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

Average Error: 24.7 → 11.8

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.3811946255390914 \cdot 10^{145}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{elif}\;z \le -2.17773616923341961 \cdot 10^{-82}:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{elif}\;z \le -7.5007130778928983 \cdot 10^{-263}:\\ \;\;\;\;x + \frac{1}{a - z} \cdot \left(\left(y - z\right) \cdot \left(t - x\right)\right)\\ \mathbf{elif}\;z \le 6.866555469574397 \cdot 10^{171}:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -7.381194625539091e+145)) {
		VAR = ((double) (t + ((double) (y * ((double) (((double) (x / z)) - ((double) (t / z))))))));
	} else {
		double VAR_1;
		if ((z <= -2.1777361692334196e-82)) {
			VAR_1 = ((double) (x + ((double) (((double) (y - z)) * ((double) (((double) (t - x)) * ((double) (1.0 / ((double) (a - z))))))))));
		} else {
			double VAR_2;
			if ((z <= -7.500713077892898e-263)) {
				VAR_2 = ((double) (x + ((double) (((double) (1.0 / ((double) (a - z)))) * ((double) (((double) (y - z)) * ((double) (t - x))))))));
			} else {
				double VAR_3;
				if ((z <= 6.866555469574397e+171)) {
					VAR_3 = ((double) (x + ((double) (((double) (y - z)) * ((double) (((double) (t - x)) * ((double) (1.0 / ((double) (a - z))))))))));
				} else {
					VAR_3 = ((double) (t + ((double) (y * ((double) (((double) (x / z)) - ((double) (t / z))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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	24.7
Target	12.2
Herbie	11.8

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -7.3811946255390914e145 or 6.866555469574397e171 < z

   1. Initial program 47.1

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

   2. Simplified 29.7

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

   3. Taylor expanded around inf 24.8

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

   4. Simplified 15.7

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

   if -7.3811946255390914e145 < z < -2.17773616923341961e-82 or -7.5007130778928983e-263 < z < 6.866555469574397e171

   1. Initial program 17.8

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

   2. Simplified 10.7

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

   4. Applied div-inv 10.7

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

   if -2.17773616923341961e-82 < z < -7.5007130778928983e-263

   1. Initial program 8.2

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

   2. Simplified 7.1

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

   4. Applied div-inv 7.1

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

   5. Applied associate-*r* 8.3

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.3811946255390914 \cdot 10^{145}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{elif}\;z \le -2.17773616923341961 \cdot 10^{-82}:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{elif}\;z \le -7.5007130778928983 \cdot 10^{-263}:\\ \;\;\;\;x + \frac{1}{a - z} \cdot \left(\left(y - z\right) \cdot \left(t - x\right)\right)\\ \mathbf{elif}\;z \le 6.866555469574397 \cdot 10^{171}:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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