Average Error: 6.4 → 1.7
Time: 17.6s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -7.768603561893780181171682032240737260521 \cdot 10^{305}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -5.989061428235972049640443908724325403146 \cdot 10^{-77}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -0.0:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{\sqrt[3]{z}}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 3.534109248982643983253870825363988567111 \cdot 10^{298}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot y}{z} \le -7.768603561893780181171682032240737260521 \cdot 10^{305}:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;\frac{x \cdot y}{z} \le -5.989061428235972049640443908724325403146 \cdot 10^{-77}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;\frac{x \cdot y}{z} \le -0.0:\\
\;\;\;\;\frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{\sqrt[3]{z}}{y}}\\

\mathbf{elif}\;\frac{x \cdot y}{z} \le 3.534109248982643983253870825363988567111 \cdot 10^{298}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r521905 = x;
        double r521906 = y;
        double r521907 = r521905 * r521906;
        double r521908 = z;
        double r521909 = r521907 / r521908;
        return r521909;
}
↓
double f(double x, double y, double z) {
        double r521910 = x;
        double r521911 = y;
        double r521912 = r521910 * r521911;
        double r521913 = z;
        double r521914 = r521912 / r521913;
        double r521915 = -7.76860356189378e+305;
        bool r521916 = r521914 <= r521915;
        double r521917 = r521911 / r521913;
        double r521918 = r521910 * r521917;
        double r521919 = -5.989061428235972e-77;
        bool r521920 = r521914 <= r521919;
        double r521921 = -0.0;
        bool r521922 = r521914 <= r521921;
        double r521923 = cbrt(r521913);
        double r521924 = r521923 * r521923;
        double r521925 = r521910 / r521924;
        double r521926 = r521923 / r521911;
        double r521927 = r521925 / r521926;
        double r521928 = 3.534109248982644e+298;
        bool r521929 = r521914 <= r521928;
        double r521930 = r521913 / r521911;
        double r521931 = r521910 / r521930;
        double r521932 = r521929 ? r521914 : r521931;
        double r521933 = r521922 ? r521927 : r521932;
        double r521934 = r521920 ? r521914 : r521933;
        double r521935 = r521916 ? r521918 : r521934;
        return r521935;
}

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
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (* x y) z) < -7.76860356189378e+305

    1. Initial program 62.6

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

    3. Applied *-un-lft-identity62.6

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

    if -7.76860356189378e+305 < (/ (* x y) z) < -5.989061428235972e-77 or -0.0 < (/ (* x y) z) < 3.534109248982644e+298

    1. Initial program 2.2

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

    if -5.989061428235972e-77 < (/ (* x y) z) < -0.0

    1. Initial program 5.7

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

    3. Applied associate-/l*6.4

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

    5. Applied *-un-lft-identity6.4

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

    6. Applied add-cube-cbrt7.0

      \[\leadsto \frac{x}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{1 \cdot y}}\]

    7. Applied times-frac7.0

      \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1} \cdot \frac{\sqrt[3]{z}}{y}}}\]

    8. Applied associate-/r*5.3

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1}}}{\frac{\sqrt[3]{z}}{y}}}\]

    9. Simplified5.3

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\frac{\sqrt[3]{z}}{y}}\]

    if 3.534109248982644e+298 < (/ (* x y) z)

    1. Initial program 57.5

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

    3. Applied associate-/l*1.5

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -7.768603561893780181171682032240737260521 \cdot 10^{305}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -5.989061428235972049640443908724325403146 \cdot 10^{-77}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -0.0:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{\sqrt[3]{z}}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 3.534109248982643983253870825363988567111 \cdot 10^{298}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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 1.70421306606504721e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))