(- (* (* (* a x) b) y) (* (* (* a z) b) w))

Average Error: 14.8 → 6.1

Time: 2.3s

Precision: binary64

Math

\[\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w \le -5.9086083022026796 \cdot 10^{304} \lor \neg \left(\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w \le -4.6981988480296783 \cdot 10^{-288} \lor \neg \left(\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w \le 4.7715395290706475 \cdot 10^{-99} \lor \neg \left(\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w \le 7.2413267307502334 \cdot 10^{277}\right)\right)\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot \left(x \cdot y - z \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w\\ \end{array}\]

C

double code(double a, double x, double b, double y, double z, double w) {
	return ((double) (((double) (((double) (((double) (a * x)) * b)) * y)) - ((double) (((double) (((double) (a * z)) * b)) * w))));
}

↓

double code(double a, double x, double b, double y, double z, double w) {
	double VAR;
	if (((((double) (((double) (((double) (((double) (a * x)) * b)) * y)) - ((double) (((double) (((double) (a * z)) * b)) * w)))) <= -5.90860830220268e+304) || !((((double) (((double) (((double) (((double) (a * x)) * b)) * y)) - ((double) (((double) (((double) (a * z)) * b)) * w)))) <= -4.698198848029678e-288) || !((((double) (((double) (((double) (((double) (a * x)) * b)) * y)) - ((double) (((double) (((double) (a * z)) * b)) * w)))) <= 4.7715395290706475e-99) || !(((double) (((double) (((double) (((double) (a * x)) * b)) * y)) - ((double) (((double) (((double) (a * z)) * b)) * w)))) <= 7.241326730750233e+277))))) {
		VAR = ((double) (((double) (b * a)) * ((double) (((double) (x * y)) - ((double) (z * w))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (a * x)) * b)) * y)) - ((double) (((double) (((double) (a * z)) * b)) * w))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Bits error versus b

Bits error versus y

Bits error versus z

Bits error versus w

Derivation

1. Split input into 2 regimes

2. if (- (* (* (* a x) b) y) (* (* (* a z) b) w)) < -5.9086083022026796e304 or -4.6981988480296783e-288 < (- (* (* (* a x) b) y) (* (* (* a z) b) w)) < 4.7715395290706475e-99 or 7.2413267307502334e277 < (- (* (* (* a x) b) y) (* (* (* a z) b) w))

   1. Initial program 32.1

      \[\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w\]

   2. Simplified 11.7

      \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot \left(x \cdot y - z \cdot w\right)}\]

   if -5.9086083022026796e304 < (- (* (* (* a x) b) y) (* (* (* a z) b) w)) < -4.6981988480296783e-288 or 4.7715395290706475e-99 < (- (* (* (* a x) b) y) (* (* (* a z) b) w)) < 7.2413267307502334e277

   1. Initial program 2.0

      \[\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w \le -5.9086083022026796 \cdot 10^{304} \lor \neg \left(\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w \le -4.6981988480296783 \cdot 10^{-288} \lor \neg \left(\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w \le 4.7715395290706475 \cdot 10^{-99} \lor \neg \left(\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w \le 7.2413267307502334 \cdot 10^{277}\right)\right)\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot \left(x \cdot y - z \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot b\right) \cdot y - \left(\left(a \cdot z\right) \cdot b\right) \cdot w\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (a x b y z w)
  :name "(- (* (* (* a x) b) y) (* (* (* a z) b) w))"
  :precision binary64
  (- (* (* (* a x) b) y) (* (* (* a z) b) w)))