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

Average Error: 6.3 → 0.4

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t_1 \leq 2.2508171281552814 \cdot 10^{+304}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a}\right) - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) a)))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z t) a) x)
     (if (<= t_1 2.2508171281552814e+304)
       (- (+ x (/ (* y z) a)) (/ (* y t) a))
       (fma (/ y a) (- z t) x)))))

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 t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2.2508171281552814e+304) {
		tmp = (x + ((y * z) / a)) - ((y * t) / a);
	} else {
		tmp = fma((y / a), (z - t), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.3
Target	0.7
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.2

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.2508171281552814e304

   1. Initial program 0.4

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

   2. Simplified 7.0

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

   3. Taylor expanded in y around 0 0.4

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

   if 2.2508171281552814e304 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 60.2

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

   2. Simplified 1.9

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

   3. Taylor expanded in y around 0 60.2

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

   4. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2.2508171281552814 \cdot 10^{+304}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{a}\right) - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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 (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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