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

Average Error: 7.5 → 3.2

Time: 3.6s

Precision: binary64

Math

\[\frac{x + y}{1 - \frac{y}{z}}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.9104075020926874 \cdot 10^{116} \lor \neg \left(y \le 3.90865984670371717 \cdot 10^{199}\right):\\ \;\;\;\;\frac{1}{\frac{1}{y + x} - \sqrt{\frac{y}{y + x}} \cdot \frac{\sqrt{\frac{y}{y + x}}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{y}{z}}{y + x}}\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((double) (((double) (x + y)) / ((double) (1.0 - ((double) (y / z))))));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if (((y <= -1.9104075020926874e+116) || !(y <= 3.908659846703717e+199))) {
		VAR = ((double) (1.0 / ((double) (((double) (1.0 / ((double) (y + x)))) - ((double) (((double) sqrt(((double) (y / ((double) (y + x)))))) * ((double) (((double) sqrt(((double) (y / ((double) (y + x)))))) / z))))))));
	} else {
		VAR = ((double) (1.0 / ((double) (((double) (1.0 - ((double) (y / z)))) / ((double) (y + x))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.5
Target	4.1
Herbie	3.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -1.9104075020926874e116 or 3.90865984670371717e199 < y

   1. Initial program 21.4

      \[\frac{x + y}{1 - \frac{y}{z}}\]
   2. Using strategy rm

   3. Applied clear-num 21.5

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}}\]
   4. Using strategy rm

   5. Applied div-sub 21.5

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

   6. Simplified 21.5

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

   7. Simplified 16.9

      \[\leadsto \frac{1}{\frac{1}{y + x} - \color{blue}{\frac{y}{z \cdot \left(y + x\right)}}}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 16.9

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

   10. Applied times-frac 0.2

       \[\leadsto \frac{1}{\frac{1}{y + x} - \color{blue}{\frac{1}{z} \cdot \frac{y}{y + x}}}\]
   11. Using strategy rm

   12. Applied add-sqr-sqrt 3.7

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

   13. Applied associate-*r* 3.7

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

   14. Simplified 3.7

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

   if -1.9104075020926874e116 < y < 3.90865984670371717e199

   1. Initial program 2.9

      \[\frac{x + y}{1 - \frac{y}{z}}\]
   2. Using strategy rm

   3. Applied clear-num 3.0

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.9104075020926874 \cdot 10^{116} \lor \neg \left(y \le 3.90865984670371717 \cdot 10^{199}\right):\\ \;\;\;\;\frac{1}{\frac{1}{y + x} - \sqrt{\frac{y}{y + x}} \cdot \frac{\sqrt{\frac{y}{y + x}}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{y}{z}}{y + x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (neg y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (neg y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))