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

Average Error: 7.4 → 0.3

Time: 7.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.847567984179807 \cdot 10^{-300} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{\frac{1}{1 - \frac{y}{z}}}{\frac{1}{x + y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((x + y) / (1.0 - (y / z))) <= -2.847567984179807e-300) || !(((x + y) / (1.0 - (y / z))) <= 0.0))) {
		VAR = ((1.0 / (1.0 - (y / z))) / (1.0 / (x + y)));
	} else {
		VAR = (1.0 / ((1.0 / (x + y)) - (y / ((x + y) * z))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.4
Target	3.9
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x y) (- 1.0 (/ y z))) < -2.847567984179807e-300 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

   1. Initial program 0.1

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

   3. Applied clear-num 0.3

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

   5. Applied div-inv 0.3

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

   6. Applied associate-/r* 0.3

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

   if -2.847567984179807e-300 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

   1. Initial program 59.8

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

   3. Applied clear-num 59.8

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

   5. Applied div-sub 59.8

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

   7. Applied div-inv 59.8

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

   8. Applied associate-/l* 0.4

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

   9. Simplified 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.847567984179807 \cdot 10^{-300} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{\frac{1}{1 - \frac{y}{z}}}{\frac{1}{x + y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(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) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1 (/ y z))))