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

Average Error: 15.3 → 0.5

Time: 1.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -2.76363593143672944 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 4.03989522891504801 \cdot 10^{-229}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.5045101283005079 \cdot 10^{295}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{-z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((y / z) <= -inf.0)) {
		VAR = (y * (x / z));
	} else {
		double VAR_1;
		if (((y / z) <= -2.7636359314367294e-155)) {
			VAR_1 = (x / (z / y));
		} else {
			double VAR_2;
			if (((y / z) <= 4.039895228915048e-229)) {
				VAR_2 = (y * (x / z));
			} else {
				double VAR_3;
				if (((y / z) <= 6.504510128300508e+295)) {
					VAR_3 = (x / (z / y));
				} else {
					VAR_3 = ((x * -y) / -z);
				}
				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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -inf.0 or -2.7636359314367294e-155 < (/ y z) < 4.039895228915048e-229

   1. Initial program 21.4

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

   2. Simplified 14.6

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

   4. Applied frac-2neg 14.6

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

   5. Applied associate-*r/ 0.8

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

   7. Applied neg-mul-1 0.8

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

   8. Applied *-commutative 0.8

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

   9. Applied times-frac 1.0

      \[\leadsto \color{blue}{\frac{-y}{-1} \cdot \frac{x}{z}}\]

   10. Simplified 1.0

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

   if -inf.0 < (/ y z) < -2.7636359314367294e-155 or 4.039895228915048e-229 < (/ y z) < 6.504510128300508e+295

   1. Initial program 10.2

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

   2. Simplified 0.3

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

   4. Applied clear-num 0.3

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{z}{y}}}\]

   5. Applied un-div-inv 0.2

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

   if 6.504510128300508e+295 < (/ y z)

   1. Initial program 60.2

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

   2. Simplified 56.9

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

   4. Applied frac-2neg 56.9

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

   5. Applied associate-*r/ 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -2.76363593143672944 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 4.03989522891504801 \cdot 10^{-229}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.5045101283005079 \cdot 10^{295}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{-z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  :precision binary64
  (* x (/ (* (/ y z) t) t)))