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

Average Error: 13.9 → 1.6

Time: 54.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.226279784595816 \cdot 10^{271}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{-z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.0435942227704809 \cdot 10^{-301}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 2.40252814846900272 \cdot 10^{-189}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{y}{\sqrt[3]{z}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x}}}\\ \end{array}\]

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)) <= -1.226279784595816e+271)) {
		VAR = ((double) (((double) (x * ((double) -(y)))) / ((double) -(z))));
	} else {
		double VAR_1;
		if ((((double) (y / z)) <= -1.0435942227704809e-301)) {
			VAR_1 = ((double) (x * ((double) (y / z))));
		} else {
			double VAR_2;
			if ((((double) (y / z)) <= 2.4025281484690027e-189)) {
				VAR_2 = ((double) (((double) (x * ((double) -(y)))) / ((double) -(z))));
			} else {
				VAR_2 = ((double) (((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) * ((double) (y / ((double) cbrt(z)))))) / ((double) (((double) (((double) cbrt(z)) * ((double) cbrt(z)))) / ((double) cbrt(x))))));
			}
			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	13.9
Target	1.4
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -1.226279784595816e+271 or -1.0435942227704809e-301 < (/ y z) < 2.4025281484690027e-189

   1. Initial program 21.9

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

   2. Simplified 16.4

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

   4. Applied frac-2neg 16.4

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

   5. Applied associate-*r/ 0.3

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

   if -1.226279784595816e+271 < (/ y z) < -1.0435942227704809e-301

   1. Initial program 9.3

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

   2. Simplified 0.2

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

   if 2.4025281484690027e-189 < (/ y z)

   1. Initial program 13.0

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

   2. Simplified 4.8

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

   4. Applied add-cube-cbrt 5.8

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

   5. Applied *-un-lft-identity 5.8

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

   6. Applied times-frac 5.8

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

   7. Applied associate-*r* 6.9

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

   8. Simplified 6.9

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

   10. Applied add-cube-cbrt 7.1

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

   11. Applied associate-/l* 7.1

       \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x}}}} \cdot \frac{y}{\sqrt[3]{z}}\]

   12. Applied associate-*l/ 4.0

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.226279784595816 \cdot 10^{271}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{-z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.0435942227704809 \cdot 10^{-301}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 2.40252814846900272 \cdot 10^{-189}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{y}{\sqrt[3]{z}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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)))