Average Error: 6.0 → 3.4
Time: 7.0s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -2.4829212502580601 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \le 4.09359032056483698 \cdot 10^{-135}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.48436680561384441 \cdot 10^{38}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -2.4829212502580601 \cdot 10^{-130}:\\
\;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right)\\

\mathbf{elif}\;x \cdot y \le 4.09359032056483698 \cdot 10^{-135}:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;x \cdot y \le 1.48436680561384441 \cdot 10^{38}:\\
\;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r733980 = x;
        double r733981 = y;
        double r733982 = r733980 * r733981;
        double r733983 = z;
        double r733984 = r733982 / r733983;
        return r733984;
}
↓
double f(double x, double y, double z) {
        double r733985 = x;
        double r733986 = y;
        double r733987 = r733985 * r733986;
        double r733988 = -2.48292125025806e-130;
        bool r733989 = r733987 <= r733988;
        double r733990 = 1.0;
        double r733991 = z;
        double r733992 = r733990 / r733991;
        double r733993 = r733992 * r733987;
        double r733994 = 4.093590320564837e-135;
        bool r733995 = r733987 <= r733994;
        double r733996 = r733986 / r733991;
        double r733997 = r733985 * r733996;
        double r733998 = 1.4843668056138444e+38;
        bool r733999 = r733987 <= r733998;
        double r734000 = r733991 / r733986;
        double r734001 = r733985 / r734000;
        double r734002 = r733999 ? r733993 : r734001;
        double r734003 = r733995 ? r733997 : r734002;
        double r734004 = r733989 ? r733993 : r734003;
        return r734004;
}

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

Original6.0
Target6.6
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.70421306606504721 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -2.48292125025806e-130 or 4.093590320564837e-135 < (* x y) < 1.4843668056138444e+38

    1. Initial program 3.7

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

    3. Applied associate-/l*10.2

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

    5. Applied clear-num10.5

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{y}}{x}}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity10.5

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

    8. Applied div-inv10.5

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

    9. Applied times-frac3.8

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

    10. Applied add-cube-cbrt3.8

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

    11. Applied times-frac4.0

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

    12. Simplified4.0

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

    13. Simplified3.8

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

    if -2.48292125025806e-130 < (* x y) < 4.093590320564837e-135

    1. Initial program 7.5

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

    3. Applied associate-/l*1.5

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

    5. Applied *-un-lft-identity1.5

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

    6. Applied *-un-lft-identity1.5

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

    7. Applied times-frac1.5

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

    8. Applied *-un-lft-identity1.5

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

    9. Applied times-frac1.5

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

    10. Simplified1.5

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

    11. Simplified1.4

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

    if 1.4843668056138444e+38 < (* x y)

    1. Initial program 9.3

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

    3. Applied associate-/l*6.2

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -2.4829212502580601 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \le 4.09359032056483698 \cdot 10^{-135}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.48436680561384441 \cdot 10^{38}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))