Average Error: 8.3 → 0.1
Time: 2.0s
Precision: 64
\[\frac{x \cdot y}{y + 1}\]
↓
\[\frac{x}{\mathsf{fma}\left(1, \frac{1}{y}, 1\right)}\]
\frac{x \cdot y}{y + 1}
↓
\frac{x}{\mathsf{fma}\left(1, \frac{1}{y}, 1\right)}
double code(double x, double y) {
	return ((x * y) / (y + 1.0));
}
↓
double code(double x, double y) {
	return (x / fma(1.0, (1.0 / y), 1.0));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.3
Target0.0
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Initial program 8.3

    \[\frac{x \cdot y}{y + 1}\]
  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. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020078 +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)))