Average Error: 6.1 → 1.4
Time: 16.0s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -3.20189178708899581657271843297725010796 \cdot 10^{136}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -4.762336977697826968317141793899825216849 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{y}}{x}}\\ \mathbf{elif}\;x \cdot y \le 5.272950596448448823066987117820011404591 \cdot 10^{145}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -3.20189178708899581657271843297725010796 \cdot 10^{136}:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;x \cdot y \le -4.762336977697826968317141793899825216849 \cdot 10^{-104}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;x \cdot y \le 0.0:\\
\;\;\;\;\frac{1}{\frac{\frac{z}{y}}{x}}\\

\mathbf{elif}\;x \cdot y \le 5.272950596448448823066987117820011404591 \cdot 10^{145}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r40128 = x;
        double r40129 = y;
        double r40130 = r40128 * r40129;
        double r40131 = z;
        double r40132 = r40130 / r40131;
        return r40132;
}
↓
double f(double x, double y, double z) {
        double r40133 = x;
        double r40134 = y;
        double r40135 = r40133 * r40134;
        double r40136 = -3.201891787088996e+136;
        bool r40137 = r40135 <= r40136;
        double r40138 = z;
        double r40139 = r40134 / r40138;
        double r40140 = r40133 * r40139;
        double r40141 = -4.762336977697827e-104;
        bool r40142 = r40135 <= r40141;
        double r40143 = r40135 / r40138;
        double r40144 = 0.0;
        bool r40145 = r40135 <= r40144;
        double r40146 = 1.0;
        double r40147 = r40138 / r40134;
        double r40148 = r40147 / r40133;
        double r40149 = r40146 / r40148;
        double r40150 = 5.272950596448449e+145;
        bool r40151 = r40135 <= r40150;
        double r40152 = r40133 / r40138;
        double r40153 = r40152 * r40134;
        double r40154 = r40151 ? r40143 : r40153;
        double r40155 = r40145 ? r40149 : r40154;
        double r40156 = r40142 ? r40143 : r40155;
        double r40157 = r40137 ? r40140 : r40156;
        return r40157;
}

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

Original6.1
Target6.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* x y) < -3.201891787088996e+136

    1. Initial program 17.0

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

    3. Applied *-un-lft-identity17.0

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

    4. Applied times-frac2.0

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

    5. Simplified2.0

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

    if -3.201891787088996e+136 < (* x y) < -4.762336977697827e-104 or 0.0 < (* x y) < 5.272950596448449e+145

    1. Initial program 0.4

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

    if -4.762336977697827e-104 < (* x y) < 0.0

    1. Initial program 9.7

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

    3. Applied associate-/l*2.0

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

    5. Applied div-inv2.0

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

    6. Applied associate-/r*1.4

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{1}{y}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity1.4

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

    9. Applied *-un-lft-identity1.4

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

    10. Applied times-frac1.4

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

    11. Applied associate-/l*2.1

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

    12. Simplified2.7

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

    if 5.272950596448449e+145 < (* x y)

    1. Initial program 17.6

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

    3. Applied associate-/l*2.8

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

    5. Applied associate-/r/2.5

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -3.20189178708899581657271843297725010796 \cdot 10^{136}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -4.762336977697826968317141793899825216849 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{y}}{x}}\\ \mathbf{elif}\;x \cdot y \le 5.272950596448448823066987117820011404591 \cdot 10^{145}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019310 
(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 1.70421306606504721e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))