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

Average Error: 1.4 → 0.8

Time: 7.0s

Precision: 64

Math

\[x + y \cdot \frac{z - t}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.6662664711442952 \cdot 10^{-34} \lor \neg \left(y \le 1.9897176974016923 \cdot 10^{-212}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t} - \frac{1}{\frac{z - t}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{a - t} \cdot \left(y \cdot \left(z - t\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if (((y <= -1.6662664711442952e-34) || !(y <= 1.9897176974016923e-212))) {
		temp = (x + (y / ((a / (z - t)) - (1.0 / ((z - t) / t)))));
	} else {
		temp = (x + ((1.0 / (a - t)) * (y * (z - t))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.4
Target	0.5
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -1.6662664711442952e-34 or 1.9897176974016923e-212 < y

   1. Initial program 0.9

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

   3. Applied clear-num 1.0

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

   4. Applied un-div-inv 0.9

      \[\leadsto x + \color{blue}{\frac{y}{\frac{a - t}{z - t}}}\]
   5. Using strategy rm

   6. Applied div-sub 0.9

      \[\leadsto x + \frac{y}{\color{blue}{\frac{a}{z - t} - \frac{t}{z - t}}}\]
   7. Using strategy rm

   8. Applied clear-num 0.9

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

   if -1.6662664711442952e-34 < y < 1.9897176974016923e-212

   1. Initial program 2.5

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

   3. Applied clear-num 2.5

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

   4. Applied un-div-inv 2.3

      \[\leadsto x + \color{blue}{\frac{y}{\frac{a - t}{z - t}}}\]
   5. Using strategy rm

   6. Applied div-inv 2.3

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

   7. Applied *-un-lft-identity 2.3

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

   8. Applied times-frac 0.5

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

   9. Simplified 0.4

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.6662664711442952 \cdot 10^{-34} \lor \neg \left(y \le 1.9897176974016923 \cdot 10^{-212}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t} - \frac{1}{\frac{z - t}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{a - t} \cdot \left(y \cdot \left(z - t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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