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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.18878643735303595 \cdot 10^{-172} \lor \neg \left(x \le 3.9235379008614165 \cdot 10^{-302}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t} - \frac{x}{t}, z, x\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -2.18878643735303595 \cdot 10^{-172} \lor \neg \left(x \le 3.9235379008614165 \cdot 10^{-302}\right):\\
\;\;\;\;\frac{z}{t} \cdot \left(y - x\right) + x\\

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

\end{array}

double f(double x, double y, double z, double t) {
        double r1190650 = x;
        double r1190651 = y;
        double r1190652 = r1190651 - r1190650;
        double r1190653 = z;
        double r1190654 = r1190652 * r1190653;
        double r1190655 = t;
        double r1190656 = r1190654 / r1190655;
        double r1190657 = r1190650 + r1190656;
        return r1190657;
}

↓

double f(double x, double y, double z, double t) {
        double r1190658 = x;
        double r1190659 = -2.188786437353036e-172;
        bool r1190660 = r1190658 <= r1190659;
        double r1190661 = 3.9235379008614165e-302;
        bool r1190662 = r1190658 <= r1190661;
        double r1190663 = !r1190662;
        bool r1190664 = r1190660 || r1190663;
        double r1190665 = z;
        double r1190666 = t;
        double r1190667 = r1190665 / r1190666;
        double r1190668 = y;
        double r1190669 = r1190668 - r1190658;
        double r1190670 = r1190667 * r1190669;
        double r1190671 = r1190670 + r1190658;
        double r1190672 = r1190668 / r1190666;
        double r1190673 = r1190658 / r1190666;
        double r1190674 = r1190672 - r1190673;
        double r1190675 = fma(r1190674, r1190665, r1190658);
        double r1190676 = r1190664 ? r1190671 : r1190675;
        return r1190676;
}

Original	6.8
Target	2.1
Herbie	2.1

Derivation

Split input into 2 regimes
if x < -2.188786437353036e-172 or 3.9235379008614165e-302 < x
1. Initial program 7.1
  \[x + \frac{\left(y - x\right) \cdot z}{t}\]
2. Simplified6.6
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
3. Using strategy rm
4. Applied div-sub6.6
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{t} - \frac{x}{t}}, z, x\right)\]
5. Taylor expanded around inf 7.1
  \[\leadsto \color{blue}{\left(\frac{z \cdot y}{t} + x\right) - \frac{x \cdot z}{t}}\]
6. Simplified5.8
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{x \cdot z}{t}\right)}\]
7. Taylor expanded around 0 7.1
  \[\leadsto \color{blue}{\left(\frac{z \cdot y}{t} + x\right) - \frac{x \cdot z}{t}}\]
8. Simplified1.6
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right) + x}\]
if -2.188786437353036e-172 < x < 3.9235379008614165e-302
1. Initial program 5.1
  \[x + \frac{\left(y - x\right) \cdot z}{t}\]
2. Simplified5.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
3. Using strategy rm
4. Applied div-sub5.1
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{t} - \frac{x}{t}}, z, x\right)\]
Recombined 2 regimes into one program.
Final simplification2.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.18878643735303595 \cdot 10^{-172} \lor \neg \left(x \le 3.9235379008614165 \cdot 10^{-302}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t} - \frac{x}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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

Error

Target

Derivation

`if x < -2.188786437353036e-172 or 3.9235379008614165e-302 < x`

`if -2.188786437353036e-172 < x < 3.9235379008614165e-302`

Reproduce