Average Error: 6.7 → 1.0
Time: 14.4s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \le 7.47453856091499786 \cdot 10^{296}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \le 7.47453856091499786 \cdot 10^{296}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r621730 = x;
        double r621731 = y;
        double r621732 = r621731 - r621730;
        double r621733 = z;
        double r621734 = r621732 * r621733;
        double r621735 = t;
        double r621736 = r621734 / r621735;
        double r621737 = r621730 + r621736;
        return r621737;
}
↓
double f(double x, double y, double z, double t) {
        double r621738 = x;
        double r621739 = y;
        double r621740 = r621739 - r621738;
        double r621741 = z;
        double r621742 = r621740 * r621741;
        double r621743 = t;
        double r621744 = r621742 / r621743;
        double r621745 = r621738 + r621744;
        double r621746 = -inf.0;
        bool r621747 = r621745 <= r621746;
        double r621748 = 7.474538560914998e+296;
        bool r621749 = r621745 <= r621748;
        double r621750 = !r621749;
        bool r621751 = r621747 || r621750;
        double r621752 = r621740 / r621743;
        double r621753 = fma(r621752, r621741, r621738);
        double r621754 = r621751 ? r621753 : r621745;
        return r621754;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.7
Target2.3
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ (* (- y x) z) t)) < -inf.0 or 7.474538560914998e+296 < (+ x (/ (* (- y x) z) t))

    1. Initial program 58.1

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

    2. Simplified2.8

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

    if -inf.0 < (+ x (/ (* (- y x) z) t)) < 7.474538560914998e+296

    1. Initial program 0.8

      \[x + \frac{\left(y - x\right) \cdot z}{t}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \le 7.47453856091499786 \cdot 10^{296}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))