Average Error: 12.6 → 3.2
Time: 1.9s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{x}{\frac{z}{y + z}}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{x}{\frac{z}{y + z}}
double f(double x, double y, double z) {
        double r420872 = x;
        double r420873 = y;
        double r420874 = z;
        double r420875 = r420873 + r420874;
        double r420876 = r420872 * r420875;
        double r420877 = r420876 / r420874;
        return r420877;
}
↓
double f(double x, double y, double z) {
        double r420878 = x;
        double r420879 = z;
        double r420880 = y;
        double r420881 = r420880 + r420879;
        double r420882 = r420879 / r420881;
        double r420883 = r420878 / r420882;
        return r420883;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.6
Target3.2
Herbie3.2
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.6

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

  3. Applied associate-/l*3.2

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

  4. Final simplification3.2

    \[\leadsto \frac{x}{\frac{z}{y + z}}\]

Reproduce

herbie shell --seed 2020083 
(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))