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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.3965576020640518 \cdot 10^{99}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}} + t\\ \mathbf{elif}\;y \le 1.47008867651433422 \cdot 10^{-30}:\\ \;\;\;\;\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array}\]

\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 1.47008867651433422 \cdot 10^{-30}:\\
\;\;\;\;\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right) + t\\

\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 r397031 = x;
        double r397032 = y;
        double r397033 = r397031 / r397032;
        double r397034 = z;
        double r397035 = t;
        double r397036 = r397034 - r397035;
        double r397037 = r397033 * r397036;
        double r397038 = r397037 + r397035;
        return r397038;
}

↓

double f(double x, double y, double z, double t) {
        double r397039 = y;
        double r397040 = -2.396557602064052e+99;
        bool r397041 = r397039 <= r397040;
        double r397042 = x;
        double r397043 = z;
        double r397044 = t;
        double r397045 = r397043 - r397044;
        double r397046 = r397039 / r397045;
        double r397047 = r397042 / r397046;
        double r397048 = r397047 + r397044;
        double r397049 = 1.4700886765143342e-30;
        bool r397050 = r397039 <= r397049;
        double r397051 = r397042 * r397043;
        double r397052 = r397051 / r397039;
        double r397053 = r397044 * r397042;
        double r397054 = r397053 / r397039;
        double r397055 = r397052 - r397054;
        double r397056 = r397055 + r397044;
        double r397057 = r397042 / r397039;
        double r397058 = fma(r397057, r397045, r397044);
        double r397059 = r397050 ? r397056 : r397058;
        double r397060 = r397041 ? r397048 : r397059;
        return r397060;
}

Original	2.2
Target	2.5
Herbie	1.6

Derivation

Split input into 3 regimes
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 < 1.4700886765143342e-30
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
  \[\leadsto \color{blue}{\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right)} + t\]
if 1.4700886765143342e-30 < y
1. Initial program 1.1
  \[\frac{x}{y} \cdot \left(z - t\right) + t\]
2. Simplified1.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)}\]
Recombined 3 regimes into one program.
Final simplification1.6
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.3965576020640518 \cdot 10^{99}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}} + t\\ \mathbf{elif}\;y \le 1.47008867651433422 \cdot 10^{-30}:\\ \;\;\;\;\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array}\]

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))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

`if y < -2.396557602064052e+99`

`if -2.396557602064052e+99 < y < 1.4700886765143342e-30`

`if 1.4700886765143342e-30 < y`

Reproduce