Average Error: 6.0 → 0.7
Time: 12.0s
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 r28983693 = x;
        double r28983694 = y;
        double r28983695 = r28983693 * r28983694;
        double r28983696 = z;
        double r28983697 = r28983695 / r28983696;
        return r28983697;
}
↓
double f(double x, double y, double z) {
        double r28983698 = x;
        double r28983699 = y;
        double r28983700 = r28983698 * r28983699;
        double r28983701 = -4.783271960069769e+134;
        bool r28983702 = r28983700 <= r28983701;
        double r28983703 = z;
        double r28983704 = r28983698 / r28983703;
        double r28983705 = r28983704 * r28983699;
        double r28983706 = -6.768744269614454e-216;
        bool r28983707 = r28983700 <= r28983706;
        double r28983708 = r28983700 / r28983703;
        double r28983709 = -0.0;
        bool r28983710 = r28983700 <= r28983709;
        double r28983711 = 1.2767606644191683e+158;
        bool r28983712 = r28983700 <= r28983711;
        double r28983713 = r28983699 / r28983703;
        double r28983714 = r28983713 * r28983698;
        double r28983715 = r28983712 ? r28983708 : r28983714;
        double r28983716 = r28983710 ? r28983705 : r28983715;
        double r28983717 = r28983707 ? r28983708 : r28983716;
        double r28983718 = r28983702 ? r28983705 : r28983717;
        return r28983718;
}

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))