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

Average Error: 14.2 → 0.4

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -5.77198757729635198 \cdot 10^{244}:\\ \;\;\;\;{\left(y \cdot \frac{x}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le -2.09577494757252 \cdot 10^{-174}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.2250350955876178 \cdot 10^{-236}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 3.34650733303407685 \cdot 10^{227}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((y / z) <= -5.771987577296352e+244)) {
		VAR = pow((y * (x / z)), 1.0);
	} else {
		double VAR_1;
		if (((y / z) <= -2.09577494757252e-174)) {
			VAR_1 = (x * (y / z));
		} else {
			double VAR_2;
			if (((y / z) <= 1.2250350955876178e-236)) {
				VAR_2 = pow(((x * y) / z), 1.0);
			} else {
				double VAR_3;
				if (((y / z) <= 3.346507333034077e+227)) {
					VAR_3 = (x * (y / z));
				} else {
					VAR_3 = ((x * y) * (1.0 / 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 4 regimes

2. if (/ y z) < -5.771987577296352e+244

   1. Initial program 49.3

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

   2. Simplified 35.0

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

   4. Applied *-un-lft-identity 35.0

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

   5. Applied add-cube-cbrt 35.5

      \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot z}\]

   6. Applied times-frac 35.5

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{z}\right)}\]

   7. Applied associate-*r* 8.7

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{z}}\]

   8. Simplified 8.7

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x\right)} \cdot \frac{\sqrt[3]{y}}{z}\]
   9. Using strategy rm

   10. Applied associate-*l* 9.0

       \[\leadsto \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x \cdot \frac{\sqrt[3]{y}}{z}\right)}\]
   11. Using strategy rm

   12. Applied pow1 9.0

       \[\leadsto \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x \cdot \color{blue}{{\left(\frac{\sqrt[3]{y}}{z}\right)}^{1}}\right)\]

   13. Applied pow1 9.0

       \[\leadsto \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\color{blue}{{x}^{1}} \cdot {\left(\frac{\sqrt[3]{y}}{z}\right)}^{1}\right)\]

   14. Applied pow-prod-down 9.0

       \[\leadsto \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \color{blue}{{\left(x \cdot \frac{\sqrt[3]{y}}{z}\right)}^{1}}\]

   15. Applied pow1 9.0

       \[\leadsto \left(\sqrt[3]{y} \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right) \cdot {\left(x \cdot \frac{\sqrt[3]{y}}{z}\right)}^{1}\]

   16. Applied pow1 9.0

       \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {\left(x \cdot \frac{\sqrt[3]{y}}{z}\right)}^{1}\]

   17. Applied pow-prod-down 9.0

       \[\leadsto \color{blue}{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}} \cdot {\left(x \cdot \frac{\sqrt[3]{y}}{z}\right)}^{1}\]

   18. Applied pow-prod-down 9.0

       \[\leadsto \color{blue}{{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x \cdot \frac{\sqrt[3]{y}}{z}\right)\right)}^{1}}\]

   19. Simplified 0.6

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

   if -5.771987577296352e+244 < (/ y z) < -2.09577494757252e-174 or 1.2250350955876178e-236 < (/ y z) < 3.346507333034077e+227

   1. Initial program 8.5

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

   2. Simplified 0.2

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

   if -2.09577494757252e-174 < (/ y z) < 1.2250350955876178e-236

   1. Initial program 16.1

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

   2. Simplified 10.1

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

   4. Applied *-un-lft-identity 10.1

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

   5. Applied add-cube-cbrt 10.4

      \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot z}\]

   6. Applied times-frac 10.4

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{z}\right)}\]

   7. Applied associate-*r* 2.7

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{z}}\]

   8. Simplified 2.7

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x\right)} \cdot \frac{\sqrt[3]{y}}{z}\]
   9. Using strategy rm

   10. Applied pow1 2.7

       \[\leadsto \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x\right) \cdot \color{blue}{{\left(\frac{\sqrt[3]{y}}{z}\right)}^{1}}\]

   11. Applied pow1 2.7

       \[\leadsto \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \color{blue}{{x}^{1}}\right) \cdot {\left(\frac{\sqrt[3]{y}}{z}\right)}^{1}\]

   12. Applied pow1 2.7

       \[\leadsto \left(\left(\sqrt[3]{y} \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right) \cdot {x}^{1}\right) \cdot {\left(\frac{\sqrt[3]{y}}{z}\right)}^{1}\]

   13. Applied pow1 2.7

       \[\leadsto \left(\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {x}^{1}\right) \cdot {\left(\frac{\sqrt[3]{y}}{z}\right)}^{1}\]

   14. Applied pow-prod-down 2.7

       \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}} \cdot {x}^{1}\right) \cdot {\left(\frac{\sqrt[3]{y}}{z}\right)}^{1}\]

   15. Applied pow-prod-down 2.7

       \[\leadsto \color{blue}{{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x\right)}^{1}} \cdot {\left(\frac{\sqrt[3]{y}}{z}\right)}^{1}\]

   16. Applied pow-prod-down 2.7

       \[\leadsto \color{blue}{{\left(\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x\right) \cdot \frac{\sqrt[3]{y}}{z}\right)}^{1}}\]

   17. Simplified 0.6

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

   if 3.346507333034077e+227 < (/ y z)

   1. Initial program 46.5

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

   2. Simplified 32.2

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

   4. Applied div-inv 32.3

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

   5. Applied associate-*r* 1.0

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -5.77198757729635198 \cdot 10^{244}:\\ \;\;\;\;{\left(y \cdot \frac{x}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le -2.09577494757252 \cdot 10^{-174}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.2250350955876178 \cdot 10^{-236}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z}\right)}^{1}\\ \mathbf{elif}\;\frac{y}{z} \le 3.34650733303407685 \cdot 10^{227}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(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)))