Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.7 → 6.1

Time: 7.3s

Precision: binary64

Math

\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -5.92276022538686 \cdot 10^{-274}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{z - t}{a - t}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 5.894528428619599 \cdot 10^{-299}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{z - t}{\frac{a - t}{y}}\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a) :precision binary64 (- (+ x y) (/ (* (- z t) y) (- a t))))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (+ x y) (/ (* y (- z t)) (- a t))) -5.92276022538686e-274)
   (+ x (* y (- 1.0 (/ (- z t) (- a t)))))
   (if (<= (- (+ x y) (/ (* y (- z t)) (- a t))) 5.894528428619599e-299)
     (+ x (* y (/ z t)))
     (+ x (- y (/ (- z t) (/ (- a t) y)))))))

C

double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - t));
}

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + y) - ((y * (z - t)) / (a - t))) <= -5.92276022538686e-274) {
		tmp = x + (y * (1.0 - ((z - t) / (a - t))));
	} else if (((x + y) - ((y * (z - t)) / (a - t))) <= 5.894528428619599e-299) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x + (y - ((z - t) / ((a - t) / y)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.7
Target	8.3
Herbie	6.1

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -5.9227602253868597e-274

   1. Initial program 12.9

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied associate-/l*_binary64_17755 7.6

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
   4. Using strategy rm

   5. Applied associate--l+_binary64_17747 5.8

      \[\leadsto \color{blue}{x + \left(y - \frac{z - t}{\frac{a - t}{y}}\right)}\]
   6. Using strategy rm

   7. Applied associate-/r/_binary64_17756 4.9

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

   8. Applied *-un-lft-identity_binary64_17810 4.9

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

   9. Applied distribute-rgt-out--_binary64_17764 4.9

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

   if -5.9227602253868597e-274 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 5.8945284286195986e-299

   1. Initial program 60.3

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied associate-/l*_binary64_17755 60.9

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
   4. Using strategy rm

   5. Applied associate--l+_binary64_17747 44.2

      \[\leadsto \color{blue}{x + \left(y - \frac{z - t}{\frac{a - t}{y}}\right)}\]
   6. Using strategy rm

   7. Applied associate-/r/_binary64_17756 33.4

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

   8. Applied *-un-lft-identity_binary64_17810 33.4

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

   9. Applied distribute-rgt-out--_binary64_17764 33.4

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

   10. Taylor expanded around inf 16.3

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

   if 5.8945284286195986e-299 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 12.5

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied associate-/l*_binary64_17755 7.4

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
   4. Using strategy rm

   5. Applied associate--l+_binary64_17747 5.5

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

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -5.92276022538686 \cdot 10^{-274}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{z - t}{a - t}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 5.894528428619599 \cdot 10^{-299}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{z - t}{\frac{a - t}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))