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

Average Error: 14.8 → 0.8

Time: 1.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.8517075312100841 \cdot 10^{206}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.28974222975093373 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 0.0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.2588805863233109 \cdot 10^{162}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array}\]

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) <= -4.851707531210084e+206)) {
		VAR = (y * (x / z));
	} else {
		double VAR_1;
		if (((y / z) <= -6.289742229750934e-132)) {
			VAR_1 = (x * (y / z));
		} else {
			double VAR_2;
			if (((y / z) <= 0.0)) {
				VAR_2 = (y / (z / x));
			} else {
				double VAR_3;
				if (((y / z) <= 1.258880586323311e+162)) {
					VAR_3 = (x / (z / y));
				} else {
					VAR_3 = (y / (z / x));
				}
				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 4 regimes

2. if (/ y z) < -4.851707531210084e+206

   1. Initial program 41.0

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

   2. Simplified 26.7

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

   4. Applied div-inv 26.8

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

   5. Applied associate-*r* 1.3

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]
   6. Using strategy rm

   7. Applied *-commutative 1.3

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

   8. Applied associate-*l* 1.1

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

   9. Simplified 1.0

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

   if -4.851707531210084e+206 < (/ y z) < -6.289742229750934e-132

   1. Initial program 7.1

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

   2. Simplified 0.3

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

   if -6.289742229750934e-132 < (/ y z) < 0.0 or 1.258880586323311e+162 < (/ y z)

   1. Initial program 21.8

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

   2. Simplified 13.2

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

   4. Applied associate-*r/ 1.5

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

   6. Applied *-commutative 1.5

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

   7. Applied associate-/l* 1.1

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

   if 0.0 < (/ y z) < 1.258880586323311e+162

   1. Initial program 9.0

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

   2. Simplified 0.4

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

   4. Applied clear-num 0.9

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

   5. Applied un-div-inv 0.8

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.8517075312100841 \cdot 10^{206}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -6.28974222975093373 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 0.0:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{y}{z} \le 1.2588805863233109 \cdot 10^{162}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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)))