Average Error: 12.7 → 2.0
Time: 1.9s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.025060815940221284599523414245967851062 \cdot 10^{-116} \lor \neg \left(x \le 2.033791867065685906604667525203590533578 \cdot 10^{-67}\right):\\ \;\;\;\;x \cdot \frac{y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.025060815940221284599523414245967851062 \cdot 10^{-116} \lor \neg \left(x \le 2.033791867065685906604667525203590533578 \cdot 10^{-67}\right):\\
\;\;\;\;x \cdot \frac{y}{z} + x\\

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

\end{array}
double f(double x, double y, double z) {
        double r432750 = x;
        double r432751 = y;
        double r432752 = z;
        double r432753 = r432751 + r432752;
        double r432754 = r432750 * r432753;
        double r432755 = r432754 / r432752;
        return r432755;
}
↓
double f(double x, double y, double z) {
        double r432756 = x;
        double r432757 = -1.0250608159402213e-116;
        bool r432758 = r432756 <= r432757;
        double r432759 = 2.033791867065686e-67;
        bool r432760 = r432756 <= r432759;
        double r432761 = !r432760;
        bool r432762 = r432758 || r432761;
        double r432763 = y;
        double r432764 = z;
        double r432765 = r432763 / r432764;
        double r432766 = r432756 * r432765;
        double r432767 = r432766 + r432756;
        double r432768 = r432756 * r432763;
        double r432769 = r432768 / r432764;
        double r432770 = r432769 + r432756;
        double r432771 = r432762 ? r432767 : r432770;
        return r432771;
}

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

Original12.7
Target3.2
Herbie2.0
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0250608159402213e-116 or 2.033791867065686e-67 < x

    1. Initial program 16.9

      \[\frac{x \cdot \left(y + z\right)}{z}\]

    2. Taylor expanded around 0 5.9

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

    4. Applied *-un-lft-identity5.9

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

    5. Applied times-frac0.7

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

    6. Simplified0.7

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

    if -1.0250608159402213e-116 < x < 2.033791867065686e-67

    1. Initial program 6.5

      \[\frac{x \cdot \left(y + z\right)}{z}\]

    2. Taylor expanded around 0 3.9

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.025060815940221284599523414245967851062 \cdot 10^{-116} \lor \neg \left(x \le 2.033791867065685906604667525203590533578 \cdot 10^{-67}\right):\\ \;\;\;\;x \cdot \frac{y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

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

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