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

Average Error: 24.5 → 11.0

Time: 20.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} = -\infty:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.2575042853915238 \cdot 10^{-298}:\\ \;\;\;\;x + \frac{1}{\frac{a - z}{\left(y - z\right) \cdot \left(t - x\right)}}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 3.7317631195897233 \cdot 10^{-242}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 3.0859848969069718 \cdot 10^{305}:\\ \;\;\;\;x + \frac{1}{\frac{a - z}{\left(y - z\right) \cdot \left(t - x\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \end{array}\]

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 ((((double) (x + ((double) (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z)))))) <= -inf.0)) {
		VAR = ((double) (((double) (y * ((double) (((double) (x / z)) - ((double) (t / z)))))) + t));
	} else {
		double VAR_1;
		if ((((double) (x + ((double) (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z)))))) <= -1.2575042853915238e-298)) {
			VAR_1 = ((double) (x + ((double) (1.0 / ((double) (((double) (a - z)) / ((double) (((double) (y - z)) * ((double) (t - x))))))))));
		} else {
			double VAR_2;
			if ((((double) (x + ((double) (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z)))))) <= 3.7317631195897233e-242)) {
				VAR_2 = ((double) (((double) (((double) (((double) (x * y)) / z)) + t)) - ((double) (((double) (t * y)) / z))));
			} else {
				double VAR_3;
				if ((((double) (x + ((double) (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z)))))) <= 3.085984896906972e+305)) {
					VAR_3 = ((double) (x + ((double) (1.0 / ((double) (((double) (a - z)) / ((double) (((double) (y - z)) * ((double) (t - x))))))))));
				} else {
					VAR_3 = ((double) (((double) (y * ((double) (((double) (x / z)) - ((double) (t / z)))))) + t));
				}
				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	24.5
Target	12.1
Herbie	11.0

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -inf.0 or 3.085984896906972e+305 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 63.9

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

   3. Applied add-cube-cbrt 63.9

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

   4. Applied times-frac 17.6

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 17.8

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

   7. Taylor expanded around inf 40.3

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

   8. Simplified 27.3

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

   if -inf.0 < (+ x (/ (* (- y z) (- t x)) (- a z))) < -1.2575042853915238e-298 or 3.7317631195897233e-242 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 3.085984896906972e+305

   1. Initial program 2.1

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

   3. Applied clear-num 2.2

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

   if -1.2575042853915238e-298 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 3.7317631195897233e-242

   1. Initial program 55.4

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

   2. Taylor expanded around inf 20.2

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} = -\infty:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.2575042853915238 \cdot 10^{-298}:\\ \;\;\;\;x + \frac{1}{\frac{a - z}{\left(y - z\right) \cdot \left(t - x\right)}}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 3.7317631195897233 \cdot 10^{-242}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 3.0859848969069718 \cdot 10^{305}:\\ \;\;\;\;x + \frac{1}{\frac{a - z}{\left(y - z\right) \cdot \left(t - x\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(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))))