Average Error: 8.2 → 0.1
Time: 1.5s
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 r704710 = x;
        double r704711 = y;
        double r704712 = r704710 * r704711;
        double r704713 = 1.0;
        double r704714 = r704711 + r704713;
        double r704715 = r704712 / r704714;
        return r704715;
}
↓
double f(double x, double y) {
        double r704716 = x;
        double r704717 = 1.0;
        double r704718 = 1.0;
        double r704719 = y;
        double r704720 = r704718 / r704719;
        double r704721 = r704717 * r704720;
        double r704722 = r704721 + r704718;
        double r704723 = r704716 / r704722;
        return r704723;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.2
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 8.2

    \[\frac{x \cdot y}{y + 1}\]
  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 2019354 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891) (/ (* x y) (+ y 1)) (- (/ x (* y y)) (- (/ x y) x))))

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