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

Average Error: 15.2 → 2.1

Time: 3.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -3.69264767565036338 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 8.1313930047618145 \cdot 10^{-276}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

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 temp;
	if (((y / z) <= -inf.0)) {
		temp = ((y * x) * (1.0 / z));
	} else {
		double temp_1;
		if (((y / z) <= -3.6926476756503634e-178)) {
			temp_1 = (x / (z / y));
		} else {
			double temp_2;
			if (((y / z) <= 8.131393004761814e-276)) {
				temp_2 = ((y * x) * (1.0 / z));
			} else {
				temp_2 = (x * (y / z));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	15.2
Target	1.6
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -inf.0 or -3.6926476756503634e-178 < (/ y z) < 8.131393004761814e-276

   1. Initial program 22.5

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

   2. Simplified 17.3

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

   4. Applied div-inv 17.3

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

   5. Applied associate-*r* 0.6

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

   6. Simplified 0.6

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

   if -inf.0 < (/ y z) < -3.6926476756503634e-178

   1. Initial program 9.9

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   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 8.131393004761814e-276 < (/ y z)

   1. Initial program 14.5

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

   2. Simplified 4.5

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -3.69264767565036338 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 8.1313930047618145 \cdot 10^{-276}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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)))