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

Average Error: 23.8 → 10.1

Time: 5.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -4.55707230449148269 \cdot 10^{-145}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \le 2.94058806947867771 \cdot 10^{-209}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \end{array}\]

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 ((a <= -4.557072304491483e-145)) {
		VAR = ((double) (x + ((double) (((double) (y - x)) * ((double) (((double) (z - t)) / ((double) (a - t))))))));
	} else {
		double VAR_1;
		if ((a <= 2.9405880694786777e-209)) {
			VAR_1 = ((double) (((double) (y + ((double) (((double) (x * z)) / t)))) - ((double) (((double) (z * y)) / t))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (y - x)) / ((double) (((double) (a - t)) / ((double) (z - t))))))));
		}
		VAR = VAR_1;
	}
	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	23.8
Target	9.3
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if a < -4.55707230449148269e-145

   1. Initial program 22.0

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

   3. Applied *-un-lft-identity 22.0

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

   4. Applied times-frac 9.3

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

   5. Simplified 9.3

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

   if -4.55707230449148269e-145 < a < 2.94058806947867771e-209

   1. Initial program 30.0

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

   2. Taylor expanded around inf 12.1

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

   if 2.94058806947867771e-209 < a

   1. Initial program 22.8

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

   3. Applied associate-/l* 10.0

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -4.55707230449148269 \cdot 10^{-145}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \le 2.94058806947867771 \cdot 10^{-209}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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))))