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

Average Error: 14.7 → 0.2

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -6.229867897237334 \cdot 10^{-289}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 1.5569973242013258 \cdot 10^{-271}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 1.3395959204687884 \cdot 10^{+267}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((y / z) <= ((double) -(((double) INFINITY))))) {
		VAR = ((double) (((double) (y * x)) * (1.0 / z)));
	} else {
		double VAR_1;
		if (((y / z) <= -6.229867897237334e-289)) {
			VAR_1 = ((double) ((y / z) * x));
		} else {
			double VAR_2;
			if (((y / z) <= 1.5569973242013258e-271)) {
				VAR_2 = ((double) (((double) (y * x)) * (1.0 / z)));
			} else {
				double VAR_3;
				if (((y / z) <= 1.3395959204687884e+267)) {
					VAR_3 = ((double) ((y / z) * x));
				} else {
					VAR_3 = (1.0 / ((z / x) / y));
				}
				VAR_2 = VAR_3;
			}
			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	1.6
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} < -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -inf.0 or -6.22986789723733411e-289 < (/ y z) < 1.5569973242013258e-271

   1. Initial program 24.5

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

   2. Simplified 21.8

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

   4. Applied div-inv 21.8

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

   5. Applied associate-*r* 0.2

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

   if -inf.0 < (/ y z) < -6.22986789723733411e-289 or 1.5569973242013258e-271 < (/ y z) < 1.3395959204687884e267

   1. Initial program 10.3

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

   2. Simplified 0.2

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

   if 1.3395959204687884e267 < (/ y z)

   1. Initial program 51.1

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

   2. Simplified 38.3

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

   4. Applied associate-*r/ 0.2

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

   6. Applied clear-num 0.3

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]
   7. Using strategy rm

   8. Applied associate-/r* 0.5

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -6.229867897237334 \cdot 10^{-289}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 1.5569973242013258 \cdot 10^{-271}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 1.3395959204687884 \cdot 10^{+267}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))