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

Average Error: 13.8 → 2.8

Time: 4.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.287923571909922 \cdot 10^{100}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.3790989800665981 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 8.2442590521453074 \cdot 10^{-97}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(y \cdot \frac{\sqrt[3]{x}}{z}\right)\\ \end{array}\]

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)) <= -4.2879235719099215e+100)) {
		VAR = ((double) (((double) (y * x)) * ((double) (1.0 / z))));
	} else {
		double VAR_1;
		if ((((double) (y / z)) <= -1.379098980066598e-229)) {
			VAR_1 = ((double) (((double) (y / z)) * x));
		} else {
			double VAR_2;
			if ((((double) (y / z)) <= 8.244259052145307e-97)) {
				VAR_2 = ((double) (((double) (y * x)) / z));
			} else {
				VAR_2 = ((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) * ((double) (y * ((double) (((double) cbrt(x)) / z))))));
			}
			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	13.8
Target	1.5
Herbie	2.8

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

Derivation

1. Split input into 4 regimes

2. if (/ y z) < -4.287923571909922e100

   1. Initial program 27.1

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

   2. Simplified 12.3

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

   4. Applied div-inv 12.4

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

   5. Applied associate-*r* 4.6

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

   if -4.287923571909922e100 < (/ y z) < -1.3790989800665981e-229

   1. Initial program 7.1

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

   2. Simplified 0.2

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

   if -1.3790989800665981e-229 < (/ y z) < 8.2442590521453074e-97

   1. Initial program 15.0

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

   2. Simplified 8.1

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

   4. Applied associate-*r/ 1.7

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

   if 8.2442590521453074e-97 < (/ y z)

   1. Initial program 12.9

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

   2. Simplified 4.9

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

   4. Applied add-cube-cbrt 6.0

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

   5. Applied associate-*l* 6.0

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

   6. Simplified 5.9

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -4.287923571909922 \cdot 10^{100}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.3790989800665981 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 8.2442590521453074 \cdot 10^{-97}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(y \cdot \frac{\sqrt[3]{x}}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))