Average Error: 1.7 → 1.7
Time: 3.1s
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 f(double x, double y, double z) {
        double r105669 = x;
        double r105670 = 4.0;
        double r105671 = r105669 + r105670;
        double r105672 = y;
        double r105673 = r105671 / r105672;
        double r105674 = r105669 / r105672;
        double r105675 = z;
        double r105676 = r105674 * r105675;
        double r105677 = r105673 - r105676;
        double r105678 = fabs(r105677);
        return r105678;
}
↓
double f(double x, double y, double z) {
        double r105679 = 4.0;
        double r105680 = 1.0;
        double r105681 = y;
        double r105682 = r105680 / r105681;
        double r105683 = x;
        double r105684 = r105683 / r105681;
        double r105685 = fma(r105679, r105682, r105684);
        double r105686 = z;
        double r105687 = r105684 * r105686;
        double r105688 = r105685 - r105687;
        double r105689 = fabs(r105688);
        return r105689;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.7

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

  2. Taylor expanded around 0 1.7

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

  3. Simplified1.7

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

  4. Final simplification1.7

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))