Average Error: 6.2 → 1.9
Time: 2.5s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.07003647724086002 \cdot 10^{-303} \lor \neg \left(t \le 7.7584251326065252 \cdot 10^{-53}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -7.07003647724086002 \cdot 10^{-303} \lor \neg \left(t \le 7.7584251326065252 \cdot 10^{-53}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r283646 = x;
        double r283647 = y;
        double r283648 = z;
        double r283649 = r283648 - r283646;
        double r283650 = r283647 * r283649;
        double r283651 = t;
        double r283652 = r283650 / r283651;
        double r283653 = r283646 + r283652;
        return r283653;
}
↓
double f(double x, double y, double z, double t) {
        double r283654 = t;
        double r283655 = -7.07003647724086e-303;
        bool r283656 = r283654 <= r283655;
        double r283657 = 7.758425132606525e-53;
        bool r283658 = r283654 <= r283657;
        double r283659 = !r283658;
        bool r283660 = r283656 || r283659;
        double r283661 = y;
        double r283662 = r283661 / r283654;
        double r283663 = z;
        double r283664 = x;
        double r283665 = r283663 - r283664;
        double r283666 = fma(r283662, r283665, r283664);
        double r283667 = 1.0;
        double r283668 = r283661 * r283665;
        double r283669 = r283654 / r283668;
        double r283670 = r283667 / r283669;
        double r283671 = r283664 + r283670;
        double r283672 = r283660 ? r283666 : r283671;
        return r283672;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.2
Target2.2
Herbie1.9
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if t < -7.07003647724086e-303 or 7.758425132606525e-53 < t

    1. Initial program 6.9

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

    2. Simplified1.8

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

    if -7.07003647724086e-303 < t < 7.758425132606525e-53

    1. Initial program 2.3

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

    3. Applied clear-num2.3

      \[\leadsto x + \color{blue}{\frac{1}{\frac{t}{y \cdot \left(z - x\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.07003647724086002 \cdot 10^{-303} \lor \neg \left(t \le 7.7584251326065252 \cdot 10^{-53}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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