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

Average Error: 5.9 → 0.8

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -5.7077180361223938 \cdot 10^{171}:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 7.14285184082454088 \cdot 10^{130}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a} + \mathsf{fma}\left(-t, \frac{y}{a}, x\right)\\ \end{array}\]

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 (((y * (z - t)) <= -5.707718036122394e+171)) {
		temp = ((y * ((z - t) / a)) + x);
	} else {
		double temp_1;
		if (((y * (z - t)) <= 7.142851840824541e+130)) {
			temp_1 = (x + ((y * (z - t)) / a));
		} else {
			temp_1 = ((z * (y / a)) + fma(-t, (y / a), 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	5.9
Target	0.8
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (* y (- z t)) < -5.707718036122394e+171

   1. Initial program 23.8

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

   2. Simplified 1.2

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

   4. Applied fma-udef 1.2

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

   6. Applied div-inv 1.3

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

   7. Applied associate-*l* 1.8

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

   8. Simplified 1.7

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

   if -5.707718036122394e+171 < (* y (- z t)) < 7.142851840824541e+130

   1. Initial program 0.5

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

   if 7.142851840824541e+130 < (* y (- z t))

   1. Initial program 18.5

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

   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 1.7

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

   6. Applied sub-neg 1.7

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

   7. Applied distribute-rgt-in 1.7

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

   8. Applied associate-+l+ 1.7

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

   9. Simplified 1.7

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -5.7077180361223938 \cdot 10^{171}:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 7.14285184082454088 \cdot 10^{130}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a} + \mathsf{fma}\left(-t, \frac{y}{a}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))