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

Average Error: 6.6 → 1.6

Time: 1.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -8.04065500205821708 \cdot 10^{99}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)\\ \mathbf{elif}\;t \le 5.3464729145636056 \cdot 10^{-37}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - x}{\frac{t}{y}} + x\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((t <= -8.040655002058217e+99)) {
		VAR = fma(((z - x) / t), y, x);
	} else {
		double VAR_1;
		if ((t <= 5.3464729145636056e-37)) {
			VAR_1 = (((y * (z - x)) / t) + x);
		} else {
			VAR_1 = (((z - x) / (t / y)) + x);
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	2.0
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if t < -8.040655002058217e+99

   1. Initial program 11.8

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

   2. Simplified 1.0

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

   4. Applied fma-udef 1.0

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

   6. Applied clear-num 1.1

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

   7. Applied associate-*l/ 1.0

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

   8. Simplified 1.0

      \[\leadsto \frac{\color{blue}{z - x}}{\frac{t}{y}} + x\]
   9. Using strategy rm

   10. Applied associate-/r/ 1.3

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

   11. Applied fma-def 1.3

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

   if -8.040655002058217e+99 < t < 5.3464729145636056e-37

   1. Initial program 2.2

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

   2. Simplified 3.2

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

   4. Applied fma-udef 3.2

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

   6. Applied associate-*l/ 2.2

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

   if 5.3464729145636056e-37 < t

   1. Initial program 8.6

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

   2. Simplified 1.1

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

   4. Applied fma-udef 1.1

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

   6. Applied clear-num 1.2

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

   7. Applied associate-*l/ 1.2

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

   8. Simplified 1.2

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.04065500205821708 \cdot 10^{99}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)\\ \mathbf{elif}\;t \le 5.3464729145636056 \cdot 10^{-37}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - x}{\frac{t}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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