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

Average Error: 10.4 → 1.0

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.10784404302477939 \cdot 10^{84} \lor \neg \left(y \le 5.5802422866714181 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a - t} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \end{array}\]

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 <= -1.1078440430247794e+84) || !(y <= 5.580242286671418e-161))) {
		temp = ((((z - t) / (a - t)) * y) + x);
	} else {
		temp = (((1.0 / (a - t)) * ((z - t) * y)) + x);
	}
	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	10.4
Target	1.3
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if y < -1.1078440430247794e+84 or 5.580242286671418e-161 < y

   1. Initial program 18.5

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

   2. Simplified 3.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)}\]
   3. Using strategy rm

   4. Applied clear-num 3.6

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, z - t, x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 3.6

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

   7. Simplified 3.2

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

   9. Applied associate-/r/ 0.7

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

   if -1.1078440430247794e+84 < y < 5.580242286671418e-161

   1. Initial program 1.4

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

   2. Simplified 3.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)}\]
   3. Using strategy rm

   4. Applied clear-num 3.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, z - t, x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 3.2

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

   7. Simplified 3.2

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

   9. Applied div-inv 3.2

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

   10. Applied *-un-lft-identity 3.2

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

   11. Applied times-frac 1.4

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

   12. Simplified 1.4

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.10784404302477939 \cdot 10^{84} \lor \neg \left(y \le 5.5802422866714181 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a - t} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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