Average Error: 6.4 → 0.5
Time: 2.0s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.122825852725341740510374421793401440482 \cdot 10^{174}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -1.187138743731467067277490994856280614263 \cdot 10^{-188}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 2.131572119135182027868179571511010877863 \cdot 10^{-317}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.379689488789308046398090942052262660111 \cdot 10^{229}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -1.122825852725341740510374421793401440482 \cdot 10^{174}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;x \cdot y \le -1.187138743731467067277490994856280614263 \cdot 10^{-188}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;x \cdot y \le 2.131572119135182027868179571511010877863 \cdot 10^{-317}:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;x \cdot y \le 1.379689488789308046398090942052262660111 \cdot 10^{229}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{z} \cdot y\\

\end{array}
double f(double x, double y, double z) {
        double r852452 = x;
        double r852453 = y;
        double r852454 = r852452 * r852453;
        double r852455 = z;
        double r852456 = r852454 / r852455;
        return r852456;
}
↓
double f(double x, double y, double z) {
        double r852457 = x;
        double r852458 = y;
        double r852459 = r852457 * r852458;
        double r852460 = -1.1228258527253417e+174;
        bool r852461 = r852459 <= r852460;
        double r852462 = z;
        double r852463 = r852462 / r852458;
        double r852464 = r852457 / r852463;
        double r852465 = -1.187138743731467e-188;
        bool r852466 = r852459 <= r852465;
        double r852467 = r852459 / r852462;
        double r852468 = 2.1315721191352e-317;
        bool r852469 = r852459 <= r852468;
        double r852470 = r852458 / r852462;
        double r852471 = r852457 * r852470;
        double r852472 = 1.379689488789308e+229;
        bool r852473 = r852459 <= r852472;
        double r852474 = r852457 / r852462;
        double r852475 = r852474 * r852458;
        double r852476 = r852473 ? r852467 : r852475;
        double r852477 = r852469 ? r852471 : r852476;
        double r852478 = r852466 ? r852467 : r852477;
        double r852479 = r852461 ? r852464 : r852478;
        return r852479;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.4
Target6.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* x y) < -1.1228258527253417e+174

    1. Initial program 22.8

      \[\frac{x \cdot y}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*1.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]

    if -1.1228258527253417e+174 < (* x y) < -1.187138743731467e-188 or 2.1315721191352e-317 < (* x y) < 1.379689488789308e+229

    1. Initial program 0.3

      \[\frac{x \cdot y}{z}\]

    if -1.187138743731467e-188 < (* x y) < 2.1315721191352e-317

    1. Initial program 13.5

      \[\frac{x \cdot y}{z}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity13.5

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac0.6

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y}{z}}\]

    5. Simplified0.6

      \[\leadsto \color{blue}{x} \cdot \frac{y}{z}\]

    if 1.379689488789308e+229 < (* x y)

    1. Initial program 31.3

      \[\frac{x \cdot y}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
    4. Using strategy rm

    5. Applied associate-/r/0.8

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.122825852725341740510374421793401440482 \cdot 10^{174}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -1.187138743731467067277490994856280614263 \cdot 10^{-188}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 2.131572119135182027868179571511010877863 \cdot 10^{-317}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.379689488789308046398090942052262660111 \cdot 10^{229}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))