Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.8 → 1.6

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -3.24141495336215111 \cdot 10^{295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -1.79180666670937744 \cdot 10^{-193}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right) + x\\ \end{array}\]

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 ((((double) (x + ((double) (((double) (((double) (y - x)) * z)) / t)))) <= -3.241414953362151e+295)) {
		VAR = ((double) fma(((double) (((double) (y - x)) / t)), z, x));
	} else {
		double VAR_1;
		if ((((double) (x + ((double) (((double) (((double) (y - x)) * z)) / t)))) <= -1.7918066667093774e-193)) {
			VAR_1 = ((double) (x + ((double) (((double) (((double) (y - x)) * z)) / t))));
		} else {
			VAR_1 = ((double) (((double) (((double) (z / t)) * ((double) (y - x)))) + 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	6.8
Target	2.1
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) z) t)) < -3.241414953362151e+295

   1. Initial program 50.4

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

   2. Simplified 5.5

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

   if -3.241414953362151e+295 < (+ x (/ (* (- y x) z) t)) < -1.7918066667093774e-193

   1. Initial program 0.3

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

   if -1.7918066667093774e-193 < (+ x (/ (* (- y x) z) t))

   1. Initial program 6.6

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

   2. Simplified 6.2

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

   4. Applied div-inv 6.3

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

   5. Taylor expanded around 0 6.6

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

   6. Simplified 5.4

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

   7. Taylor expanded around 0 6.6

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

   8. Simplified 2.1

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -3.24141495336215111 \cdot 10^{295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -1.79180666670937744 \cdot 10^{-193}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(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)))