Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Average Error: 24.4 → 8.7

Time: 5.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -2.6179859008682989 \cdot 10^{-253} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{z}{t} \cdot \left(x - y\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (z - t)))) / ((double) (a - t)))))) <= -2.617985900868299e-253) || !(((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (z - t)))) / ((double) (a - t)))))) <= 0.0))) {
		VAR = ((double) (x + ((double) (((double) (y - x)) * ((double) (((double) (z - t)) / ((double) (a - t))))))));
	} else {
		VAR = ((double) (y + ((double) (((double) (z / t)) * ((double) (x - y))))));
	}
	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	24.4
Target	9.4
Herbie	8.7

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -2.6179859008682989e-253 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 21.2

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

   2. Simplified 7.5

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

   if -2.6179859008682989e-253 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 57.0

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

   2. Simplified 56.7

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

   4. Applied add-cube-cbrt 56.7

      \[\leadsto x + \left(y - x\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{z - t}{a - t}} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\frac{z - t}{a - t}}\right)}\]

   5. Applied associate-*r* 56.7

      \[\leadsto x + \color{blue}{\left(\left(y - x\right) \cdot \left(\sqrt[3]{\frac{z - t}{a - t}} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right)\right) \cdot \sqrt[3]{\frac{z - t}{a - t}}}\]

   6. Taylor expanded around inf 21.2

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

   7. Simplified 21.1

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -2.6179859008682989 \cdot 10^{-253} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{z}{t} \cdot \left(x - y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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