fabs fraction 1

Average Error: 1.6 → 0.1

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.12093481890030686:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{-y}, \frac{x + 4}{y}\right) + \frac{z}{-y} \cdot \left(x + \left(-x\right)\right)\right|\\ \mathbf{elif}\;x \le 3.1968188028043897 \cdot 10^{47}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -\left(x + 4\right)\right)}{-y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

C

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 VAR;
	if ((x <= -0.12093481890030686)) {
		VAR = fabs((fma(x, (z / -y), ((x + 4.0) / y)) + ((z / -y) * (x + -x))));
	} else {
		double VAR_1;
		if ((x <= 3.1968188028043897e+47)) {
			VAR_1 = fabs((fma(z, x, -(x + 4.0)) / -y));
		} else {
			VAR_1 = fabs((((x + 4.0) / y) - (z / (y / x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Split input into 3 regimes

2. if x < -0.12093481890030686

   1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
   2. Using strategy rm

   3. Applied clear-num 0.2

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

   4. Applied associate-*l/ 0.1

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

   5. Simplified 0.1

      \[\leadsto \left|\frac{x + 4}{y} - \frac{\color{blue}{z}}{\frac{y}{x}}\right|\]
   6. Using strategy rm

   7. Applied frac-2neg 0.1

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

   8. Applied associate-/r/ 0.1

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

   9. Applied div-inv 0.3

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

   10. Applied prod-diff 0.3

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

   11. Simplified 0.1

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

   12. Simplified 0.1

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

   if -0.12093481890030686 < x < 3.1968188028043897e+47

   1. Initial program 2.4

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
   2. Using strategy rm

   3. Applied frac-2neg 2.4

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

   4. Applied associate-*l/ 0.2

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

   5. Applied frac-2neg 0.2

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

   6. Applied sub-div 0.2

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

   7. Simplified 0.2

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

   if 3.1968188028043897e+47 < x

   1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
   2. Using strategy rm

   3. Applied clear-num 0.2

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

   4. Applied associate-*l/ 0.1

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

   5. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.12093481890030686:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{-y}, \frac{x + 4}{y}\right) + \frac{z}{-y} \cdot \left(x + \left(-x\right)\right)\right|\\ \mathbf{elif}\;x \le 3.1968188028043897 \cdot 10^{47}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -\left(x + 4\right)\right)}{-y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))