Average Error: 6.5 → 0.5
Time: 1.5s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -6.1932913106938277 \cdot 10^{-129}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 8.60947675331187076 \cdot 10^{-202}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 3.340969170147564 \cdot 10^{255}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y = -\infty:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;x \cdot y \le -6.1932913106938277 \cdot 10^{-129}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;x \cdot y \le 8.60947675331187076 \cdot 10^{-202}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;x \cdot y \le 3.340969170147564 \cdot 10^{255}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r702033 = x;
        double r702034 = y;
        double r702035 = r702033 * r702034;
        double r702036 = z;
        double r702037 = r702035 / r702036;
        return r702037;
}
↓
double f(double x, double y, double z) {
        double r702038 = x;
        double r702039 = y;
        double r702040 = r702038 * r702039;
        double r702041 = -inf.0;
        bool r702042 = r702040 <= r702041;
        double r702043 = z;
        double r702044 = r702039 / r702043;
        double r702045 = r702038 * r702044;
        double r702046 = -6.193291310693828e-129;
        bool r702047 = r702040 <= r702046;
        double r702048 = r702040 / r702043;
        double r702049 = 8.60947675331187e-202;
        bool r702050 = r702040 <= r702049;
        double r702051 = r702043 / r702039;
        double r702052 = r702038 / r702051;
        double r702053 = 3.340969170147564e+255;
        bool r702054 = r702040 <= r702053;
        double r702055 = r702038 / r702043;
        double r702056 = r702055 * r702039;
        double r702057 = r702054 ? r702048 : r702056;
        double r702058 = r702050 ? r702052 : r702057;
        double r702059 = r702047 ? r702048 : r702058;
        double r702060 = r702042 ? r702045 : r702059;
        return r702060;
}

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

Original6.5
Target6.3
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.70421306606504721 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* x y) < -inf.0

    1. Initial program 64.0

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

    3. Applied *-un-lft-identity64.0

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

    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 -inf.0 < (* x y) < -6.193291310693828e-129 or 8.60947675331187e-202 < (* x y) < 3.340969170147564e+255

    1. Initial program 0.2

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

    if -6.193291310693828e-129 < (* x y) < 8.60947675331187e-202

    1. Initial program 9.6

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

    3. Applied associate-/l*0.9

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

    if 3.340969170147564e+255 < (* x y)

    1. Initial program 38.7

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

    3. Applied associate-/l*0.5

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

    5. Applied associate-/r/0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -6.1932913106938277 \cdot 10^{-129}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 8.60947675331187076 \cdot 10^{-202}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 3.340969170147564 \cdot 10^{255}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))