Average Error: 6.3 → 1.5
Time: 3.6s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2117257699034317484720128 \lor \neg \left(t \le 4.298768646380980053593446645561660090143 \cdot 10^{-219}\right):\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z \cdot y}{t} - \frac{x \cdot y}{t}\right) + x\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -2117257699034317484720128 \lor \neg \left(t \le 4.298768646380980053593446645561660090143 \cdot 10^{-219}\right):\\
\;\;\;\;\frac{y}{t} \cdot \left(z - x\right) + x\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r322676 = x;
        double r322677 = y;
        double r322678 = z;
        double r322679 = r322678 - r322676;
        double r322680 = r322677 * r322679;
        double r322681 = t;
        double r322682 = r322680 / r322681;
        double r322683 = r322676 + r322682;
        return r322683;
}
↓
double f(double x, double y, double z, double t) {
        double r322684 = t;
        double r322685 = -2.1172576990343175e+24;
        bool r322686 = r322684 <= r322685;
        double r322687 = 4.29876864638098e-219;
        bool r322688 = r322684 <= r322687;
        double r322689 = !r322688;
        bool r322690 = r322686 || r322689;
        double r322691 = y;
        double r322692 = r322691 / r322684;
        double r322693 = z;
        double r322694 = x;
        double r322695 = r322693 - r322694;
        double r322696 = r322692 * r322695;
        double r322697 = r322696 + r322694;
        double r322698 = r322693 * r322691;
        double r322699 = r322698 / r322684;
        double r322700 = r322694 * r322691;
        double r322701 = r322700 / r322684;
        double r322702 = r322699 - r322701;
        double r322703 = r322702 + r322694;
        double r322704 = r322690 ? r322697 : r322703;
        return r322704;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.3
Target2.0
Herbie1.5
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if t < -2.1172576990343175e+24 or 4.29876864638098e-219 < t

    1. Initial program 7.8

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

    2. Simplified1.4

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

    4. Applied fma-udef1.4

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

    if -2.1172576990343175e+24 < t < 4.29876864638098e-219

    1. Initial program 1.9

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

    2. Simplified3.9

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

    4. Applied fma-udef3.9

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

    6. Applied add-cube-cbrt4.6

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) \cdot \sqrt[3]{\frac{y}{t}}\right)} \cdot \left(z - x\right) + x\]

    7. Applied associate-*l*4.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) \cdot \left(\sqrt[3]{\frac{y}{t}} \cdot \left(z - x\right)\right)} + x\]

    8. Taylor expanded around 0 1.9

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2117257699034317484720128 \lor \neg \left(t \le 4.298768646380980053593446645561660090143 \cdot 10^{-219}\right):\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z \cdot y}{t} - \frac{x \cdot y}{t}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))