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

Average Error: 10.4 → 0.4

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 4.4139637721785413 \cdot 10^{278}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{z - a}{y}} + x\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if ((((y * (z - t)) / (z - a)) <= -inf.0)) {
		temp = fma((y / (z - a)), (z - t), x);
	} else {
		double temp_1;
		if ((((y * (z - t)) / (z - a)) <= 4.4139637721785413e+278)) {
			temp_1 = (x + ((y * (z - t)) / (z - a)));
		} else {
			temp_1 = (((z - t) / ((z - a) / y)) + x);
		}
		temp = temp_1;
	}
	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	1.4
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (* y (- z t)) (- z a)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.2

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

   if -inf.0 < (/ (* y (- z t)) (- z a)) < 4.4139637721785413e+278

   1. Initial program 0.3

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

   if 4.4139637721785413e+278 < (/ (* y (- z t)) (- z a))

   1. Initial program 59.7

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

   2. Simplified 1.5

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

   4. Applied clear-num 1.6

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

   6. Applied fma-udef 1.6

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

   7. Simplified 1.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 4.4139637721785413 \cdot 10^{278}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{z - a}{y}} + x\\ \end{array}\]

Reproduce

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

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

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