(/ (- (* a b) (* c d)) (* a b))

Average Error: 15.4 → 4.8

Time: 1.6s

Precision: binary64

Math

\[\frac{a \cdot b - c \cdot d}{a \cdot b}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot b = -inf.0 \lor \neg \left(a \cdot b \le -2.1650279700724893 \cdot 10^{-76} \lor \neg \left(a \cdot b \le 8.30588671625312569 \cdot 10^{-187} \lor \neg \left(a \cdot b \le 3.01882169834954777 \cdot 10^{142}\right)\right)\right):\\ \;\;\;\;\frac{b - \frac{c \cdot d}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot b - c \cdot d}{a \cdot b}\\ \end{array}\]

C

double code(double a, double b, double c, double d) {
	return ((double) (((double) (((double) (a * b)) - ((double) (c * d)))) / ((double) (a * b))));
}

↓

double code(double a, double b, double c, double d) {
	double VAR;
	if (((((double) (a * b)) <= -inf.0) || !((((double) (a * b)) <= -2.1650279700724893e-76) || !((((double) (a * b)) <= 8.305886716253126e-187) || !(((double) (a * b)) <= 3.018821698349548e+142))))) {
		VAR = ((double) (((double) (b - ((double) (((double) (c * d)) / a)))) / b));
	} else {
		VAR = ((double) (((double) (((double) (a * b)) - ((double) (c * d)))) / ((double) (a * b))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

1. Split input into 2 regimes

2. if (* a b) < -inf.0 or -2.1650279700724893e-76 < (* a b) < 8.30588671625312569e-187 or 3.01882169834954777e142 < (* a b)

   1. Initial program 29.3

      \[\frac{a \cdot b - c \cdot d}{a \cdot b}\]

   2. Simplified 6.7

      \[\leadsto \color{blue}{\frac{b - \frac{c \cdot d}{a}}{b}}\]

   if -inf.0 < (* a b) < -2.1650279700724893e-76 or 8.30588671625312569e-187 < (* a b) < 3.01882169834954777e142

   1. Initial program 3.0

      \[\frac{a \cdot b - c \cdot d}{a \cdot b}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b = -inf.0 \lor \neg \left(a \cdot b \le -2.1650279700724893 \cdot 10^{-76} \lor \neg \left(a \cdot b \le 8.30588671625312569 \cdot 10^{-187} \lor \neg \left(a \cdot b \le 3.01882169834954777 \cdot 10^{142}\right)\right)\right):\\ \;\;\;\;\frac{b - \frac{c \cdot d}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot b - c \cdot d}{a \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (a b c d)
  :name "(/ (- (* a b) (* c d)) (* a b))"
  :precision binary64
  (/ (- (* a b) (* c d)) (* a b)))