Average Error: 12.6 → 3.5
Time: 1.9s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[x \cdot \frac{y + z}{z}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
x \cdot \frac{y + z}{z}
double f(double x, double y, double z) {
        double r438641 = x;
        double r438642 = y;
        double r438643 = z;
        double r438644 = r438642 + r438643;
        double r438645 = r438641 * r438644;
        double r438646 = r438645 / r438643;
        return r438646;
}
↓
double f(double x, double y, double z) {
        double r438647 = x;
        double r438648 = y;
        double r438649 = z;
        double r438650 = r438648 + r438649;
        double r438651 = r438650 / r438649;
        double r438652 = r438647 * r438651;
        return r438652;
}

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
Herbie3.5
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.6

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

  3. Applied *-un-lft-identity12.6

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

  4. Applied times-frac3.5

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

  5. Simplified3.5

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

  6. Final simplification3.5

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

Reproduce

herbie shell --seed 2020025 
(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))