -1.614250424175128305979823788907008463268 \cdot 10^{-177}:\\ \;\;\;\;1 \cdot \frac{\left(z - t\right) \cdot x}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.173002117854331202174503352915459420841 \cdot 10^{-109}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{y}, x, t\right)\\

\mathbf{elif}\;x \le -1.614250424175128305979823788907008463268 \cdot 10^{-177}:\\
\;\;\;\;1 \cdot \frac{\left(z - t\right) \cdot x}{y} + t\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r579203 = x;
        double r579204 = y;
        double r579205 = r579203 / r579204;
        double r579206 = z;
        double r579207 = t;
        double r579208 = r579206 - r579207;
        double r579209 = r579205 * r579208;
        double r579210 = r579209 + r579207;
        return r579210;
}
↓
double f(double x, double y, double z, double t) {
        double r579211 = x;
        double r579212 = -2.173002117854331e-109;
        bool r579213 = r579211 <= r579212;
        double r579214 = z;
        double r579215 = t;
        double r579216 = r579214 - r579215;
        double r579217 = y;
        double r579218 = r579216 / r579217;
        double r579219 = fma(r579218, r579211, r579215);
        double r579220 = -1.6142504241751283e-177;
        bool r579221 = r579211 <= r579220;
        double r579222 = 1.0;
        double r579223 = r579216 * r579211;
        double r579224 = r579223 / r579217;
        double r579225 = r579222 * r579224;
        double r579226 = r579225 + r579215;
        double r579227 = r579211 / r579217;
        double r579228 = r579227 * r579216;
        double r579229 = r579228 + r579215;
        double r579230 = r579221 ? r579226 : r579229;
        double r579231 = r579213 ? r579219 : r579230;
        return r579231;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.2
Target2.4
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.173002117854331e-109

    1. Initial program 3.0

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

    3. Applied *-un-lft-identity3.0

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

    4. Applied associate-*l*3.0

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

    5. Simplified10.1

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

    6. Taylor expanded around 0 10.1

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

    7. Simplified2.3

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

    if -2.173002117854331e-109 < x < -1.6142504241751283e-177

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

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

    4. Applied associate-*l*0.8

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

    5. Simplified0.8

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

    if -1.6142504241751283e-177 < x

    1. Initial program 2.0

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.173002117854331202174503352915459420841 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{y}, x, t\right)\\ \mathbf{elif}\;x \le -1.614250424175128305979823788907008463268 \cdot 10^{-177}:\\ \;\;\;\;1 \cdot \frac{\left(z - t\right) \cdot x}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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))