Average Error: 1.6 → 0.8
Time: 5.5s
Precision: binary64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.6209712599203426 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq -3.864679112187972 \cdot 10^{-280}:\\ \;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array} \]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.6209712599203426 \cdot 10^{+31}:\\
\;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\

\mathbf{elif}\;x \leq -3.864679112187972 \cdot 10^{-280}:\\
\;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\


\end{array}

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6209712599203426e+31)
   (fabs (* (/ x y) (- 1.0 z)))
   (if (<= x -3.864679112187972e-280)
     (fabs (/ (- x (fma x z -4.0)) y))
     (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))))
double code(double x, double y, double z) {
	return fabs(((x + 4.0) / y) - ((x / y) * z));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6209712599203426e+31) {
		tmp = fabs((x / y) * (1.0 - z));
	} else if (x <= -3.864679112187972e-280) {
		tmp = fabs((x - fma(x, z, -4.0)) / y);
	} else {
		tmp = fabs(((x + 4.0) / y) - ((x / y) * z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 3 regimes

  2. if x < -1.6209712599203426e31

    1. Initial program 0.1

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

    2. Taylor expanded in x around inf 0.3

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

    3. Simplified0.1

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

    if -1.6209712599203426e31 < x < -3.8646791121879721e-280

    1. Initial program 2.5

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

    2. Simplified0.1

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

    if -3.8646791121879721e-280 < x

    1. Initial program 1.5

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

    2. Applied clear-num_binary641.6

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

    3. Taylor expanded in y around 0 1.5

      \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x}{y} \cdot z\right| \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6209712599203426 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq -3.864679112187972 \cdot 10^{-280}:\\ \;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))