Average Error: 6.5 → 0.6
Time: 17.6s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.827618842210325856116378407212167275336 \cdot 10^{252}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -1.416922114786241732626195364860208906831 \cdot 10^{-291}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.654394754607905895780595947422963951718 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 1.866392507649868309583567517907241893789 \cdot 10^{207}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -1.827618842210325856116378407212167275336 \cdot 10^{252}:\\
\;\;\;\;\frac{x}{z} \cdot y\\

\mathbf{elif}\;x \cdot y \le -1.416922114786241732626195364860208906831 \cdot 10^{-291}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;x \cdot y \le 1.654394754607905895780595947422963951718 \cdot 10^{-114}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;x \cdot y \le 1.866392507649868309583567517907241893789 \cdot 10^{207}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r173532657 = x;
        double r173532658 = y;
        double r173532659 = r173532657 * r173532658;
        double r173532660 = z;
        double r173532661 = r173532659 / r173532660;
        return r173532661;
}
↓
double f(double x, double y, double z) {
        double r173532662 = x;
        double r173532663 = y;
        double r173532664 = r173532662 * r173532663;
        double r173532665 = -1.8276188422103259e+252;
        bool r173532666 = r173532664 <= r173532665;
        double r173532667 = z;
        double r173532668 = r173532662 / r173532667;
        double r173532669 = r173532668 * r173532663;
        double r173532670 = -1.4169221147862417e-291;
        bool r173532671 = r173532664 <= r173532670;
        double r173532672 = r173532664 / r173532667;
        double r173532673 = 1.654394754607906e-114;
        bool r173532674 = r173532664 <= r173532673;
        double r173532675 = r173532667 / r173532663;
        double r173532676 = r173532662 / r173532675;
        double r173532677 = 1.8663925076498683e+207;
        bool r173532678 = r173532664 <= r173532677;
        double r173532679 = r173532663 / r173532667;
        double r173532680 = r173532662 * r173532679;
        double r173532681 = r173532678 ? r173532672 : r173532680;
        double r173532682 = r173532674 ? r173532676 : r173532681;
        double r173532683 = r173532671 ? r173532672 : r173532682;
        double r173532684 = r173532666 ? r173532669 : r173532683;
        return r173532684;
}

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

Original6.5
Target6.3
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* x y) < -1.8276188422103259e+252

    1. Initial program 40.2

      \[\frac{x \cdot y}{z}\]
    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/0.2

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

    if -1.8276188422103259e+252 < (* x y) < -1.4169221147862417e-291 or 1.654394754607906e-114 < (* x y) < 1.8663925076498683e+207

    1. Initial program 0.2

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

    if -1.4169221147862417e-291 < (* x y) < 1.654394754607906e-114

    1. Initial program 10.2

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

    3. Applied associate-/l*1.3

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

    if 1.8663925076498683e+207 < (* x y)

    1. Initial program 31.5

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

    3. Applied *-un-lft-identity31.5

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

    4. Applied times-frac0.5

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

    5. Simplified0.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.827618842210325856116378407212167275336 \cdot 10^{252}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -1.416922114786241732626195364860208906831 \cdot 10^{-291}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.654394754607905895780595947422963951718 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 1.866392507649868309583567517907241893789 \cdot 10^{207}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"

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

  (/ (* x y) z))