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

Average Error: 7.7 → 0.1

Time: 7.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.5484069609159813 \cdot 10^{-285}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\left(\frac{x \cdot z}{y} + \left(z + \frac{{z}^{2}}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (+ x y) (- 1.0 (/ y z))) -1.5484069609159813e-285)
   (+ (/ x (- 1.0 (/ y z))) (/ y (- 1.0 (/ y z))))
   (if (<= (/ (+ x y) (- 1.0 (/ y z))) 0.0)
     (- (+ (/ (* x z) y) (+ z (/ (pow z 2.0) y))))
     (* (+ x y) (/ 1.0 (- 1.0 (/ y z)))))))

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 tmp;
	if (((x + y) / (1.0 - (y / z))) <= -1.5484069609159813e-285) {
		tmp = (x / (1.0 - (y / z))) + (y / (1.0 - (y / z)));
	} else if (((x + y) / (1.0 - (y / z))) <= 0.0) {
		tmp = -(((x * z) / y) + (z + (pow(z, 2.0) / y)));
	} else {
		tmp = (x + y) * (1.0 / (1.0 - (y / z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	4.0
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.54840696091598132e-285

   1. Initial program 0.1

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

   2. Taylor expanded around 0 0.1

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

   if -1.54840696091598132e-285 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 58.5

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

   3. Applied clear-num_binary64_16104 58.5

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

   4. Simplified 58.5

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

   5. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(\frac{x \cdot z}{y} + \left(z + \frac{{z}^{2}}{y}\right)\right)}\]

   if -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 0.1

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

   3. Applied div-inv_binary64_16102 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1.5484069609159813 \cdot 10^{-285}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\left(\frac{x \cdot z}{y} + \left(z + \frac{{z}^{2}}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(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) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

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