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

Average Error: 15.3 → 0.5

Time: 1.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -2.76363593143672944 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 4.03989522891504801 \cdot 10^{-229}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.5045101283005079 \cdot 10^{295}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

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 VAR;
	if (((y / z) <= -inf.0)) {
		VAR = (y * (x / z));
	} else {
		double VAR_1;
		if (((y / z) <= -2.7636359314367294e-155)) {
			VAR_1 = (x / (z / y));
		} else {
			double VAR_2;
			if (((y / z) <= 4.039895228915048e-229)) {
				VAR_2 = (y * (x / z));
			} else {
				double VAR_3;
				if (((y / z) <= 6.504510128300508e+295)) {
					VAR_3 = (x / (z / y));
				} else {
					VAR_3 = ((x * y) / 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

Target

Original	15.3
Target	1.5
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -inf.0 or -2.7636359314367294e-155 < (/ y z) < 4.039895228915048e-229

   1. Initial program 21.4

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

   2. Simplified 14.6

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

   4. Applied associate-*r/ 0.8

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

   6. Applied *-commutative 0.8

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

   7. Applied associate-/l* 0.9

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

   9. Applied div-inv 1.1

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

   10. Simplified 1.0

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

   if -inf.0 < (/ y z) < -2.7636359314367294e-155 or 4.039895228915048e-229 < (/ y z) < 6.504510128300508e+295

   1. Initial program 10.2

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   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 6.504510128300508e+295 < (/ y z)

   1. Initial program 60.2

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

   2. Simplified 56.9

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

   4. Applied associate-*r/ 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} = -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -2.76363593143672944 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 4.03989522891504801 \cdot 10^{-229}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.5045101283005079 \cdot 10^{295}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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)))