Average Error: 1.7 → 0.1
Time: 3.9s
Precision: binary64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
↓
\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{if}\;t_0 \leq 1.733408754029193 \cdot 10^{-55}:\\ \;\;\;\;\left|\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
t_0 := \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\
\mathbf{if}\;t_0 \leq 1.733408754029193 \cdot 10^{-55}:\\
\;\;\;\;\left|\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z)))))
   (if (<= t_0 1.733408754029193e-55)
     (fabs (- (+ (/ x y) (* 4.0 (/ 1.0 y))) (/ (* x z) y)))
     t_0)))
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 t_0 = fabs(((x + 4.0) / y) - ((x / y) * z));
	double tmp;
	if (t_0 <= 1.733408754029193e-55) {
		tmp = fabs(((x / y) + (4.0 * (1.0 / y))) - ((x * z) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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. Split input into 2 regimes

  2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 1.7334087540291929e-55

    1. Initial program 5.2

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

    2. Taylor expanded in x around 0 0.1

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

    if 1.7334087540291929e-55 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)))

    1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 1.733408754029193 \cdot 10^{-55}:\\ \;\;\;\;\left|\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))