Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 14.7 → 2.5

Time: 3.5s

Precision: binary64

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.76554137336039552 \cdot 10^{-91}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 8.7758563984809059 \cdot 10^{-213}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.9951289073166991 \cdot 10^{197}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((((double) (y / z)) <= -2.7655413733603955e-91)) {
		VAR = ((double) (x * ((double) (y / z))));
	} else {
		double VAR_1;
		if ((((double) (y / z)) <= 8.775856398480906e-213)) {
			VAR_1 = ((double) (((double) (x * y)) / z));
		} else {
			double VAR_2;
			if ((((double) (y / z)) <= 7.995128907316699e+197)) {
				VAR_2 = ((double) (x * ((double) (y / z))));
			} else {
				VAR_2 = ((double) (((double) (x * y)) * ((double) (1.0 / z))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.7
Target	1.4
Herbie	2.5

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045005 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390633 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.65895442315341522 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -2.76554137336039552e-91 or 8.7758563984809059e-213 < (/ y z) < 7.9951289073166991e197

   1. Initial program 11.6

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 3.2

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

   if -2.76554137336039552e-91 < (/ y z) < 8.7758563984809059e-213

   1. Initial program 15.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 8.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 1.7

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

   if 7.9951289073166991e197 < (/ y z)

   1. Initial program 40.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 24.6

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied div-inv 24.8

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

   5. Applied associate-*r* 1.0

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.76554137336039552 \cdot 10^{-91}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 8.7758563984809059 \cdot 10^{-213}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.9951289073166991 \cdot 10^{197}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

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