Average Error: 6.2 → 0.8
Time: 16.1s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.04346702924937464405655528797078472535 \cdot 10^{184}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -6.767888822225492857201731483039858827192 \cdot 10^{-97}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;x \cdot y \le 3.151161814343171226593651794549303754932 \cdot 10^{282}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -1.04346702924937464405655528797078472535 \cdot 10^{184}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;x \cdot y \le -6.767888822225492857201731483039858827192 \cdot 10^{-97}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;x \cdot y \le 0.0:\\
\;\;\;\;\frac{y}{z} \cdot x\\

\mathbf{elif}\;x \cdot y \le 3.151161814343171226593651794549303754932 \cdot 10^{282}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r31253030 = x;
        double r31253031 = y;
        double r31253032 = r31253030 * r31253031;
        double r31253033 = z;
        double r31253034 = r31253032 / r31253033;
        return r31253034;
}
↓
double f(double x, double y, double z) {
        double r31253035 = x;
        double r31253036 = y;
        double r31253037 = r31253035 * r31253036;
        double r31253038 = -1.0434670292493746e+184;
        bool r31253039 = r31253037 <= r31253038;
        double r31253040 = z;
        double r31253041 = r31253040 / r31253036;
        double r31253042 = r31253035 / r31253041;
        double r31253043 = -6.767888822225493e-97;
        bool r31253044 = r31253037 <= r31253043;
        double r31253045 = r31253037 / r31253040;
        double r31253046 = 0.0;
        bool r31253047 = r31253037 <= r31253046;
        double r31253048 = r31253036 / r31253040;
        double r31253049 = r31253048 * r31253035;
        double r31253050 = 3.151161814343171e+282;
        bool r31253051 = r31253037 <= r31253050;
        double r31253052 = r31253035 / r31253040;
        double r31253053 = r31253052 * r31253036;
        double r31253054 = r31253051 ? r31253045 : r31253053;
        double r31253055 = r31253047 ? r31253049 : r31253054;
        double r31253056 = r31253044 ? r31253045 : r31253055;
        double r31253057 = r31253039 ? r31253042 : r31253056;
        return r31253057;
}

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

Original6.2
Target6.1
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* x y) < -1.0434670292493746e+184

    1. Initial program 22.1

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

    3. Applied associate-/l*1.7

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

    if -1.0434670292493746e+184 < (* x y) < -6.767888822225493e-97 or 0.0 < (* x y) < 3.151161814343171e+282

    1. Initial program 0.3

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

    3. Applied associate-/l*9.0

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

    5. Applied *-un-lft-identity9.0

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

    6. Applied add-cube-cbrt9.8

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

    7. Applied times-frac9.8

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

    8. Applied add-cube-cbrt10.1

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

    9. Applied times-frac4.3

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

    10. Simplified4.3

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

    11. Taylor expanded around 0 0.3

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

    if -6.767888822225493e-97 < (* x y) < 0.0

    1. Initial program 9.8

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

    3. Applied *-un-lft-identity9.8

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

    4. Applied times-frac1.6

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

    5. Simplified1.6

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

    if 3.151161814343171e+282 < (* x y)

    1. Initial program 51.0

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

    3. Applied associate-/l*0.2

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

    5. Applied associate-/r/0.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.04346702924937464405655528797078472535 \cdot 10^{184}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -6.767888822225492857201731483039858827192 \cdot 10^{-97}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;x \cdot y \le 3.151161814343171226593651794549303754932 \cdot 10^{282}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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))