Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E

Average Error: 6.4 → 1.1

Time: 2.5s

Precision: 64

Math

\[x + \frac{y \cdot \left(z - t\right)}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.96605357789947992 \cdot 10^{-23}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a} + x\\ \mathbf{elif}\;a \le 1.23690458140457419 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return (x + ((y * (z - t)) / a));
}

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if ((a <= -1.96605357789948e-23)) {
		temp = (((z - t) * (y / a)) + x);
	} else {
		double temp_1;
		if ((a <= 1.2369045814045742e-16)) {
			temp_1 = ((((z - t) * y) / a) + x);
		} else {
			temp_1 = (1.0 * fma(((z - t) / a), y, x));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.4
Target	0.7
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -1.96605357789948e-23

   1. Initial program 9.3

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]

   2. Simplified 1.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 1.9

      \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(z - t\right) + x}\]
   5. Using strategy rm

   6. Applied *-commutative 1.9

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

   if -1.96605357789948e-23 < a < 1.2369045814045742e-16

   1. Initial program 0.9

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]

   2. Simplified 3.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 3.7

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

   5. Taylor expanded around 0 0.9

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

   6. Simplified 0.9

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

   if 1.2369045814045742e-16 < a

   1. Initial program 9.4

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]

   2. Simplified 1.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 1.6

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

   5. Taylor expanded around 0 9.4

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

   6. Simplified 9.4

      \[\leadsto \color{blue}{\frac{\left(z - t\right) \cdot y}{a}} + x\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 9.4

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

   9. Applied *-un-lft-identity 9.4

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

   10. Applied distribute-lft-out 9.4

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

   11. Simplified 0.5

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.96605357789947992 \cdot 10^{-23}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a} + x\\ \mathbf{elif}\;a \le 1.23690458140457419 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))