Average Error: 8.1 → 0.1
Time: 1.8s
Precision: 64
\[\frac{x \cdot y}{y + 1}\]
↓
\[\frac{x}{1 \cdot \frac{1}{y} + 1}\]
\frac{x \cdot y}{y + 1}
↓
\frac{x}{1 \cdot \frac{1}{y} + 1}
double f(double x, double y) {
        double r939969 = x;
        double r939970 = y;
        double r939971 = r939969 * r939970;
        double r939972 = 1.0;
        double r939973 = r939970 + r939972;
        double r939974 = r939971 / r939973;
        return r939974;
}
↓
double f(double x, double y) {
        double r939975 = x;
        double r939976 = 1.0;
        double r939977 = 1.0;
        double r939978 = y;
        double r939979 = r939977 / r939978;
        double r939980 = r939976 * r939979;
        double r939981 = r939980 + r939977;
        double r939982 = r939975 / r939981;
        return r939982;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.1
Target0.0
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Initial program 8.1

    \[\frac{x \cdot y}{y + 1}\]
  2. Using strategy rm

  3. Applied associate-/l*0.1

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

  4. Taylor expanded around 0 0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019318 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891003) (/ (* x y) (+ y 1)) (- (/ x (* y y)) (- (/ x y) x))))

  (/ (* x y) (+ y 1)))