Average Error: 6.4 → 0.7
Time: 4.6s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.394495778794157817173263708359068376665 \cdot 10^{144}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -2.091115834215142392182824625708863732465 \cdot 10^{-215}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.107151871760683778945393640864932427286 \cdot 10^{-309}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 8.013896806549786117243748175613103383758 \cdot 10^{151}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -1.394495778794157817173263708359068376665 \cdot 10^{144}:\\
\;\;\;\;\frac{x}{z} \cdot y\\

\mathbf{elif}\;x \cdot y \le -2.091115834215142392182824625708863732465 \cdot 10^{-215}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;x \cdot y \le 1.107151871760683778945393640864932427286 \cdot 10^{-309}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;x \cdot y \le 8.013896806549786117243748175613103383758 \cdot 10^{151}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r808003 = x;
        double r808004 = y;
        double r808005 = r808003 * r808004;
        double r808006 = z;
        double r808007 = r808005 / r808006;
        return r808007;
}
↓
double f(double x, double y, double z) {
        double r808008 = x;
        double r808009 = y;
        double r808010 = r808008 * r808009;
        double r808011 = -1.3944957787941578e+144;
        bool r808012 = r808010 <= r808011;
        double r808013 = z;
        double r808014 = r808008 / r808013;
        double r808015 = r808014 * r808009;
        double r808016 = -2.0911158342151424e-215;
        bool r808017 = r808010 <= r808016;
        double r808018 = r808010 / r808013;
        double r808019 = 1.107151871760684e-309;
        bool r808020 = r808010 <= r808019;
        double r808021 = r808013 / r808009;
        double r808022 = r808008 / r808021;
        double r808023 = 8.013896806549786e+151;
        bool r808024 = r808010 <= r808023;
        double r808025 = r808024 ? r808018 : r808015;
        double r808026 = r808020 ? r808022 : r808025;
        double r808027 = r808017 ? r808018 : r808026;
        double r808028 = r808012 ? r808015 : r808027;
        return r808028;
}

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
Target6.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -1.3944957787941578e+144 or 8.013896806549786e+151 < (* x y)

    1. Initial program 19.0

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

    3. Applied associate-/l*2.4

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

    5. Applied associate-/r/3.0

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

    if -1.3944957787941578e+144 < (* x y) < -2.0911158342151424e-215 or 1.107151871760684e-309 < (* x y) < 8.013896806549786e+151

    1. Initial program 0.2

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

    3. Applied associate-/l*9.4

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

    4. Taylor expanded around 0 0.2

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

    if -2.0911158342151424e-215 < (* x y) < 1.107151871760684e-309

    1. Initial program 14.4

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

    3. Applied associate-/l*0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.394495778794157817173263708359068376665 \cdot 10^{144}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -2.091115834215142392182824625708863732465 \cdot 10^{-215}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.107151871760683778945393640864932427286 \cdot 10^{-309}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 8.013896806549786117243748175613103383758 \cdot 10^{151}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))