Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.6 → 2.0

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 3.6000243022547191 \cdot 10^{-81} \lor \neg \left(z \le 1.95116791513166456 \cdot 10^{237}\right):\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((z <= 3.600024302254719e-81) || !(z <= 1.9511679151316646e+237))) {
		VAR = ((double) (x + ((double) (((double) (y - x)) / ((double) (t / z))))));
	} else {
		VAR = ((double) (x + ((double) (((double) (((double) (y - x)) / t)) * z))));
	}
	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	2.0

\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < 3.600024302254719e-81 or 1.9511679151316646e+237 < z

   1. Initial program 6.1

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

   3. Applied associate-/l* 2.0

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

   if 3.600024302254719e-81 < z < 1.9511679151316646e+237

   1. Initial program 8.1

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

   3. Applied associate-/l* 2.5

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

   5. Applied associate-/r/ 2.0

      \[\leadsto x + \color{blue}{\frac{y - x}{t} \cdot z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 3.6000243022547191 \cdot 10^{-81} \lor \neg \left(z \le 1.95116791513166456 \cdot 10^{237}\right):\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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