Average Error: 6.0 → 0.7
Time: 12.2s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -4.783271960069768719879279291578812109813 \cdot 10^{134}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -6.768744269614454269682340382759909093619 \cdot 10^{-216}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le -0.0:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le 1.27676066441916826729575320760818348292 \cdot 10^{158}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -4.783271960069768719879279291578812109813 \cdot 10^{134}:\\
\;\;\;\;\frac{x}{z} \cdot y\\

\mathbf{elif}\;x \cdot y \le -6.768744269614454269682340382759909093619 \cdot 10^{-216}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\mathbf{elif}\;x \cdot y \le 1.27676066441916826729575320760818348292 \cdot 10^{158}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r25391074 = x;
        double r25391075 = y;
        double r25391076 = r25391074 * r25391075;
        double r25391077 = z;
        double r25391078 = r25391076 / r25391077;
        return r25391078;
}
↓
double f(double x, double y, double z) {
        double r25391079 = x;
        double r25391080 = y;
        double r25391081 = r25391079 * r25391080;
        double r25391082 = -4.783271960069769e+134;
        bool r25391083 = r25391081 <= r25391082;
        double r25391084 = z;
        double r25391085 = r25391079 / r25391084;
        double r25391086 = r25391085 * r25391080;
        double r25391087 = -6.768744269614454e-216;
        bool r25391088 = r25391081 <= r25391087;
        double r25391089 = r25391081 / r25391084;
        double r25391090 = -0.0;
        bool r25391091 = r25391081 <= r25391090;
        double r25391092 = 1.2767606644191683e+158;
        bool r25391093 = r25391081 <= r25391092;
        double r25391094 = r25391080 / r25391084;
        double r25391095 = r25391094 * r25391079;
        double r25391096 = r25391093 ? r25391089 : r25391095;
        double r25391097 = r25391091 ? r25391086 : r25391096;
        double r25391098 = r25391088 ? r25391089 : r25391097;
        double r25391099 = r25391083 ? r25391086 : r25391098;
        return r25391099;
}

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
Target6.2
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) < -4.783271960069769e+134 or -6.768744269614454e-216 < (* x y) < -0.0

    1. Initial program 14.6

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

    3. Applied associate-/l*1.3

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

    5. Applied associate-/r/1.1

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

    if -4.783271960069769e+134 < (* x y) < -6.768744269614454e-216 or -0.0 < (* x y) < 1.2767606644191683e+158

    1. Initial program 0.3

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

    3. Applied associate-/l*8.7

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

    4. Taylor expanded around 0 0.3

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

    if 1.2767606644191683e+158 < (* x y)

    1. Initial program 21.3

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

    3. Applied *-un-lft-identity21.3

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

    4. Applied times-frac2.3

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

    5. Simplified2.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -4.783271960069768719879279291578812109813 \cdot 10^{134}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -6.768744269614454269682340382759909093619 \cdot 10^{-216}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le -0.0:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le 1.27676066441916826729575320760818348292 \cdot 10^{158}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))