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

↓

\[\frac{x}{\mathsf{fma}\left(1, \frac{1}{y}, 1\right)}\]

\frac{x \cdot y}{y + 1}

↓

\frac{x}{\mathsf{fma}\left(1, \frac{1}{y}, 1\right)}

double f(double x, double y) {
        double r636684 = x;
        double r636685 = y;
        double r636686 = r636684 * r636685;
        double r636687 = 1.0;
        double r636688 = r636685 + r636687;
        double r636689 = r636686 / r636688;
        return r636689;
}

↓

double f(double x, double y) {
        double r636690 = x;
        double r636691 = 1.0;
        double r636692 = 1.0;
        double r636693 = y;
        double r636694 = r636692 / r636693;
        double r636695 = fma(r636691, r636694, r636692);
        double r636696 = r636690 / r636695;
        return r636696;
}

Target

Original	8.2
Target	0.0
Herbie	0.1

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

Derivation

Initial program 8.2
\[\frac{x \cdot y}{y + 1}\]
Using strategy rm
Applied associate-/l*0.1
\[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y}}}\]
Taylor expanded around 0 0.1
\[\leadsto \frac{x}{\color{blue}{1 \cdot \frac{1}{y} + 1}}\]
Simplified0.1
\[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(1, \frac{1}{y}, 1\right)}}\]
Final simplification0.1
\[\leadsto \frac{x}{\mathsf{fma}\left(1, \frac{1}{y}, 1\right)}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(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)))

Error

Target

Derivation

Reproduce