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

Average Error: 16.9 → 10.3

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.5863942931958193 \cdot 10^{-254} \lor \neg \left(\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t)))))) <= -1.5863942931958193e-254) || !(((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t)))))) <= 0.0))) {
		VAR = ((double) fma(((double) (y * ((double) (1.0 / ((double) (a - t)))))), ((double) (t - z)), ((double) (x + y))));
	} else {
		VAR = x;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.9
Target	8.5
Herbie	10.3

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -1.5863942931958193e-254 or 0.0 < (- (+ x y) (/ (* (- z t) y) (- a t)))

   1. Initial program 13.0

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

   2. Simplified 8.0

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

   4. Applied div-inv 8.0

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

   if -1.5863942931958193e-254 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 0.0

   1. Initial program 59.0

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

   2. Simplified 59.6

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

   3. Taylor expanded around 0 34.5

      \[\leadsto \color{blue}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.5863942931958193 \cdot 10^{-254} \lor \neg \left(\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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 (- 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 (- a t))) y))))

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