Average Error: 1.5 → 1.5
Time: 9.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|
double code(double x, double y, double z) {
	return ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (((double) (x / y)) * z))))));
}
↓
double code(double x, double y, double z) {
	return ((double) fabs(((double) (((double) fma(4.0, ((double) (1.0 / y)), ((double) (x / y)))) - ((double) (((double) (x / y)) * z))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.5

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

  2. Taylor expanded around 0 1.5

    \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)} - \frac{x}{y} \cdot z\right|\]

  3. Simplified1.5

    \[\leadsto \left|\color{blue}{\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right)} - \frac{x}{y} \cdot z\right|\]

  4. Final simplification1.5

    \[\leadsto \left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))