Average Error: 6.4 → 1.9
Time: 15.4s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -4.784509795924599783439519186149299258721 \cdot 10^{-175}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 2.06509035370514777050973690805775237354 \cdot 10^{-309}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 2.556004766497536971746737596687333685074 \cdot 10^{232}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -4.784509795924599783439519186149299258721 \cdot 10^{-175}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;x \cdot y \le 2.06509035370514777050973690805775237354 \cdot 10^{-309}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;x \cdot y \le 2.556004766497536971746737596687333685074 \cdot 10^{232}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r35694927 = x;
        double r35694928 = y;
        double r35694929 = r35694927 * r35694928;
        double r35694930 = z;
        double r35694931 = r35694929 / r35694930;
        return r35694931;
}
↓
double f(double x, double y, double z) {
        double r35694932 = x;
        double r35694933 = y;
        double r35694934 = r35694932 * r35694933;
        double r35694935 = -4.7845097959246e-175;
        bool r35694936 = r35694934 <= r35694935;
        double r35694937 = z;
        double r35694938 = r35694934 / r35694937;
        double r35694939 = 2.06509035370515e-309;
        bool r35694940 = r35694934 <= r35694939;
        double r35694941 = r35694937 / r35694933;
        double r35694942 = r35694932 / r35694941;
        double r35694943 = 2.556004766497537e+232;
        bool r35694944 = r35694934 <= r35694943;
        double r35694945 = r35694933 / r35694937;
        double r35694946 = r35694945 * r35694932;
        double r35694947 = r35694944 ? r35694938 : r35694946;
        double r35694948 = r35694940 ? r35694942 : r35694947;
        double r35694949 = r35694936 ? r35694938 : r35694948;
        return r35694949;
}

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
Target6.2
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -4.7845097959246e-175 or 2.06509035370515e-309 < (* x y) < 2.556004766497537e+232

    1. Initial program 2.4

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

    if -4.7845097959246e-175 < (* x y) < 2.06509035370515e-309

    1. Initial program 13.5

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

    3. Applied associate-/l*0.5

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

    if 2.556004766497537e+232 < (* x y)

    1. Initial program 31.8

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

    3. Applied *-un-lft-identity31.8

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

    4. Applied times-frac0.6

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

    5. Simplified0.6

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -4.784509795924599783439519186149299258721 \cdot 10^{-175}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 2.06509035370514777050973690805775237354 \cdot 10^{-309}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 2.556004766497536971746737596687333685074 \cdot 10^{232}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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))