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

Average Error: 7.4 → 6.2

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -3.3763009443507461 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

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 temp;
	if (((((x + y) / (1.0 - (y / z))) <= -3.376300944350746e-298) || !(((x + y) / (1.0 - (y / z))) <= -0.0))) {
		temp = ((x + y) / (1.0 - (y / z)));
	} else {
		temp = (((cbrt((x + y)) * cbrt((x + y))) / (sqrt(1.0) + (sqrt(y) / sqrt(z)))) * (cbrt((x + y)) / (sqrt(1.0) - (sqrt(y) / sqrt(z)))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.4
Target	3.9
Herbie	6.2

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x y) (- 1.0 (/ y z))) < -3.376300944350746e-298 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

   1. Initial program 0.1

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

   if -3.376300944350746e-298 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

   1. Initial program 59.3

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

   3. Applied add-sqr-sqrt 61.3

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

   4. Applied add-sqr-sqrt 62.6

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

   5. Applied times-frac 62.6

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

   6. Applied add-sqr-sqrt 62.6

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

   7. Applied difference-of-squares 62.6

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

   8. Applied add-cube-cbrt 62.6

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

   9. Applied times-frac 49.5

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

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -3.3763009443507461 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))