Average Error: 12.6 → 3.0
Time: 13.2s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.244096151631290896943813259882212972022 \cdot 10^{-205}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{elif}\;z \le 1.123685082691174413471401852132580668372 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(z + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.244096151631290896943813259882212972022 \cdot 10^{-205}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\

\mathbf{elif}\;z \le 1.123685082691174413471401852132580668372 \cdot 10^{-120}:\\
\;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(z + y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\

\end{array}
double f(double x, double y, double z) {
        double r20566996 = x;
        double r20566997 = y;
        double r20566998 = z;
        double r20566999 = r20566997 + r20566998;
        double r20567000 = r20566996 * r20566999;
        double r20567001 = r20567000 / r20566998;
        return r20567001;
}
↓
double f(double x, double y, double z) {
        double r20567002 = z;
        double r20567003 = -1.244096151631291e-205;
        bool r20567004 = r20567002 <= r20567003;
        double r20567005 = x;
        double r20567006 = y;
        double r20567007 = r20567006 / r20567002;
        double r20567008 = 1.0;
        double r20567009 = r20567007 + r20567008;
        double r20567010 = r20567005 * r20567009;
        double r20567011 = 1.1236850826911744e-120;
        bool r20567012 = r20567002 <= r20567011;
        double r20567013 = r20567008 / r20567002;
        double r20567014 = r20567002 + r20567006;
        double r20567015 = r20567005 * r20567014;
        double r20567016 = r20567013 * r20567015;
        double r20567017 = r20567012 ? r20567016 : r20567010;
        double r20567018 = r20567004 ? r20567010 : r20567017;
        return r20567018;
}

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

Original12.6
Target3.1
Herbie3.0
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.244096151631291e-205 or 1.1236850826911744e-120 < z

    1. Initial program 13.1

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity13.1

      \[\leadsto \frac{x \cdot \left(y + z\right)}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac1.5

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

    5. Simplified1.5

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

    6. Taylor expanded around 0 1.5

      \[\leadsto x \cdot \color{blue}{\left(\frac{y}{z} + 1\right)}\]

    if -1.244096151631291e-205 < z < 1.1236850826911744e-120

    1. Initial program 10.2

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied div-inv10.4

      \[\leadsto \color{blue}{\left(x \cdot \left(y + z\right)\right) \cdot \frac{1}{z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.244096151631290896943813259882212972022 \cdot 10^{-205}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{elif}\;z \le 1.123685082691174413471401852132580668372 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(z + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))