Average Error: 12.1 → 2.4
Time: 11.8s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{x}{\frac{z}{z + y}}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{x}{\frac{z}{z + y}}
double f(double x, double y, double z) {
        double r19905653 = x;
        double r19905654 = y;
        double r19905655 = z;
        double r19905656 = r19905654 + r19905655;
        double r19905657 = r19905653 * r19905656;
        double r19905658 = r19905657 / r19905655;
        return r19905658;
}
↓
double f(double x, double y, double z) {
        double r19905659 = x;
        double r19905660 = z;
        double r19905661 = y;
        double r19905662 = r19905660 + r19905661;
        double r19905663 = r19905660 / r19905662;
        double r19905664 = r19905659 / r19905663;
        return r19905664;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.1
Target2.4
Herbie2.4
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.1

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

  3. Applied associate-/l*2.4

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

  4. Final simplification2.4

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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))