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

Average Error: 7.8 → 2.2

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.15796120366031069 \cdot 10^{-268} \lor \neg \left(z \le 2.55693185217176228 \cdot 10^{-243}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{x + y}}}{\sqrt[3]{x + y}} \cdot \frac{\frac{\sqrt[3]{y}}{z}}{\sqrt[3]{x + y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

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 (((z <= -2.1579612036603107e-268) || !(z <= 2.5569318521717623e-243))) {
		VAR = (1.0 / ((1.0 / (x + y)) - ((((cbrt(y) * cbrt(y)) / cbrt((x + y))) / cbrt((x + y))) * ((cbrt(y) / z) / cbrt((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	7.8
Target	4.1
Herbie	2.2

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

Derivation

1. Split input into 2 regimes

2. if z < -2.1579612036603107e-268 or 2.5569318521717623e-243 < z

   1. Initial program 5.9

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

   3. Applied clear-num 6.1

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

   5. Applied div-sub 6.1

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

   7. Applied add-cube-cbrt 6.5

      \[\leadsto \frac{1}{\frac{1}{x + y} - \frac{\frac{y}{z}}{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}}\]

   8. Applied *-un-lft-identity 6.5

      \[\leadsto \frac{1}{\frac{1}{x + y} - \frac{\frac{y}{\color{blue}{1 \cdot z}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}\]

   9. Applied add-cube-cbrt 6.3

      \[\leadsto \frac{1}{\frac{1}{x + y} - \frac{\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot z}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}\]

   10. Applied times-frac 6.3

       \[\leadsto \frac{1}{\frac{1}{x + y} - \frac{\color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{z}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}\]

   11. Applied times-frac 1.2

       \[\leadsto \frac{1}{\frac{1}{x + y} - \color{blue}{\frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\frac{\sqrt[3]{y}}{z}}{\sqrt[3]{x + y}}}}\]

   12. Simplified 1.2

       \[\leadsto \frac{1}{\frac{1}{x + y} - \color{blue}{\frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{x + y}}}{\sqrt[3]{x + y}}} \cdot \frac{\frac{\sqrt[3]{y}}{z}}{\sqrt[3]{x + y}}}\]

   if -2.1579612036603107e-268 < z < 2.5569318521717623e-243

   1. Initial program 27.1

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

   3. Applied clear-num 27.2

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

   5. Applied div-sub 27.3

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

   7. Applied div-inv 27.3

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

   8. Applied associate-/l* 12.6

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

   9. Simplified 12.6

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.15796120366031069 \cdot 10^{-268} \lor \neg \left(z \le 2.55693185217176228 \cdot 10^{-243}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{x + y}}}{\sqrt[3]{x + y}} \cdot \frac{\frac{\sqrt[3]{y}}{z}}{\sqrt[3]{x + y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(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))))