Average Error: 5.8 → 2.9
Time: 1.6s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -2.0212306242717315 \cdot 10^{115}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -2.08794536631012826 \cdot 10^{-85}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -2.0212306242717315 \cdot 10^{115}:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;x \cdot y \le -2.08794536631012826 \cdot 10^{-85}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\

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

\mathbf{else}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r717788 = x;
        double r717789 = y;
        double r717790 = r717788 * r717789;
        double r717791 = z;
        double r717792 = r717790 / r717791;
        return r717792;
}
↓
double f(double x, double y, double z) {
        double r717793 = x;
        double r717794 = y;
        double r717795 = r717793 * r717794;
        double r717796 = -2.0212306242717315e+115;
        bool r717797 = r717795 <= r717796;
        double r717798 = z;
        double r717799 = r717794 / r717798;
        double r717800 = r717793 * r717799;
        double r717801 = -2.0879453663101283e-85;
        bool r717802 = r717795 <= r717801;
        double r717803 = 1.0;
        double r717804 = r717803 / r717798;
        double r717805 = r717795 * r717804;
        double r717806 = 0.0;
        bool r717807 = r717795 <= r717806;
        double r717808 = r717793 / r717798;
        double r717809 = r717808 * r717794;
        double r717810 = r717807 ? r717809 : r717805;
        double r717811 = r717802 ? r717805 : r717810;
        double r717812 = r717797 ? r717800 : r717811;
        return r717812;
}

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

Original5.8
Target5.9
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.70421306606504721 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -2.0212306242717315e+115

    1. Initial program 13.5

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

    3. Applied *-un-lft-identity13.5

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

    4. Applied times-frac3.0

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

    5. Simplified3.0

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

    if -2.0212306242717315e+115 < (* x y) < -2.0879453663101283e-85 or 0.0 < (* x y)

    1. Initial program 3.1

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

    3. Applied div-inv3.1

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

    if -2.0879453663101283e-85 < (* x y) < 0.0

    1. Initial program 9.0

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

    3. Applied associate-/l*1.9

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

    5. Applied associate-/r/2.2

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -2.0212306242717315 \cdot 10^{115}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -2.08794536631012826 \cdot 10^{-85}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))